importance of formal logic in critical thinking

Introduction to Logic and Critical Thinking

(10 reviews)

Matthew Van Cleave, Lansing Community College

Copyright Year: 2016

Publisher: Matthew J. Van Cleave

Language: English

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Reviewed by "yusef" Alexander Hayes, Professor, North Shore Community College on 6/9/21

Formal and informal reasoning, argument structure, and fallacies are covered comprehensively, meeting the author's goal of both depth and succinctness. read more

Comprehensiveness rating: 5 see less

Formal and informal reasoning, argument structure, and fallacies are covered comprehensively, meeting the author's goal of both depth and succinctness.

Content Accuracy rating: 5

The book is accurate.

Relevance/Longevity rating: 5

While many modern examples are used, and they are helpful, they are not necessarily needed. The usefulness of logical principles and skills have proved themselves, and this text presents them clearly with many examples.

Clarity rating: 5

It is obvious that the author cares about their subject, audience, and students. The text is comprehensible and interesting.

Consistency rating: 5

The format is easy to understand and is consistent in framing.

Modularity rating: 5

This text would be easy to adapt.

Organization/Structure/Flow rating: 5

The organization is excellent, my one suggestion would be a concluding chapter.

Interface rating: 5

I accessed the PDF version and it would be easy to work with.

Grammatical Errors rating: 5

The writing is excellent.

Cultural Relevance rating: 5

This is not an offensive text.

Reviewed by Susan Rottmann, Part-time Lecturer, University of Southern Maine on 3/2/21

I reviewed this book for a course titled "Creative and Critical Inquiry into Modern Life." It won't meet all my needs for that course, but I haven't yet found a book that would. I wanted to review this one because it states in the preface that it... read more

Comprehensiveness rating: 4 see less

I reviewed this book for a course titled "Creative and Critical Inquiry into Modern Life." It won't meet all my needs for that course, but I haven't yet found a book that would. I wanted to review this one because it states in the preface that it fits better for a general critical thinking course than for a true logic course. I'm not sure that I'd agree. I have been using Browne and Keeley's "Asking the Right Questions: A Guide to Critical Thinking," and I think that book is a better introduction to critical thinking for non-philosophy majors. However, the latter is not open source so I will figure out how to get by without it in the future. Overall, the book seems comprehensive if the subject is logic. The index is on the short-side, but fine. However, one issue for me is that there are no page numbers on the table of contents, which is pretty annoying if you want to locate particular sections.

Content Accuracy rating: 4

I didn't find any errors. In general the book uses great examples. However, they are very much based in the American context, not for an international student audience. Some effort to broaden the chosen examples would make the book more widely applicable.

Relevance/Longevity rating: 4

I think the book will remain relevant because of the nature of the material that it addresses, however there will be a need to modify the examples in future editions and as the social and political context changes.

Clarity rating: 3

The text is lucid, but I think it would be difficult for introductory-level students who are not philosophy majors. For example, in Browne and Keeley's "Asking the Right Questions: A Guide to Critical Thinking," the sub-headings are very accessible, such as "Experts cannot rescue us, despite what they say" or "wishful thinking: perhaps the biggest single speed bump on the road to critical thinking." By contrast, Van Cleave's "Introduction to Logic and Critical Thinking" has more subheadings like this: "Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form" or "Propositional logic and the four basic truth functional connectives." If students are prepared very well for the subject, it would work fine, but for students who are newly being introduced to critical thinking, it is rather technical.

It seems to be very consistent in terms of its terminology and framework.

Modularity rating: 4

The book is divided into 4 chapters, each having many sub-chapters. In that sense, it is readily divisible and modular. However, as noted above, there are no page numbers on the table of contents, which would make assigning certain parts rather frustrating. Also, I'm not sure why the book is only four chapter and has so many subheadings (for instance 17 in Chapter 2) and a length of 242 pages. Wouldn't it make more sense to break up the book into shorter chapters? I think this would make it easier to read and to assign in specific blocks to students.

Organization/Structure/Flow rating: 4

The organization of the book is fine overall, although I think adding page numbers to the table of contents and breaking it up into more separate chapters would help it to be more easily navigable.

Interface rating: 4

The book is very simply presented. In my opinion it is actually too simple. There are few boxes or diagrams that highlight and explain important points.

The text seems fine grammatically. I didn't notice any errors.

The book is written with an American audience in mind, but I did not notice culturally insensitive or offensive parts.

Overall, this book is not for my course, but I think it could work well in a philosophy course.

Reviewed by Daniel Lee, Assistant Professor of Economics and Leadership, Sweet Briar College on 11/11/19

This textbook is not particularly comprehensive (4 chapters long), but I view that as a benefit. In fact, I recommend it for use outside of traditional logic classes, but rather interdisciplinary classes that evaluate argument read more

Comprehensiveness rating: 3 see less

This textbook is not particularly comprehensive (4 chapters long), but I view that as a benefit. In fact, I recommend it for use outside of traditional logic classes, but rather interdisciplinary classes that evaluate argument

To the best of my ability, I regard this content as accurate, error-free, and unbiased

The book is broadly relevant and up-to-date, with a few stray temporal references (sydney olympics, particular presidencies). I don't view these time-dated examples as problematic as the logical underpinnings are still there and easily assessed

Clarity rating: 4

My only pushback on clarity is I didn't find the distinction between argument and explanation particularly helpful/useful/easy to follow. However, this experience may have been unique to my class.

To the best of my ability, I regard this content as internally consistent

I found this text quite modular, and was easily able to integrate other texts into my lessons and disregard certain chapters or sub-sections

The book had a logical and consistent structure, but to the extent that there are only 4 chapters, there isn't much scope for alternative approaches here

No problems with the book's interface

The text is grammatically sound

Cultural Relevance rating: 4

Perhaps the text could have been more universal in its approach. While I didn't find the book insensitive per-se, logic can be tricky here because the point is to evaluate meaningful (non-trivial) arguments, but any argument with that sense of gravity can also be traumatic to students (abortion, death penalty, etc)

No additional comments

Reviewed by Lisa N. Thomas-Smith, Graduate Part-time Instructor, CU Boulder on 7/1/19

The text covers all the relevant technical aspects of introductory logic and critical thinking, and covers them well. A separate glossary would be quite helpful to students. However, the terms are clearly and thoroughly explained within the text,... read more

The text covers all the relevant technical aspects of introductory logic and critical thinking, and covers them well. A separate glossary would be quite helpful to students. However, the terms are clearly and thoroughly explained within the text, and the index is very thorough.

The content is excellent. The text is thorough and accurate with no errors that I could discern. The terminology and exercises cover the material nicely and without bias.

The text should easily stand the test of time. The exercises are excellent and would be very helpful for students to internalize correct critical thinking practices. Because of the logical arrangement of the text and the many sub-sections, additional material should be very easy to add.

The text is extremely clearly and simply written. I anticipate that a diligent student could learn all of the material in the text with little additional instruction. The examples are relevant and easy to follow.

The text did not confuse terms or use inconsistent terminology, which is very important in a logic text. The discipline often uses multiple terms for the same concept, but this text avoids that trap nicely.

The text is fairly easily divisible. Since there are only four chapters, those chapters include large blocks of information. However, the chapters themselves are very well delineated and could be easily broken up so that parts could be left out or covered in a different order from the text.

The flow of the text is excellent. All of the information is handled solidly in an order that allows the student to build on the information previously covered.

The PDF Table of Contents does not include links or page numbers which would be very helpful for navigation. Other than that, the text was very easy to navigate. All the images, charts, and graphs were very clear

I found no grammatical errors in the text.

Cultural Relevance rating: 3

The text including examples and exercises did not seem to be offensive or insensitive in any specific way. However, the examples included references to black and white people, but few others. Also, the text is very American specific with many examples from and for an American audience. More diversity, especially in the examples, would be appropriate and appreciated.

Reviewed by Leslie Aarons, Associate Professor of Philosophy, CUNY LaGuardia Community College on 5/16/19

This is an excellent introductory (first-year) Logic and Critical Thinking textbook. The book covers the important elementary information, clearly discussing such things as the purpose and basic structure of an argument; the difference between an... read more

This is an excellent introductory (first-year) Logic and Critical Thinking textbook. The book covers the important elementary information, clearly discussing such things as the purpose and basic structure of an argument; the difference between an argument and an explanation; validity; soundness; and the distinctions between an inductive and a deductive argument in accessible terms in the first chapter. It also does a good job introducing and discussing informal fallacies (Chapter 4). The incorporation of opportunities to evaluate real-world arguments is also very effective. Chapter 2 also covers a number of formal methods of evaluating arguments, such as Venn Diagrams and Propositional logic and the four basic truth functional connectives, but to my mind, it is much more thorough in its treatment of Informal Logic and Critical Thinking skills, than it is of formal logic. I also appreciated that Van Cleave’s book includes exercises with answers and an index, but there is no glossary; which I personally do not find detracts from the book's comprehensiveness.

Overall, Van Cleave's book is error-free and unbiased. The language used is accessible and engaging. There were no glaring inaccuracies that I was able to detect.

Van Cleave's Textbook uses relevant, contemporary content that will stand the test of time, at least for the next few years. Although some examples use certain subjects like former President Obama, it does so in a useful manner that inspires the use of critical thinking skills. There are an abundance of examples that inspire students to look at issues from many different political viewpoints, challenging students to practice evaluating arguments, and identifying fallacies. Many of these exercises encourage students to critique issues, and recognize their own inherent reader-biases and challenge their own beliefs--hallmarks of critical thinking.

As mentioned previously, the author has an accessible style that makes the content relatively easy to read and engaging. He also does a suitable job explaining jargon/technical language that is introduced in the textbook.

Van Cleave uses terminology consistently and the chapters flow well. The textbook orients the reader by offering effective introductions to new material, step-by-step explanations of the material, as well as offering clear summaries of each lesson.

This textbook's modularity is really quite good. Its language and structure are not overly convoluted or too-lengthy, making it convenient for individual instructors to adapt the materials to suit their methodological preferences.

The topics in the textbook are presented in a logical and clear fashion. The structure of the chapters are such that it is not necessary to have to follow the chapters in their sequential order, and coverage of material can be adapted to individual instructor's preferences.

The textbook is free of any problematic interface issues. Topics, sections and specific content are accessible and easy to navigate. Overall it is user-friendly.

I did not find any significant grammatical issues with the textbook.

The textbook is not culturally insensitive, making use of a diversity of inclusive examples. Materials are especially effective for first-year critical thinking/logic students.

I intend to adopt Van Cleave's textbook for a Critical Thinking class I am teaching at the Community College level. I believe that it will help me facilitate student-learning, and will be a good resource to build additional classroom activities from the materials it provides.

Reviewed by Jennie Harrop, Chair, Department of Professional Studies, George Fox University on 3/27/18

While the book is admirably comprehensive, its extensive details within a few short chapters may feel overwhelming to students. The author tackles an impressive breadth of concepts in Chapter 1, 2, 3, and 4, which leads to 50-plus-page chapters... read more

While the book is admirably comprehensive, its extensive details within a few short chapters may feel overwhelming to students. The author tackles an impressive breadth of concepts in Chapter 1, 2, 3, and 4, which leads to 50-plus-page chapters that are dense with statistical analyses and critical vocabulary. These topics are likely better broached in manageable snippets rather than hefty single chapters.

The ideas addressed in Introduction to Logic and Critical Thinking are accurate but at times notably political. While politics are effectively used to exemplify key concepts, some students may be distracted by distinct political leanings.

The terms and definitions included are relevant, but the examples are specific to the current political, cultural, and social climates, which could make the materials seem dated in a few years without intentional and consistent updates.

While the reasoning is accurate, the author tends to complicate rather than simplify -- perhaps in an effort to cover a spectrum of related concepts. Beginning readers are likely to be overwhelmed and under-encouraged by his approach.

Consistency rating: 3

The four chapters are somewhat consistent in their play of definition, explanation, and example, but the structure of each chapter varies according to the concepts covered. In the third chapter, for example, key ideas are divided into sub-topics numbering from 3.1 to 3.10. In the fourth chapter, the sub-divisions are further divided into sub-sections numbered 4.1.1-4.1.5, 4.2.1-4.2.2, and 4.3.1 to 4.3.6. Readers who are working quickly to master new concepts may find themselves mired in similarly numbered subheadings, longing for a grounded concepts on which to hinge other key principles.

Modularity rating: 3

The book's four chapters make it mostly self-referential. The author would do well to beak this text down into additional subsections, easing readers' accessibility.

The content of the book flows logically and well, but the information needs to be better sub-divided within each larger chapter, easing the student experience.

The book's interface is effective, allowing readers to move from one section to the next with a single click. Additional sub-sections would ease this interplay even further.

Grammatical Errors rating: 4

Some minor errors throughout.

For the most part, the book is culturally neutral, avoiding direct cultural references in an effort to remain relevant.

Reviewed by Yoichi Ishida, Assistant Professor of Philosophy, Ohio University on 2/1/18

This textbook covers enough topics for a first-year course on logic and critical thinking. Chapter 1 covers the basics as in any standard textbook in this area. Chapter 2 covers propositional logic and categorical logic. In propositional logic,... read more

This textbook covers enough topics for a first-year course on logic and critical thinking. Chapter 1 covers the basics as in any standard textbook in this area. Chapter 2 covers propositional logic and categorical logic. In propositional logic, this textbook does not cover suppositional arguments, such as conditional proof and reductio ad absurdum. But other standard argument forms are covered. Chapter 3 covers inductive logic, and here this textbook introduces probability and its relationship with cognitive biases, which are rarely discussed in other textbooks. Chapter 4 introduces common informal fallacies. The answers to all the exercises are given at the end. However, the last set of exercises is in Chapter 3, Section 5. There are no exercises in the rest of the chapter. Chapter 4 has no exercises either. There is index, but no glossary.

The textbook is accurate.

The content of this textbook will not become obsolete soon.

The textbook is written clearly.

The textbook is internally consistent.

The textbook is fairly modular. For example, Chapter 3, together with a few sections from Chapter 1, can be used as a short introduction to inductive logic.

The textbook is well-organized.

There are no interface issues.

I did not find any grammatical errors.

This textbook is relevant to a first semester logic or critical thinking course.

Reviewed by Payal Doctor, Associate Professro, LaGuardia Community College on 2/1/18

This text is a beginner textbook for arguments and propositional logic. It covers the basics of identifying arguments, building arguments, and using basic logic to construct propositions and arguments. It is quite comprehensive for a beginner... read more

This text is a beginner textbook for arguments and propositional logic. It covers the basics of identifying arguments, building arguments, and using basic logic to construct propositions and arguments. It is quite comprehensive for a beginner book, but seems to be a good text for a course that needs a foundation for arguments. There are exercises on creating truth tables and proofs, so it could work as a logic primer in short sessions or with the addition of other course content.

The books is accurate in the information it presents. It does not contain errors and is unbiased. It covers the essential vocabulary clearly and givens ample examples and exercises to ensure the student understands the concepts

The content of the book is up to date and can be easily updated. Some examples are very current for analyzing the argument structure in a speech, but for this sort of text understandable examples are important and the author uses good examples.

The book is clear and easy to read. In particular, this is a good text for community college students who often have difficulty with reading comprehension. The language is straightforward and concepts are well explained.

The book is consistent in terminology, formatting, and examples. It flows well from one topic to the next, but it is also possible to jump around the text without loosing the voice of the text.

The books is broken down into sub units that make it easy to assign short blocks of content at a time. Later in the text, it does refer to a few concepts that appear early in that text, but these are all basic concepts that must be used to create a clear and understandable text. No sections are too long and each section stays on topic and relates the topic to those that have come before when necessary.

The flow of the text is logical and clear. It begins with the basic building blocks of arguments, and practice identifying more and more complex arguments is offered. Each chapter builds up from the previous chapter in introducing propositional logic, truth tables, and logical arguments. A select number of fallacies are presented at the end of the text, but these are related to topics that were presented before, so it makes sense to have these last.

The text is free if interface issues. I used the PDF and it worked fine on various devices without loosing formatting.

1. The book contains no grammatical errors.

The text is culturally sensitive, but examples used are a bit odd and may be objectionable to some students. For instance, President Obama's speech on Syria is used to evaluate an extended argument. This is an excellent example and it is explained well, but some who disagree with Obama's policies may have trouble moving beyond their own politics. However, other examples look at issues from all political viewpoints and ask students to evaluate the argument, fallacy, etc. and work towards looking past their own beliefs. Overall this book does use a variety of examples that most students can understand and evaluate.

My favorite part of this book is that it seems to be written for community college students. My students have trouble understanding readings in the New York Times, so it is nice to see a logic and critical thinking text use real language that students can understand and follow without the constant need of a dictionary.

Reviewed by Rebecca Owen, Adjunct Professor, Writing, Chemeketa Community College on 6/20/17

This textbook is quite thorough--there are conversational explanations of argument structure and logic. I think students will be happy with the conversational style this author employs. Also, there are many examples and exercises using current... read more

This textbook is quite thorough--there are conversational explanations of argument structure and logic. I think students will be happy with the conversational style this author employs. Also, there are many examples and exercises using current events, funny scenarios, or other interesting ways to evaluate argument structure and validity. The third section, which deals with logical fallacies, is very clear and comprehensive. My only critique of the material included in the book is that the middle section may be a bit dense and math-oriented for learners who appreciate the more informal, informative style of the first and third section. Also, the book ends rather abruptly--it moves from a description of a logical fallacy to the answers for the exercises earlier in the text.

The content is very reader-friendly, and the author writes with authority and clarity throughout the text. There are a few surface-level typos (Starbuck's instead of Starbucks, etc.). None of these small errors detract from the quality of the content, though.

One thing I really liked about this text was the author's wide variety of examples. To demonstrate different facets of logic, he used examples from current media, movies, literature, and many other concepts that students would recognize from their daily lives. The exercises in this text also included these types of pop-culture references, and I think students will enjoy the familiarity--as well as being able to see the logical structures behind these types of references. I don't think the text will need to be updated to reflect new instances and occurrences; the author did a fine job at picking examples that are relatively timeless. As far as the subject matter itself, I don't think it will become obsolete any time soon.

The author writes in a very conversational, easy-to-read manner. The examples used are quite helpful. The third section on logical fallacies is quite easy to read, follow, and understand. A student in an argument writing class could benefit from this section of the book. The middle section is less clear, though. A student learning about the basics of logic might have a hard time digesting all of the information contained in chapter two. This material might be better in two separate chapters. I think the author loses the balance of a conversational, helpful tone and focuses too heavily on equations.

Consistency rating: 4

Terminology in this book is quite consistent--the key words are highlighted in bold. Chapters 1 and 3 follow a similar organizational pattern, but chapter 2 is where the material becomes more dense and equation-heavy. I also would have liked a closing passage--something to indicate to the reader that we've reached the end of the chapter as well as the book.

I liked the overall structure of this book. If I'm teaching an argumentative writing class, I could easily point the students to the chapters where they can identify and practice identifying fallacies, for instance. The opening chapter is clear in defining the necessary terms, and it gives the students an understanding of the toolbox available to them in assessing and evaluating arguments. Even though I found the middle section to be dense, smaller portions could be assigned.

The author does a fine job connecting each defined term to the next. He provides examples of how each defined term works in a sentence or in an argument, and then he provides practice activities for students to try. The answers for each question are listed in the final pages of the book. The middle section feels like the heaviest part of the whole book--it would take the longest time for a student to digest if assigned the whole chapter. Even though this middle section is a bit heavy, it does fit the overall structure and flow of the book. New material builds on previous chapters and sub-chapters. It ends abruptly--I didn't realize that it had ended, and all of a sudden I found myself in the answer section for those earlier exercises.

The simple layout is quite helpful! There is nothing distracting, image-wise, in this text. The table of contents is clearly arranged, and each topic is easy to find.

Tiny edits could be made (Starbuck's/Starbucks, for one). Otherwise, it is free of distracting grammatical errors.

This text is quite culturally relevant. For instance, there is one example that mentions the rumors of Barack Obama's birthplace as somewhere other than the United States. This example is used to explain how to analyze an argument for validity. The more "sensational" examples (like the Obama one above) are helpful in showing argument structure, and they can also help students see how rumors like this might gain traction--as well as help to show students how to debunk them with their newfound understanding of argument and logic.

The writing style is excellent for the subject matter, especially in the third section explaining logical fallacies. Thank you for the opportunity to read and review this text!

Reviewed by Laurel Panser, Instructor, Riverland Community College on 6/20/17

This is a review of Introduction to Logic and Critical Thinking, an open source book version 1.4 by Matthew Van Cleave. The comparison book used was Patrick J. Hurley’s A Concise Introduction to Logic 12th Edition published by Cengage as well as... read more

This is a review of Introduction to Logic and Critical Thinking, an open source book version 1.4 by Matthew Van Cleave. The comparison book used was Patrick J. Hurley’s A Concise Introduction to Logic 12th Edition published by Cengage as well as the 13th edition with the same title. Lori Watson is the second author on the 13th edition.

Competing with Hurley is difficult with respect to comprehensiveness. For example, Van Cleave’s book is comprehensive to the extent that it probably covers at least two-thirds or more of what is dealt with in most introductory, one-semester logic courses. Van Cleave’s chapter 1 provides an overview of argumentation including discerning non-arguments from arguments, premises versus conclusions, deductive from inductive arguments, validity, soundness and more. Much of Van Cleave’s chapter 1 parallel’s Hurley’s chapter 1. Hurley’s chapter 3 regarding informal fallacies is comprehensive while Van Cleave’s chapter 4 on this topic is less extensive. Categorical propositions are a topic in Van Cleave’s chapter 2; Hurley’s chapters 4 and 5 provide more instruction on this, however. Propositional logic is another topic in Van Cleave’s chapter 2; Hurley’s chapters 6 and 7 provide more information on this, though. Van Cleave did discuss messy issues of language meaning briefly in his chapter 1; that is the topic of Hurley’s chapter 2.

Van Cleave’s book includes exercises with answers and an index. A glossary was not included.

Reviews of open source textbooks typically include criteria besides comprehensiveness. These include comments on accuracy of the information, whether the book will become obsolete soon, jargon-free clarity to the extent that is possible, organization, navigation ease, freedom from grammar errors and cultural relevance; Van Cleave’s book is fine in all of these areas. Further criteria for open source books includes modularity and consistency of terminology. Modularity is defined as including blocks of learning material that are easy to assign to students. Hurley’s book has a greater degree of modularity than Van Cleave’s textbook. The prose Van Cleave used is consistent.

Van Cleave’s book will not become obsolete soon.

Van Cleave’s book has accessible prose.

Van Cleave used terminology consistently.

Van Cleave’s book has a reasonable degree of modularity.

Van Cleave’s book is organized. The structure and flow of his book is fine.

Problems with navigation are not present.

Grammar problems were not present.

Van Cleave’s book is culturally relevant.

Van Cleave’s book is appropriate for some first semester logic courses.

Table of Contents

Chapter 1: Reconstructing and analyzing arguments

• 1.1 What is an argument?
• 1.2 Identifying arguments
• 1.3 Arguments vs. explanations
• 1.4 More complex argument structures
• 1.5 Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form
• 1.6 Validity
• 1.7 Soundness
• 1.8 Deductive vs. inductive arguments
• 1.9 Arguments with missing premises
• 1.10 Assuring, guarding, and discounting
• 1.11 Evaluative language
• 1.12 Evaluating a real-life argument

Chapter 2: Formal methods of evaluating arguments

• 2.1 What is a formal method of evaluation and why do we need them?
• 2.2 Propositional logic and the four basic truth functional connectives
• 2.3 Negation and disjunction
• 2.4 Using parentheses to translate complex sentences
• 2.5 “Not both” and “neither nor”
• 2.6 The truth table test of validity
• 2.7 Conditionals
• 2.8 “Unless”
• 2.9 Material equivalence
• 2.10 Tautologies, contradictions, and contingent statements
• 2.11 Proofs and the 8 valid forms of inference
• 2.12 How to construct proofs
• 2.13 Short review of propositional logic
• 2.14 Categorical logic
• 2.15 The Venn test of validity for immediate categorical inferences
• 2.16 Universal statements and existential commitment
• 2.17 Venn validity for categorical syllogisms

Chapter 3: Evaluating inductive arguments and probabilistic and statistical fallacies

• 3.1 Inductive arguments and statistical generalizations
• 3.2 Inference to the best explanation and the seven explanatory virtues
• 3.3 Analogical arguments
• 3.4 Causal arguments
• 3.5 Probability
• 3.6 The conjunction fallacy
• 3.7 The base rate fallacy
• 3.8 The small numbers fallacy
• 3.9 Regression to the mean fallacy
• 3.10 Gambler's fallacy

Chapter 4: Informal fallacies

• 4.1 Formal vs. informal fallacies
• 4.1.1 Composition fallacy
• 4.1.2 Division fallacy
• 4.1.3 Begging the question fallacy
• 4.1.4 False dichotomy
• 4.1.5 Equivocation
• 4.2 Slippery slope fallacies
• 4.2.1 Conceptual slippery slope
• 4.2.2 Causal slippery slope
• 4.3 Fallacies of relevance
• 4.3.1 Ad hominem
• 4.3.2 Straw man
• 4.3.3 Tu quoque
• 4.3.4 Genetic
• 4.3.5 Appeal to consequences
• 4.3.6 Appeal to authority

Answers to exercises Glossary/Index

Ancillary Material

About the book.

This is an introductory textbook in logic and critical thinking. The goal of the textbook is to provide the reader with a set of tools and skills that will enable them to identify and evaluate arguments. The book is intended for an introductory course that covers both formal and informal logic. As such, it is not a formal logic textbook, but is closer to what one would find marketed as a “critical thinking textbook.”

About the Contributors

Matthew Van Cleave ,   PhD, Philosophy, University of Cincinnati, 2007.  VAP at Concordia College (Moorhead), 2008-2012.  Assistant Professor at Lansing Community College, 2012-2016. Professor at Lansing Community College, 2016-

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Curtis Silver

The Importance of Logic and Critical Thinking

"Critical thinking is a desire to seek, patience to doubt, fondness to meditate, slowness to assert, readiness to consider, carefulness to dispose and set in order; and hatred for every kind of imposture." - Francis Bacon (1605)

As parents, we are tasked with instilling a plethora of different values into our children. While some parents in the world choose to instill a lack of values in their kids, those of us that don't want our children growing up to be criminals and various misfits try a bit harder. Values and morality are one piece of the pie. These are important things to mold into a child's mind, but there are also other items in life to focus on as well. It starts with looking both ways to cross the street and either progresses from there, or stops.

If you stopped explaining the world to your children after they learned to cross the street, then perhaps you should stop reading and go back to surfing for funny pictures of cats. I may use some larger words that you might not understand, making you angry and causing you to leave troll-like comments full of bad grammar and moronic thought processes. However, if you looked at the crossing the street issue as I did – as a logical problem with cause and effect and a probable solution – then carry on. You are my target audience.

Or perhaps the opposite is true, as the former are the people that could benefit from letting some critical thinking into their lives. So what exactly is critical thinking? This bit by Linda Elder in a paper on CriticalThinking.org pretty much sums it up:

Through critical thinking, as I understand it, we acquire a means of assessing and upgrading our ability to judge well. It enables us to go into virtually any situation and to figure out the logic of whatever is happening in that situation. It provides a way for us to learn from new experiences through the process of continual self-assessment. Critical thinking, then, enables us to form sound beliefs and judgments, and in doing so, provides us with a basis for a 'rational and reasonable' emotional life. — Inquiry: Critical Thinking Across the Disciplines, Winter, 1996. Vol. XVI, No. 2.

The rationality of the world is what is at risk. Too many people are taken advantage of because of their lack of critical thinking, logic and deductive reasoning. These same people are raising children without these same skills, creating a whole new generation of clueless people.

Brian Barrett

Brendan I. Koerner

Steven Levy

Scott Gilbertson

To wit, a personal tale of deductive reasoning:

Recently I needed a new transmission for the family van. The warranty on the power train covers the transmission up to 100,000 miles. The van has around 68,000 miles on it. Therefore, even the logic-less dimwit could easily figure that the transmission was covered. Well, this was true until the dealership told me that it wasn't, stating that because we didn't get the scheduled transmission service (which is basically a fluid change) at 30,000 and 60,000 miles the warranty was no longer valid. Now, there are many people that would argue this point, but many more that would shrug, panic, and accept the full cost of repairs.

I read the warranty book. I had a receipt that said the fluid was checked at 60,000 but not replaced. A friend on Twitter pointed out the fact that they were using 100,000 mile transmission fluid. So logically, the fluid would not have to be replaced under 100,000 miles if it wasn't needed, right? So why the stipulation that it needed to be replaced at 60,000 and the loose assumption that not doing that would void the warranty? So I asked the warranty guy to show me in the book where the two items are related. Where it explicitly says that if you don't get the service, the transmission isn't covered. There were portions where it said the service was recommended, but never connecting to actual repairs. Finally the warranty guy shrugged, admitted I was right and said the service was covered.

In this case, valid logic equaled truth and a sound argument. I used very simple reasoning and logic to determine that I was being inadvertently screwed. I say "inadvertently" because I truly believe based on their behavior that they were not intentionally trying to screw me. They believed the two items were related, they had had this argument many times before and were not prepared to be questioned. While both the service manager and the warranty guy seemed at least junior college educated, proving my argument to them took longer than it should have between three adults.

However, valid logic does not always guarantee truth or a sound argument. This is where it gets a little funky. Valid logic is when the structure of logic is correct in the way of syntax and semantics rather than truth. Truth comes from deductive reasoning of said logic. For example:

All transmissions are covered parts. All covered parts are free. Therefore, all transmissions are free. This logic is technically valid, and if the premises are true, then of course the conclusion must be true. You can see here however that it's not always true, though in some situations it could be. While the logic is valid, not all transmissions are free, only those covered by the warranty. So based on that, saying all transmissions are free is not sound logic.

To take it one step further:

All Daleks are brown. Some brown things are Cylons . Therefore, some Daleks are Cylons. Sci-fi fan or not, you probably know that this is not true. The basic lesson here is that, while the logic above might seem valid because of the structure of the statement, it takes a further understanding to figure out why it's not necessarily true: That is, based on the first two statements it's possible that some Daleks are Cylons, but it's not logically concludable. That's where deductive reasoning comes on top of the logic. The underlying lesson here is not to immediately assume everything you read or are told is true, something all children need to and should learn.

This is the direct lesson that needs to be passed on to our children: that of not accepting the immediately visible logic. While not all problems are complex enough to require the scientific method, some of them need some deduction to determine if they are true. Take the example above — how many kids would immediately be satisfied with the false conclusion? Sure, it's a bit geeky with the examples, but switch out bears for Daleks and puppies for Cylons. That makes it easier, and takes the actual research out of it (to find out what Daleks and Cylons are respectively) but many people would just accept that in fact some bears are puppies, if presented with this problem in the context of a textbook or word problem.

Maybe I'm being paranoid or thinking too doomsday, whatever, but I think this is an epidemic. Children are becoming lazier and not as self sufficient because their parents have a problem with watching a three year old cry after they tell her to remove her own jeans, or ask her to put away her own toys (yes, organizational logic falls under the main topic). These are the same parents who do their kid's science project while the kid is playing video games. These kids grow up lacking the simple problem solving skills that make navigating life much easier. Remember when you were growing up and you had the plastic stacking toys ? Well, instead of toys for early development like that, parents are just plopping their kids down in front of the television. While there is some educational type programming on television, it's just not the same as hands-on experience.

My father is an engineer, and he taught me logic and reasoning by making me solve simple, then complex, problems on my own. Or at least giving me the opportunity to solve them on my own. This helped develop critical thinking and problem solving skills, something a lot of children lack these days. Too often I see children that are not allowed to solve problems on their own; instead their parents simply do it for them without argument or discussion. Hell, I am surrounded by adults every day that are unable to solve simple problems, instead choosing to immediately ask me at which point I have to fill the role that their parents never did and – knowing the solution – tell them to solve it themselves, or at least try first.

One of the things I like to work on with my kids is math. There is nothing that teaches deductive reasoning and logic better than math word problems. They are at the age where basic algebra can come into play, which sharpens their reasoning skills because they start to view real world issues with algebraic solutions. Another thing is logic puzzles , crossword puzzles and first person shooters. Actually, not that last one. That's just the reward.

Since I weeded out the folks that don't teach their kids logic in the first two paragraphs, as representatives of the real world it's up to the rest of us to spread the knowledge. It won't be easy. The best thing we can do is teach these thought processes to our children, so that they may look at other children with looks of bewilderment when other children are unable to solve simple tasks. Hopefully, they will not simply do the task for them, but teach them to think. I'm not saying we need to build a whole new generation of project managers and analysts, but it would be better than a generation of task-oriented mindless office drones with untied shoelaces, shoving on a door at the Midvale School for the Gifted .

h/t to @aubreygirl22 for the logical conversation. Image: Flickr user William Notowidagdo. Used under Creative Commons License.

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1: Introduction to Critical Thinking, Reasoning, and Logic

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What is thinking? It may seem strange to begin a logic textbook with this question. ‘Thinking’ is perhaps the most intimate and personal thing that people do. Yet the more you ‘think’ about thinking, the more mysterious it can appear. It is the sort of thing that one intuitively or naturally understands, and yet cannot describe to others without great difficulty. Many people believe that logic is very abstract, dispassionate, complicated, and even cold. But in fact the study of logic is nothing more intimidating or obscure than this: the study of good thinking.

• 1.1: Prelude to Chapter
• 1.2: Introduction and Thought Experiments- The Trolley Problem
• 1.3: Truth and Its Role in Argumentation - Certainty, Probability, and Monty Hall Only certain sorts of sentences can be used in arguments. We call these sentences propositions, statements or claims.
• 1.4: Distinction of Proof from Verification; Our Biases and the Forer Effect
• 1.5: The Scientific Method The procedure that scientists use is also a standard form of argument. Its conclusions only give you the likelihood or the probability that something is true (if your theory or hypothesis is confirmed), and not the certainty that it’s true. But when it is done correctly, the conclusions it reaches are very well-grounded in experimental evidence.
• 1.6: Diagramming Thoughts and Arguments - Analyzing News Media
• 1.7: Creating a Philosophical Outline

PHIL102: Introduction to Critical Thinking and Logic

Course introduction.

• Time: 40 hours
• College Credit Recommended ($25 Proctor Fee) -->
• Free Certificate

The course touches upon a wide range of reasoning skills, from verbal argument analysis to formal logic, visual and statistical reasoning, scientific methodology, and creative thinking. Mastering these skills will help you become a more perceptive reader and listener, a more persuasive writer and presenter, and a more effective researcher and scientist.

The first unit introduces the terrain of critical thinking and covers the basics of meaning analysis, while the second unit provides a primer for analyzing arguments. All of the material in these first units will be built upon in subsequent units, which cover informal and formal logic, Venn diagrams, scientific reasoning, and strategic and creative thinking.

Course Syllabus

First, read the course syllabus. Then, enroll in the course by clicking "Enroll me". Click Unit 1 to read its introduction and learning outcomes. You will then see the learning materials and instructions on how to use them.

Unit 1: Introduction and Meaning Analysis

Critical thinking is a broad classification for a diverse array of reasoning techniques. In general, critical thinking works by breaking arguments and claims down to their basic underlying structure so we can see them clearly and determine whether they are rational. The idea is to help us do a better job of understanding and evaluating what we read, what we hear, and what we write and say.

In this unit, we will define the broad contours of critical thinking and learn why it is a valuable and useful object of study. We will also introduce the fundamentals of meaning analysis: the difference between literal meaning and implication, the principles of definition, how to identify when a disagreement is merely verbal, the distinction between necessary and sufficient conditions, and problems with the imprecision of ordinary language.

Completing this unit should take you approximately 5 hours.

Unit 2: Argument Analysis

Arguments are the fundamental components of all rational discourse: nearly everything we read and write, like scientific reports, newspaper columns, and personal letters, as well as most of our verbal conversations, contain arguments. Picking the arguments out from the rest of our often convoluted discourse can be difficult. Once we have identified an argument, we still need to determine whether or not it is sound. Luckily, arguments obey a set of formal rules that we can use to determine whether they are good or bad.

In this unit, you will learn how to identify arguments, what makes an argument sound as opposed to unsound or merely valid, the difference between deductive and inductive reasoning, and how to map arguments to reveal their structure.

Completing this unit should take you approximately 7 hours.

Unit 3: Basic Sentential Logic

This unit introduces a topic that many students find intimidating: formal logic. Although it sounds difficult and complicated, formal (or symbolic) logic is actually a fairly straightforward way of revealing the structure of reasoning. By translating arguments into symbols, you can more readily see what is right and wrong with them and learn how to formulate better arguments. Advanced courses in formal logic focus on using rules of inference to construct elaborate proofs. Using these techniques, you can solve many complicated problems simply by manipulating symbols on the page. In this course, however, you will only be looking at the most basic properties of a system of logic. In this unit, you will learn how to turn phrases in ordinary language into well-formed formulas, draw truth tables for formulas, and evaluate arguments using those truth tables.

Completing this unit should take you approximately 13 hours.

Unit 4: Venn Diagrams

In addition to using predicate logic, the limitations of sentential logic can also be overcome by using Venn diagrams to illustrate statements and arguments. Statements that include general words like "some" or "few" as well as absolute words like "every" and "all" – so-called categorical statements – lend themselves to being represented on paper as circles that may or may not overlap.

Venn diagrams are especially helpful when dealing with logical arguments called syllogisms. Syllogisms are a special type of three-step argument with two premises and a conclusion, which involve quantifying terms. In this unit, you will learn the basic principles of Venn diagrams, how to use them to represent statements, and how to use them to evaluate arguments.

Completing this unit should take you approximately 6 hours.

Unit 5: Fallacies

Now that you have studied the necessary structure of a good argument and can represent its structure visually, you might think it would be simple to pick out bad arguments. However, identifying bad arguments can be very tricky in practice. Very often, what at first appears to be ironclad reasoning turns out to contain one or more subtle errors.

Fortunately, there are many easily identifiable fallacies (mistakes of reasoning) that you can learn to recognize by their structure or content. In this unit, you will learn about the nature of fallacies, look at a couple of different ways of classifying them, and spend some time dealing with the most common fallacies in detail.

Completing this unit should take you approximately 3 hours.

Unit 6: Scientific Reasoning

Unlike the syllogistic arguments you explored in the last unit, which are a form of deductive argument, scientific reasoning is empirical. This means that it depends on observation and evidence, not logical principles. Although some principles of deductive reasoning do apply in science, such as the principle of contradiction, scientific arguments are often inductive. For this reason, science often deals with confirmation and disconfirmation.

Nonetheless, there are general guidelines about what constitutes good scientific reasoning, and scientists are trained to be critical of their inferences and those of others in the scientific community. In this unit, you will investigate some standard methods of scientific reasoning, some principles of confirmation and disconfirmation, and some techniques for identifying and reasoning about causation.

Completing this unit should take you approximately 4 hours.

Unit 7: Strategic Reasoning and Creativity

While most of this course has focused on the types of reasoning necessary to critique and evaluate existing knowledge or to extend our knowledge following correct procedures and rules, an enormous branch of our reasoning practice runs in the opposite direction. Strategic reasoning, problem-solving, and creative thinking all rely on an ineffable component of novelty supplied by the thinker.

Despite their seemingly mystical nature, problem-solving and creative thinking are best approached by following tried and tested procedures that prompt our cognitive faculties to produce new ideas and solutions by extending our existing knowledge. In this unit, you will investigate problem-solving techniques, representing complex problems visually, making decisions in risky and uncertain scenarios, and creative thinking in general.

Completing this unit should take you approximately 2 hours.

Study Guide

This study guide will help you get ready for the final exam. It discusses the key topics in each unit, walks through the learning outcomes, and lists important vocabulary terms. It is not meant to replace the course materials!

Course Feedback Survey

Please take a few minutes to give us feedback about this course. We appreciate your feedback, whether you completed the whole course or even just a few resources. Your feedback will help us make our courses better, and we use your feedback each time we make updates to our courses.

If you come across any urgent problems, email [email protected].

Certificate Final Exam

Take this exam if you want to earn a free Course Completion Certificate.

To receive a free Course Completion Certificate, you will need to earn a grade of 70% or higher on this final exam. Your grade for the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again as many times as you want, with a 7-day waiting period between each attempt.

Once you pass this final exam, you will be awarded a free Course Completion Certificate .

Saylor Direct Credit

Take this exam if you want to earn college credit for this course . This course is eligible for college credit through Saylor Academy's Saylor Direct Credit Program .

The Saylor Direct Credit Final Exam requires a proctoring fee of $5 . To pass this course and earn a Credly Badge and official transcript , you will need to earn a grade of 70% or higher on the Saylor Direct Credit Final Exam. Your grade for this exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again a maximum of 3 times , with a 14-day waiting period between each attempt.

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Saylor Direct Credit Exam

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Success Skills

Critical thinking and logic.

Critical thinking is fundamentally a process of questioning information and data. You may question the information you read in a textbook, or you may question what a politician or a professor or a classmate says. You can also question a commonly-held belief or a new idea. With critical thinking, anything and everything is subject to question and examination.

Logic’s Relationship to Critical Thinking

The word logic comes from the Ancient Greek logike , referring to the science or art of reasoning. Using logic, a person evaluates arguments and strives to distinguish between good and bad reasoning, or between truth and falsehood. Using logic, you can evaluate ideas or claims people make, make good decisions, and form sound beliefs about the world. [1]

Questions of Logic in Critical Thinking

Let’s use a simple example of applying logic to a critical-thinking situation. In this hypothetical scenario, a man has a PhD in political science, and he works as a professor at a local college. His wife works at the college, too. They have three young children in the local school system, and their family is well known in the community.

The man is now running for political office. Are his credentials and experience sufficient for entering public office? Will he be effective in the political office? Some voters might believe that his personal life and current job, on the surface, suggest he will do well in the position, and they will vote for him.

In truth, the characteristics described don’t guarantee that the man will do a good job. The information is somewhat irrelevant. What else might you want to know? How about whether the man had already held a political office and done a good job? In this case, we want to ask, How much information is adequate in order to make a decision based on logic instead of assumptions?

The following questions, presented in Figure 1, below, are ones you may apply to formulating a logical, reasoned perspective in the above scenario or any other situation:

• What’s happening? Gather the basic information and begin to think of questions.
• Why is it important? Ask yourself why it’s significant and whether or not you agree.
• What don’t I see? Is there anything important missing?
• How do I know? Ask yourself where the information came from and how it was constructed.
• Who is saying it? What’s the position of the speaker and what is influencing them?
• What else? What if? What other ideas exist and are there other possibilities?

• "logic." Wordnik . n.d. Web. 16 Feb 2016 . ↵
• Revision, Adaptation, and Original Content. Provided by : Lumen Learning. License : CC BY: Attribution
• Thinking Critically. Authored by : UBC Learning Commons. Provided by : The University of British Columbia, Vancouver Campus. Located at : http://www.oercommons.org/courses/learning-toolkit-critical-thinking/view . License : CC BY: Attribution
• Critical Thinking Skills. Authored by : Linda Bruce. Provided by : Lumen Learning. Located at : https://courses.candelalearning.com/lumencollegesuccess/chapter/critical-thinking-skills/ . License : CC BY: Attribution

Why Study Logic? Learning Outcomes and Teaching Advice

When you think of studying logic, what comes to mind? Often, logic is one of those subjects — perhaps along with Latin and philosophy — that many associate with an outdated model of education or, if studied today, maybe even with a hint of pretension. As a homeschooling family, is it really necessary for your child to learn logic?

We believe the answer is an emphatic, “Yes!” In this post, we’ll cover the many benefits of learning logic — from developing critical thinking and decision-making skills to building good character — as well as several pieces of advice for teaching your student logic in your home school.    

Formal vs. Informal Logic

Formal logic has been called ‘math with words.’” – Leigh Bortins, The Question

Before diving into three learning outcomes that answer why we should study logic, it’s important to make a quick note on the term logic used in this post. Generally, the study of logic is categorized into informal and formal logic. The type of logic we encourage families to study is formal logic — often referred to as traditional logic — which deals with forms of reasoning. As Classical Conversations founder, Leigh Bortins, describes in her book The Question , “Formal logic has been called ‘math with words.’”

Why Study Logic? 3 Learning Outcomes of Formal Logic

Logic studies enable us to experience the world in richer, more meaningful ways; in short, logic studies make us free.” – Leigh Bortins, The Question

Studying logic is something everybody should do. This includes both homeschool students and parents . In short, there are three reasons why we should learn logic: it encourages clear thinking, empowers us to be truly in the image of God, and builds good character.

1. Studying Logic Develops Critical Thinking Skills

Studying logic involves learning the skills of critical thinking . As you and your student analyze sound reasoning through studying arguments, syllogisms, and fallacies, you’ll develop a sort of “truth compass.” In other words, you’ll be able to apply these reasoning skills to recognize truth from falsehood, whether that’s in an advertisement, a political campaign, a persuasive speech, a news article, or a social media post.

These same critical thinking skills practiced in logic can also be applied to sound decision-making, a skill every parent wants their child to develop. Finally, it’s important to study logic to become an effective communicator. After all, logic is also the backbone necessary for crafting compelling arguments in speech and writing that point others toward truth.  

2. Studying Logic Empowers Us to be Truly in the Image of God

As Christians, the God we worship is a God of form . Just look in Genesis, Chapter 1. The universe God created is the ultimate example of order, structure, and form.

Similarly, we too create forms, from math and science formulas to sentence forms to logical arguments. By using forms to indicate order from disorder and truth from uncertainty, we establish ourselves as made in the image of God.

3. Studying Logic Builds Good Character

For many parents and students, studying logic isn’t easy. Often, along with learning logic come times of frustration and befuddlement. Still, the goal of learning logic is to become better thinkers, which is a worthwhile end to strive toward no matter how strenuous the journey may become. Following through with your study of logic will empower you and your student with confidence in your abilities to learn something challenging and use critical thinking skills to make sound judgments and arrive at the truth in other areas of life.

How to Teach Logic: 3 Pieces of Advice for the Homeschool Parent

Logic trains the brain to think clearly about all subjects by ordering information into usable form. This is a skill we all need to acquire.” – Leigh Bortins, The Question

Convinced why your student should learn logic? Although the learning outcomes of studying logic are noble and inspiring, many parents struggle when it comes to actually teaching the subject. With its forms, structure, and objectivity, logic can appear intimidating. Hopefully, these three pieces of advice will help and encourage you to take on the worthy task of homeschooling your child in logic!

1. Stay Persistent!

Although this may not be what you want to hear, all difficult subjects — logic included — require persistence and hard work . Constantly remind yourself that the end goal of your student learning logic is to equip them with the skills to think critically. So, be persistent in teaching your student logic. In time, your student will learn to apply critical thinking skills to make good decisions and to detect truth from falsehood in everyday situations and encounters. It’s worth every difficult moment to see these fruits of your labor!

Times when homeschooling is hard are a natural part of this journey. Still, that doesn’t mean you have to go at it alone. Find other homeschool parents whom you can rely on for support, guidance, and advice in teaching your child logic, whether in your Classical Conversations local community , a homeschool co-op, or elsewhere. Homeschooling in isolation is never a good idea!

2. Spend Time Learning the Basics of Logic

The road to becoming a skilled logician begins with an understanding of the grammar — or foundational knowledge — of the subject. Make sure to spend time with your student repeating the basics of logic over and over before moving on to complex problems and concepts.

What are these foundations of logic? Well, there are logic vocabulary terms and definitions to commit to memory, like argument , syllogism , conclusion , major premise , minor premise , and fallacy . In addition, you and your student should understand the principles of logic, or “how logic works.” That is, spend time studying the basic rules and procedures associated with clear thinking and reasoning.

Moving on to advanced exercises and ideas before establishing a firm foundation will only lead to discouragement with this subject. For instance, don’t feel guilty if you have to spend several more weeks studying the basics of logic. In the end, this actually might end up saving you time, not to mention a good deal of frustration!

3. Apply Logic to Other Subjects

One of the tenets of classical education is the idea that all subjects are interconnected . Thus, subjects shouldn’t be studied as if they are islands, unrelated to each other.

A great benefit of learning logic is that it trains students to think clearly in all subjects by helping them organize, make connections, and draw conclusions about all types of information. So, encourage your student to utilize what they are learning in their study of logic to understand why Hester Prynne made the decisions she did in The Scarlett Letter or what events motivated American colonialists to wage war against England in the American Revolution.

The truth is that the skills of logic are applicable to all areas of life, and not just if your student goes on to study math or computer science in college. From literature and art to history and science, logic can be used everywhere. Encouraging your student to use logical reasoning in their other subjects will show them that logic is useful and an important skill to master.

The Beauty of Learning Formal Logic

Sure, learning and teaching formal logic can be intimidating. But still, there’s something equally attractive about the study of logic. Arriving at objective truth, knowing that which can be known, making good decisions — these are beautiful goals that make the study of logic well worth the effort.

If you’re on the search for a homeschool logic curriculum, consider our Traditional Logic series designed to make homeschooling logic doable with daily practice exercises to help your student develop powerful critical thinking skills.

Not yet a Classical Conversations member and interested in our community-based approach to homeschooling? We’d love to hear from you. To learn more about us, click here .

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• Informal Logic

The study of logic has often fostered the idea that its methods might be used in attempts to understand and improve thinking, reasoning, and argument as they occur in real life contexts: in public discussion and debate; in education and intellectual exchange; in interpersonal relations; and in law, medicine and other professions.

Informal logic is an attempt to build a logic suited to this purpose. It combines accounts of argument, evidence, proof and justification with an instrumental outlook which emphasizes their usefulness in the analysis of real life arguing. Blair 2015 identifies two key tasks for the informal logician: (i) the attempt to develop ways to identify (and “extract”) arguments from the exchanges in which they occur; and (ii) the attempt to develop methods and guidelines that can be used to assess their strength and cogency.

Though contributions to informal logic include studies of specific kinds or aspects of reasoning, the overriding goal is a general account of argument which can be the basis of systems of informal logic that provide ways to evaluate arguments. Such systems may be applied to arguments as they occur in contexts of reflection, inquiry, social and political debate, the news media, blogs and editorials, the internet, advertising, corporate and institutional communication, social media, and interpersonal exchange.

In the pursuit of its goals, informal logic addresses topics which include, to take only a few examples, the nature and definition of argument, criteria for argument evaluation, argumentation schemes, fallacies, notions of validity, the rhetorical and dialectical aspects of arguing, argument diagramming (“mapping”), cognitive biases, the history of argument analysis, artificial intelligence (AI), and the varying norms and rules that govern argumentative practices in different kinds of contexts.

2. Systems of Informal Logic

3. standardizing arguments, 4. testing arguments, 5. informal logic within a broader context, 6. informal logic and philosophy, other internet resources, related entries.

Puppo 2019 provides a recent collection of articles on the history of informal logic and the issues it addresses. In many ways, informal logic as we know it is a contemporary version of historical attempts to explain, systematize, assess and teach arguing for practical purposes.

In ancient times, the First Sophistic is a movement motivated by the notion that one can teach the art of logos in a way that can be effectively employed in public argument and debate. In the century that follows, Aristotle’s logical and rhetorical works — notably the Prior Analytics and the Rhetoric — provide a systematic account of logic and argument which is applicable to an impressively broad range of real life arguments. Today, they remain important works that inform discussions of informal logic.

In modern times, The Port Royal Logic (Arnauld and Nicole 1662, originally titled Logic or the Art of Thinking ) is an attempt to outline a logic that can guide everyday reasoning. It is a celebrated (and often disdained) introduction to the art of arguing which has been published in more than fifty French editions and five popular English translations. It understands “logic” as “the art of directing reason aright, in obtaining the knowledge of things, for the instruction both of ourselves and others” (25). In keeping with this, it provides a practical account of good and poor argument, discussing fallacies, syllogisms, definitions, and deductive and probable reasoning, emphasizing real rather than concocted examples of argument (see Finocchiaro 1997).

One finds other analogues of contemporary attempts to study and teach informal reasoning in nineteenth century textbooks on logic and rhetoric. Richard Whately is of special note in this regard. He began his career as a professor of political economy at Oxford, was appointed Archbishop of Dublin, and attempted to establish a national non-sectarian system of education, writing texts on reasoning that could promote this cause. In many ways, Whately’s Elements of Logic and Elements of Rhetoric (1826, 1830) are close analogues of early versions of today’s informal logic textbooks.

Another important text, critical of Whately, but in the same broad tradition is Mill’s A System of Logic (1882) (see Godden 2014). It defines logic as the “art and science of reasoning,” stipulating that “to reason is simply to infer any assertion from assertions already admitted.” The end result is a broad account of inference which is, like systems of informal logic, designed to inform real life instances of argument.

The first use of the term “informal logic” occurs in the last chapter of Gilbert Ryle’s book Dilemmas (1954). He introduces it in an attempt to distinguish between formal logic and the more varied, less strict, and less defined ways in which we need to assess many of the arguments that are used in philosophical discussion.

In North America, the rise of informal logic as it is now understood is tied to educational trends rooted in the 1960s — a time of social upheaval and protest (most notably, against the War in Vietnam) which produced calls for an education relevant to the issues of the day. This fueled an interest in the logic of everyday argument and the teaching of introductory logic, which was at the time taught with textbooks like Copi’s popular Introduction to Logic (1953).

In Logic and Contemporary Rhetoric: The Use of Reason in Everyday Life (1971), Kahane more fully embraced instances of real life arguing, discussing a wide range of examples in newspapers, the mass media, advertisements, books, and political campaigns. Other attempts to provide a general introduction to logic — by Carney and Scheer (1964), Munson (1976), and Fogelin (1978) — employed the term “informal logic” to distinguish between formal logic and other methods of argument analysis which lay outside of it.

The idea that informal logic should be developed as a distinct field of inquiry was championed by Johnson and Blair at the University of Windsor. They published a popular textbook, Logical Self-Defense (1977); organized and hosted “The First International Symposium on Informal Logic” and edited the proceedings (Blair and Johnson, 1980); established the Informal Logic Newsletter (1978-1983); and ultimately turned the newsletter into the journal Informal Logic (subtitled “Reasoning and Argumentation in Theory and Practice”). In this and other ways, they established informal logic as a field for study, research and development. Their contributions (and those of colleagues at Windsor and in Ontario and Canada) is reflected in the notion that informal logic is a Canadian (or, more narrowly, Windsor) school of argument (see Puppo 2019).

Much of the discussion that has shaped the evolution of informal logic as a field has taken place in a number of journals that have played a major role in its development. They include Informal Logic , Argumentation , Philosophy and Rhetoric , Argumentation and Advocacy (formerly the Journal of the American Forensic Association ), Teaching Philosophy , and (more recently) Cogency and Argument and Computation .

A key catalyst that promoted the development of informal logic was the “Critical Thinking” Movement within education (well described in Siegel 1988, Ennis 2011, and Blair 2021). The movement pushed (and continues to push) for educational developments which make the critical scrutiny of our beliefs and assumptions a fundamental goal, highlighting the significance of reasoning, inference, argument and critical assessment.

In 1980, a California State University Executive Order promoted the teaching of critical thinking and informal logic by requiring post secondary institutions to include formal instruction in critical thinking in their curriculum. According to the order: “Instruction in critical thinking is to be designed to achieve an understanding of the relationship of language to logic, which should lead to the ability to analyze, criticize, and advocate ideas, to reason inductively and deductively and to reach factual or judgmental conclusions based on sound inferences drawn from unambiguous statements of knowledge or belief” (Dumke 1980, Executive Order 338).

The accounts of argument that informal logic and the Critical Thinking movement provide are tied to pedagogical attempts to teach students and learners how to reason well. One result has been hundreds (perhaps thousands) of introductory textbooks used to teach logic, critical thinking, and argumentation skills to university and college students in Canada, the United States, the United Kingdom, Italy, Poland, Chile, and other countries.

Textbooks in English offer many theoretical and pedagogical innovations. Established texts include Battersby 2016 ( Is That a Fact? A Field Guide for Evaluating Statistical and Scientific Information , 2nd ed.); Bowell and Kemp 2014 ( Critical Thinking: A Concise Guide , 4th ed.); Browne and Keeley 2018 ( Asking the Right Questions , 12th ed.); Govier 2019 ( A Practical Study of Argument , 7th ed.); Groarke, Tindale and Carozza 2021 ( Good Reasoning Matters! , 7th ed.); Hughes, Lavery and Doran 2014 ( Critical Thinking: An Introduction to the Basic Skills , 7th ed.); Seay and Nuccetelli 2012 ( How to Think Logically , 2nd ed.); Weston 2018 ( A Rulebook for Arguments , 5th ed.); and Wilson 2020 ( A Guide to Good Reasoning: Cultivating Intellectual Virtues , 2nd ed.).

In Poland, Ajdukiewicz’s Pragmatic Logic (1974) is an introduction to an independently developed “pragmatic logic” which shares similar goals, teaching and promoting tools of logic as a central element of a general education. Its application aims to ensure that students think clearly and consistently, express their thoughts and ideas systematically and precisely, and justify their claims with proper inferences (see Koszowy 2010).

In part because reasoning and argument are ubiquitous and of interest across many disciplines, informal logic has in many ways been influenced by cognate fields which analyze argumentation in some way. The latter include formal logic, speech communication, rhetoric, linguistics, artificial intelligence, discourse analysis, feminism, semiotics, cognitive psychology, and computational modelling. Considered in this broader context, informal logic is a subfield of a broader multi-disciplinary attempt to develop a comprehensive account of real life arguing which is commonly called “argumentation theory.”

Informal logic and argumentation theory have been highlighted at numerous international conferences hosted by organizations committed to the study of argument. Key conferences include nine Amsterdam conferences (and a tenth conference in Zhenjiang) hosted by the International Society for the Study of Argumentation (ISSA); twelve Windsor conferences hosted by the Ontario Society for the Study of Argumentation (OSSA); six Tokyo Conferences on Argumentation hosted by the Japanese Debate Association; three meetings of the European Conference on Argumentation (ECA); eight meetings of the International Conference on Computational Models of Argument (hosted by COMMA), and many workshops on issues of argumentation, dialogue, and persuasion organized by the Polish group ArgDiaP.

1.1 Formal and Informal Logic

One of the driving forces behind the early development of informal logic was a desire for a logic which contained tools of analysis and assessment above and beyond the standard formal methods of the day: propositional logic, truth tables, syllogisms, the predicate calculus, and so on. Johnson emphasizes a “dissatisfaction with formal logic as the vehicle for teaching skill in argument evaluation and argument formation” and a “desire to provide a complete theory of reasoning that goes beyond formal deductive and inductive logic” (2014, p. 11). Though Ryle never developed a detailed account of what he meant by “informal logic,” his comments point in a similar direction.

Philosophers and logicians teaching general courses on argument have often managed the issues that this raises by mixing formal and informal methods of analysis. Though the artificial examples of argument that characterize early editions of Copi (and some other texts that emphasize formal logic) have been roundly criticized (see Johnson 1996 and Blair 2015), other authors more successfully meld classical formal logic and examples of real argument. Harrison 1969 is an example. So are the texts of Pospesel, which contain many examples of real life reasoning which illustrate the patterns of inference associated with propositional and syllogistic logic (Pospesel and Marans 1978, Rodes and Pospesel 1991, and Pospesel 2002). Little 1980 develops an approach to critical thinking and decision making that integrates propositional and syllogistic reasoning.

Some informal logicians moved in an opposite direction, developing versions of informal logic that were independent of formal logic, appealing to alternative accounts of argument borrowed from or influenced by philosophical reflection, or by other fields that study argument (notably, rhetoric, dialectics, and speech communication). Toulmin’s The Uses of Argument (1958) and Hamblin’s Fallacies (1970) became theoretical touchstones for those looking for an informal logic rooted outside of formal logic.

Even in these latter cases, systems and accounts of informal logic shared some core notions with their formal cousins. Most significantly, formal and informal logic assume (i) a premise, inference, and conclusion conception of argument; (ii) the notion that a good argument has premises which are true (or “acceptable”) and a conclusion which follows from them (making the inference the argument depends on “valid” in some way); and (iii) the idea that many arguments can be assessed by treating them as instances of more generalized forms of argument (“schemes of argument” which include standard deductive patterns of reasoning).

Today, informal logic is “informal” rather than “formal” primarily because it studies arguments as they occur in natural language discourse, and not in formal languages of the sort that characterize formal logic. The latter are notable for their rigorously defined syntax, semantics, and grammar, and precisely defined proof procedures. In contrast, arguments as they occur in real life discourse are notable for their use of everyday language; for the many different norms that apply to them in different contexts; and for the diverse ways of making meaning (using pictures, facial expressions, non-verbal sounds, music, etc.) that they employ.

The emphasis that informal logic and its historical precursors place on natural language argument is in keeping with the social and pedagogical goals that motivate their development. The latter are best served by an easily understood, broadly disseminated, understanding of the difference between strong and weak instances of real life argument. Formal methods (Venn diagrams, probability theory — see Zenker 2013, different kinds of formal logic, etc.) can at times support this goal, especially in analyses of particular kinds of argument, but the overriding aim is natural language discourse guided by the principles of successful reasoning.

1.2 Argumentation Theory

Argumentation theory is a broad interdisciplinary field that studies real life argument. Developments in argumentation theory have been greatly influenced by formal and informal logic; pragma-dialectics (developed by van Eemeren and Grootendorst and their Dutch or “Amsterdam” school of argumentation); the Critical Thinking Movement; American speech communication; rhetoric; pragmatics; and the Polish School of Argument (which has published a relevant manifesto ). In its attempt to understand argument, argumentation theory does not hesitate to make use of these and other related disciplines: cognitive psychology, computational modeling, semiotics, discourse analysis, the history of art, AI, and so on.

Above and beyond their study of general methods of argument analysis, argumentation theorists investigate arguments in a variety of ways: by exploring particular aspects of arguing (e.g., onus, burden of proof, or the norms that govern arguments in specific contexts); by analyzing arguments from historical, social, political or feminist points of view; by studying particular kinds of argument (e.g., those expressed in works of art, or those that arise in specific legal contexts); and by investigating the assumptions, conditions (philosophical, epistemological, social, political, institutional, psychological, educational, etc.) which give rise to disagreement, arguing and argument in the first place. In many ways, such endeavors intersect with informal logic.

Theoretically, argumentation theory incorporates three approaches to argument that have been associated with logic, rhetoric, and dialectics since ancient times. Logic understands an argument as an attempt to justify a conclusion, emphasizing its probative or epistemic merit. Rhetoric treats an argument as an attempt to persuade, emphasizing its persuasive force. Dialectics understands an argument as an element of an exchange between interlocutors with opposing points of view, emphasizing its place in an interaction between arguers who argue back and forth.

Like other forms of logic, informal logic emphasizes the epistemic merit of an argument. That said, its development has been greatly influenced by rhetorical and dialectical considerations, for successful real life arguments must convince their intended audiences (the public, scientists, the Members of Parliament, the readers of a particular magazine, etc.) and (as Johnson 2000 emphasizes) include answers to reasonable objections made by those with opposing points of view. Bermejo Luque 2011 has proposed a relevant theory of argument which aims to accommodate logical, rhetorical and dialectical points of view.

As Hansen (2012) emphasizes, there are many different methods that informal logicians use to analyze instances of argument. An account of informal logic needs to make room for the different approaches this implies, at the same time that it explains how they constitute a shared field of inquiry. I will attempt to provide such an account by emphasizing systems of informal logic.

I understand an informal logic system to be a collection of principles and methods designed for the analysis and assessment of real life arguments. Considered from this point of view, informal logic can be described as a field devoted to the creation, study and application of systems of informal logic (or, more simply, “informal logics”) and the issues that this raises.

The first textbook that described itself as an informal logic textbook was Munson’s The Way of Words: An Informal Logic (1976). As his title emphasizes, his book aims to teach a specific informal logic (a specific informal logic system) which outlines one approach to the analysis and assessment of real life argument. Other approaches and other systems are developed in other textbooks and in scholarly discussion.

Different systems of informal logic vary in a number of ways — often, by incorporating formal, rhetorical, dialectical and other methods of analysis to a greater or lesser extent. Many systems propose unique approaches or mix methods that they borrow from other systems. Groarke 2020 has outlined a “BLAST” approach to the identification and definition of systems of informal logic. It defines a specific informal logic system I , as I = { B , L , A , S , T }, where:

• B = the theoretical background that informs I , L = the language used to express the arguments I analyzes, A = a concept of argument, S = a way to “standardize” arguments, and T = tools and methods for testing the strength of arguments evaluated using I .

As I show below, a particular system of informal logic can be defined by outlining and explaining each of its five BLAST elements. When they have been defined, different systems of informal logic and the elements they contain can be more precisely compared, contrasted, and evaluated. Historical precursors to present day informal logic can be investigated from a similar point of view.

2.1 A Concept of Argument ( A )

In many ways, the third element in the BLAST list — a concept of argument ( A ) — is the root of all informal logics. In ordinary discourse, the word “argue” can mean “to disagree,” usually with the further implication that someone does so aggressively. Informal logics, like other logics, assume a narrower conception of argument (so called “argument-1”), which understands an argument as an attempt to resolve disagreement (or potential disagreement) by providing reasons for accepting the point of view that it advances.

This notion of argument is “evidentiary” in the sense that it understands an argument as an attempt to provide evidence in support of some conclusion. The premises in the argument convey the evidence which provides those who consider the argument reasons for accepting the conclusion. As Hitchcock 2007 puts it, an argument is “a claim-reason complex” consisting of (1) premises, (2) a conclusion, and (3) an inference from the premises to the conclusion (and the implied claim that the conclusion is true, likely true, plausible or should in some other way be accepted).

The following sentence, taken from an article in the Houston Chronicle (Devra Gartenstein 28/01/19) is a simple example of argument in this sense.

EXAMPLE 1: Small businesses are important because they provide opportunities for entrepreneurs and create meaningful jobs with greater job satisfaction than positions with larger, traditional companies.

In this example, the word “because” is an inference indicator. It tells us that the initial statement in the sentence (“Small businesses are important”) is a conclusion backed by a premise (that small businesses “provide opportunities...”) that provides a reason for believing it to be true.

The example below is taken from an opinion piece (in the Western Courier 25/10/08) which criticizes conservative groups opposed to research using human embryos.

EXAMPLE 2: This [opposition to embryonic research] is shortsighted and stubborn. The fact is, fetuses are being aborted whether conservatives like it or not. Post-abortion, the embryos are literally being thrown away when they could be used in lifesaving medical research.... Lives could be saved and vastly improved if only scientists were allowed to use embryos that are otherwise being tossed in the garbage.

We can summarize the elements of this argument as follows.

Premise : Fetuses are being aborted anyway (whether conservatives like it or not). Premise : Lives could be saved and vastly improved if scientists were allowed to use embryos that are otherwise being tossed in the garbage. Conclusion : The conservative opposition to embryonic research is shortsighted and stubborn.

This is another example of a simple argument. In real life arguing, complex arguments may contain tens (or even hundreds) of premises, and are usually made up of layers of inference that proceed from initial premises to intermediate conclusions which function as premises for further conclusions, culminating in a “main” or “principal” conclusion.

All systems of informal logic are attempts to understand and assess arguments in the premise/conclusion sense. In view of this, the core value of A in the BLAST definition is shared by all informal logics, though there are many variations that occur when informal logicians expand this conception by adding other elements to it. This can be done by understanding an argument as a premise-conclusion complex which is directed at an audience , or backed by a warrant , or provided as an answer to an opponent who has an opposing point of view. When they expand the definition of argument in these ways, systems of informal logic make an audience, warrant, and/or dialectical context an essential element of a case of argument.

2.2 Theoretical Background ( B )

Other elements recognized in the BLAST definition are notable for the extent to which they vary from one informal logic to another. The different theoretical backgrounds that inform different systems of informal logic ( B in the BLAST definition) are of special note, for they reflect the extent to which the development of informal logic has been a search for new and novel methods of analysis that can be applied to real life arguing. In a particular case, this search may end in Aristotle, in feminist theories of argument, in rhetoric, in dialogue theory, in speech communication, in formal logic, or in some combination of these and/or other theoretical points of view.

The theoretical background that characterizes systems of informal logic can be used to identify, not only specific systems, but also families of systems which share key theoretical elements. Many informal logicians have, to take one example, turned to fallacy theory in their attempt to find a logic that can account for real life arguing. In some cases, the result has been fallacy oriented textbooks which introduce tens or hundreds of fallacies used to dismiss straw man reasoning, hasty generalizations, slippery slope reasoning, and so on (see, e.g., Bennett 2018).

Systems of informal logic that adopt a fallacy approach can be summarized as instances of I in which I = { B , L , A , S , T }, where: B = Fallacy Theory, and T = a set of fallacies used to judge instances of argument. An even narrower set of (“Hamblin”) systems can be defined as systems in which B = Hamblin. This is a family of systems which is rooted in the account of fallacies proposed in Hamblin 1970.

Systems of informal logic that combine fallacy theory with other methods of testing arguments can be understood as systems which define T in a way that includes a specific a set of fallacies along with other criteria for argument evaluation (e.g., Johnson and Blair’s ARS criteria).

2.3 Language ( NL and NL+ )

One key element of informal logic is its focus on arguments as they occur within natural rather than formal languages.

Initially, informal logic texts defined natural language arguments as verbal arguments: i.e. as arguments expressed and conveyed in words associated with some established natural language. This is an important focus given that arguments of this sort are a staple in the kinds of arguing that informal logic emphasizes — in letters to the editor; parliamentary debate; court proceedings; and in essays and books written in defense of some point of view (that humans could colonize Mars, President Trump was misunderstood, socialism is the best political system, and so on).

When a system of informal logic is designed to analyze and assess verbal arguments of this sort, the language of argument they assume ( L in the BLAST definition) consists of the words and sentences (and the rules governing them) associated with some natural language (“ NL ”). This makes the language of argument a complex and nuanced system of meaning which expands the possibilities of argument beyond the “informative” statements of fact that authors like Copi took to be the only legitimate realm of argument. One notable feature of the expanded realm that informal logic studies is the extent to which metaphor and other figures of speech are used to support conclusions — something that must be accounted for in a comprehensive account of informal reasoning.

As the scope of informal logic has expanded to provide a more comprehensive account of ordinary argument, this verbal paradigm has frequently been expanded to make room for real life arguments which depend on non-verbal elements. Paradigm examples of these non-verbal elements are photographs, illustrations, drawings, videos, etc. which are used to provide evidence in support of some conclusion -- as when a photograph or video is used to prove someone’s identity, or produced in court to give the court a reason to believe that they were present at the scene of a crime. As Hitchcock (2002) notes, “a poster with a giant photograph of a starving emaciated child and the words ‘make poverty history’ can reasonably be construed as an argument.”

Arguments that employ premises and/or conclusions that arguers convey using non-verbal visual elements are commonly called “visual” arguments. I leave for elsewhere the question whether such arguments are visual in some deeper sense. In most cases, they have verbal as well as visual elements and can, in view of this, be described as arguments which are expressed as an amalgam of verbal and non-verbal (visual) elements.

The two photographs in EXAMPLE 3 (below), taken by the NASA Mars rover Phoenix, were part of a visual argument proposed by the NASA scientists who directed the Phoenix mission. The first photo provides an initial view of a dig made by the rover; the second a view of the same dig four Martian days (sols) later. When we look at the bottom left hand corner of the first photograph and compare it to the bottom left hand corner of the second photograph, we can see white crust in the first photograph but not the second. According to the NASA scientists, the only way to plausibly explain this change is by understanding the crust as water ice which evaporated when the dig exposed it to the sun.

We can summarize the argument as follows.

EXAMPLE 3: (Visual) Premise : What we see in the bottom left hand corner of the first photograph of the dig. (Visual) Premise : What we see in the bottom left hand corner of the second photograph of the dig. (Verbal) Premise : The most plausible way to explain the changes we see (the disappearance of the white crust) is by understanding it as water ice. Conclusion : There is water on the planet Mars.

This argument can be described as visual and verbal. Visual because our looking at the photographs and seeing the difference between them is a non-verbal component of the argument that provides key evidence that supports the conclusion. At the same time the argument is verbal because it contains a key verbal component provided by the scientists’ explanation of what we must be seeing. The conclusion is established by combining these visual and verbal components.

When a system of informal logic recognizes visual arguments of this sort, the value of L in its BLAST definition can be understood as NL+ , where the + indicates that the language of argument includes words and non-verbal elements like photographs and other non-verbal visual images.

One might compare the expansion of informal logic to account for visual arguments to the attempt to expand the language of formal logic (and, some argue, the notion of argument it embraces) to allow visual deductions (see Barwise and Etchemendy 1998). Visual premises can be used in geometric proofs, in legal arguments (to proving that someone committed a crime, that the murder was heinous, etc.), in reasoning about planets, stars and black holes, and in aesthetic criticism and evaluation. In ornithology, a well authenticated, clear photograph of a rare species (say, the Ivory Billed Woodpecker) is widely accepted as the proper proof that the species is not extinct.

In other cases, a visual image may serve as a conclusion which is backed by visual and/or verbal premises. A forensic artist concludes that their drawing of a suspect correctly depicts what they look like by inferring this from verbal and visual evidence which is provided by the witnesses they interview.

Kjeldsen 2015 has provided a comprehensive overview of the study of visual arguments. Informal logics that recognize visual arguments do so for the same reason that they recognize verbal arguments: in order to account for the kinds of arguing that play an important role in real life discourse and exchange. Visual arguments are increasingly significant at a time when digital communication facilitates the creation and distribution of visual images, enhancing and increasing the use of arguments which employ photographs, videos, art, political cartoons, virtual reality, 3D modeling and other kinds of visuals (see Godden, Palczewski, and Groarke, 2016).

As is the case with words, the images used in visual arguments may, as Kjeldsen and other rhetoricians point out, not be intended as literal depictions of something they represent, but function as visual metaphors which characterize some situation in an argumentatively charged way.

The cartoon of Napoleon III below (EXAMPLE 4) appeared in Punch on Feb. 19, 1859.

This is a clever caricature, but it is something more than a simple caricature, for the cartoonist uses a drawing of Napoleon-as-porcupine to convey a reason to doubt his profession of peace. We might summarize his point as the suggestion that Napoleon’s actions -- his amassing of the armaments which encircle him like quills on a porcupine -- are not the actions of someone committed to peace. The dissonance between his actions and his profession of peace invites the viewer to respond to him as they would respond to an actual porcupine -- with caution, concern and suspicion, wary of what he might do with his weapons (even though he has an “inoffensive” posture and makes an inoffensive sound).

In this case the visual image in the argument is not an attempt to literally replicate some state of affairs. The core argument we might extract from the cartoon can be understood as follows.

Premise : Napoleon declares that “The Empire embodies peace” (“L’Empire c’est la paix”). Premise : Napoleon has surrounded himself with many armaments. Conclusion : Napoleon may sound inoffensive when he says that “The Empire embodies peace,” but his build up of armaments suggests we should be wary of the empire he has built.

This is a visual argument (and the second premise a visual premise) at least in the sense that it is conveyed visually (and not with the words in our paraphrase). The verbal paraphrase summarizes the argument, though the non-verbal visual elements the cartoonist uses must be replicated if we want to exactly replicate their act of arguing. This is especially important when such arguments are open to contrary interpretations or have meanings that are difficult to convey in words.

In the case of the Punch cartoon, we might interpret it in a second way by adding to our initial paraphrase, understanding the Napoleon III-as-porcupine caricature as an allusion to another famous “Louis” King of France: Louis XII, who ruled from 1498 to 1515. Popularly known as the “porcupine” King of France, he adopted the animal as his royal emblem, portraying the French kingdom as a porcupine -- an animal he chose because it was popularly believed (to some extent now as well as then) that it could shoot its quills, symbolizing the offensive and defensive capabilities of the king’s army.

If we understand the cartoonist’s depiction as an allusion to Louis XII, the argument in the cartoon is plausibly assigned another premise that reminds the viewer of the rule of Louis XII, a king who was famous for rebuilding the French army, which he then employed in a series of wars against his neighbors (his general strategy is criticized by Machiavelli in The Prince ). The additional premise can be summarized as the suggestion that Napoleon is acting like Louis XII did — something which provides another reason to be wary of him.

In many circumstances, the visual elements of an argument (like verbal elements in arguments which are ambiguous or vague) are open to multiple interpretations. This raises issues of representation and interpretation. In the Napoleon example, it is difficult to say whether the cartoonist or his audience understood his depiction to be a reference to Louis XII. In such circumstances, the simplest strategy is one that identifies and evaluates all the plausible interpretations of an argument.

Visual metaphors are common in works of art, political propaganda, and political cartooning. One popular motif depicts politicians with a growing Pinocchio nose, suggesting that they are, like the childhood hero, preposterous liars (see Tseronis and Forceville 2017). In cases such as this, visual images can have rhetorical as well as logical significance, allowing arguers to convey their arguments in rhetorically powerful ways.

2.4 Modes of Arguing

Informal logicians who advocate for systems of informal logic that broaden the language of argument from NL to NL+ sometimes expand the standard notion of argument even further, beyond verbal and visual arguments, to make room for other ways of arguing.

The role of non-verbal carriers of meaning in real life discourse allows many different semiotic modes to contribute to informal arguments. In view of this, some argue that a fully comprehensive account of argument must include a “multimodal” account of meaning which recognizes many different modes of arguing (see Groarke 2015 ). The latter may employ gestures, facial expressions, sounds of different sorts, tastes, smells, musical notes, and a host of other non-verbal phenomena.

Groarke 2018, Kišiček 2018, and Eckstein 2018 have investigated “auditory” (or “acoustic”) modes of reasoning that depend on non-verbal sounds. Examples are situations in which the sound of a siren is pointed to as evidence that the police are on their way, or the sound of an automobile engine (or an irregular heart) is the basis of an inference to the conclusion that it has a leaky valve.

Other modes of arguing invoke the senses in other ways. In 2019, Space Cargo Unlimited and the University of Bordeaux’s wine institute (the ISVV) sent fourteen bottles of Château Petrus to the International Space Station to determine how a year in space would affect their taste and character. A year later, the wine returned to earth where it was analyzed. A key element of this analysis was a blind tasting conducted by 12 professional tasters who compared the wine from space to bottles of the same vintage which had remained on earth. The sommeliers concluded that the wine which had been in space had a different, distinct (“more evolved”) taste, inferring their conclusion from their two tastings (which functioned as ‘taste premises’).

Here is another example from a wine tasting.

EXAMPLE 5: A wine steward attempts to convince a customer that Napa Valley “Frogs Leap PS 2015” is an exceptional Petite Syrah by quoting high praise in a wine guide, and by handing them a glass, inviting them to taste it.

In this case, the steward provides evidence for their claim that Frogs Leap PS 2015 is an exceptional Petite Syrah in two ways: (i) via a verbal premise that cites the authority of a wine guide, and (ii) via a taste test that aims to support this conclusion. In the first case, the result is a verbal appeal to authority, in the second an argument by taste.

In real life arguing different modes are often mixed. In the case of verbal arguments, the meaning of oral arguments is often notable for its dependence on the sound of the human voice — which can convey meaning above and beyond that implied by words. As Gilbert (1997) notes, uttering the sentence “Fine, fine, you’re right, I’m wrong, we’ll do it your way,” “can indicate agreement with what has been said if presented flatly and intended sincerely, or, if accompanied by an expression of anger, can mean that the respondent does not agree at all, but is capitulating.” (pp. 2–3) Kišiček 2014 has highlighted the important role that the paralinguistic features of oral arguments play in many instances of verbal argument.

Gilbert 1997 was the first informal logician to suggest that there are different modes of arguing that need to be distinguished. His “multi-modal” theory of “coalescent” argument accepts the traditional conception of argument as one mode, adding to it “emotional,” “visceral” (physical) and “kisceral” (intuitive) modes of arguing. He pairs this with a very broad account of arguing which understands it as an attempt to resolve conflicting attitudes, beliefs, feelings and intuitions in a way that brings about the “coalescence” of competing points of view.

According to Gilbert’s account, an argument may consist of sentences and/or expressions of emotion, physical demonstrations and difficult to define intuitions. This allows a forlorn look or tears to count as premises in an argument insofar as they provide evidence that promotes coalescence (and inference in this broad sense). As Gilbert argues, his non-traditional modes of arguing may, in real life situations, provide effective ways of resolving the disagreement that gives rise to argument.

Gilbert’s account of modes radically expands our understanding of argument beyond “language” as it is traditionally conceived. In an account of the “anthropology” of argument, Tindale 2021 defends the alternative “logics” this makes possible. In doing so, he promotes informal logics that recognize verbal arguments (and in this sense L ) but add to it a much broader range of everyday communication which can convey the substance of an argument via expressions of emotion, physical actions (including non-verbal means of communication like so called “body language”), and difficult to define intuitions. This expands the language of argument far beyond words and sentences, in a way that maximizes the breadth of L+ .

The ultimate goal of informal logic is normative: i.e. systems of logic that provide us with tools that can be used to evaluate real life arguments and the key components they contain. Because real life discourse often fails to present the latter clearly, most systems of informal logic prepare the way for argument evaluation by explicitly identifying an argument’s premises, conclusions, and inferences. The process of doing so is commonly called “standardizing” ( S in the BLAST definition).

In the process of standardization, informal logics aim to untangle issues that obscure the structure and content of real life arguments. Arguments in ordinary discourse are, for example, frequently vague, ambiguous or in other ways unclear. Premises and conclusions can be conveyed by rhetorical questions (“Can anyone seriously believe...?” “Could the defendant have been in two places at once?”) and the key components of an argument may be interspersed with irrelevant digressions and repetition. When we extract an argument from the context in which it occurs, it can be important to recognize what is implicit but relevant at the same time that we discard what is explicit but irrelevant.

Standardizing clarifies the structure of an argument in ordinary discourse by:

• discarding irrelevant and distracting digressions, repetition, and remarks (“noise”) which do not play a role in the reasoning the argument contains;
• restating the content of rhetorical questions and other stylistic devices that may obscure the meaning of the argument’s components;
• clarifying incomplete, vague or ambiguous claims and utterances; and/or
• recognizing components of the argument which are not explicitly expressed.

Systems of informal logic may standardize arguments in a variety of ways. In its simplest form, a standardized argument is a list of premises and a conclusion (the argument’s inference is the implicit move from the premises to the conclusion). EXAMPLE 1 above can be standardized as follows.

Premise : [Small businesses] provide opportunities for entrepreneurs and create meaningful jobs with greater job satisfaction than positions with larger, traditional companies. Conclusion : Small businesses are important.

In this example, the word “because” functions as an “inference indicator” which tells us that the claim that follows is a premise offered in support of the conclusion which precedes it. In other cases, words like “for,” “given that,” “since,” are ways of introducing premises, while words like “so,” “hence,” “thus,” “therefore,” are used to introduce conclusions.

Verbal inference indicators are an important, but not necessary or sufficient, sign of argument. For indicator words can be used in other ways (“because” may indicate an explanation rather than an argument, a causal connection, emphasis of some sort, a temporal order, etc.), and are not necessary when a context makes it clear that someone is providing reasons for a conclusion.

3.1 Implicit Premises and Conclusions

One issue that is frequently addressed in standardization is an argument’s dependence on premises and conclusions which are assumed but not explicitly stated.

The following report from the New Hampshire Rockingham News (30/8/2002) comments on a court case which sent the organizer of dog fights to jail for cruelty to animals.

EXAMPLE 6: A co-ordinator for the Humane Society supported a prison sentence, claiming that the minor penalties normally associated with misdemeanor convictions are not a sufficient deterrent in this case.

We can standardize the co-ordinator’s argument as follows.

Premise : The minor penalties normally associated with misdemeanor convictions are not a sufficient deterrent in this case. Implicit Premise 1 : Penalties for crimes should have a deterrent effect. Implicit Premise 2 : Prison is a sufficient deterrent. Conclusion : A prison sentence, but not the minor penalties normally associated with misdemeanor convictions, is appropriate.

It is important to recognize the assumption which is rendered as Implicit Premise 1 (which is, in Toulmin’s words, “a warrant”) for it is a claim that the argument depends on when the arguer concludes that the penalties normally associated with misdemeanors are not a sufficient deterrent in this case. The remaining part of the conclusion — the suggestion that a prison sentence is appropriate — depends on Implicit Premise 2 , which recognizes the further assumption that this will have a deterrent effect.

Recognizing the argument’s implicit premises prepares the way for argument evaluation, for they, like the argument’s explicit premise, need to be evaluated when the argument is assessed. If we reject the implicit premises (by, e.g., arguing that the goal of criminal penalties should be retribution, not deterrence), then the argument fails to provide convincing evidence for its conclusion.

Arguments with implicit premises or conclusions are recognized in ancient discussions of enthymemes : syllogisms with unstated premises. A contemporary example is American Vice-President Dick Cheney’s defense of the Bush administration’s decision not to try foreigners charged with terrorism offenses as prisoners of war (something that would guarantee legal protections for those accused of terrorism).

EXAMPLE 7 (from a report in the New York Times , 15/11/2001): “The basic proposition here is that somebody who comes into the United States of America illegally ... is not a lawful combatant.... They don’t [therefore] deserve to be treated as a prisoner of war.”

We can standardize the argument as:

EXAMPLE 8: Premise : Somebody who comes into the United States of America illegally ... is not a lawful combatant. Implicit Premise : Someone who is not a lawful combatant doesn’t deserve to be treated as a prisoner of war. Conclusion : They don’t deserve to be treated as a prisoner of war.

As in many other cases, the argument’s implicit premise identifies the link that ties the explicit premise to the conclusion. When we evaluate the argument, it raises a number of questions. How should a “lawful combatant” be defined? Why are combatants who illegally enter a country for the purpose of war (in undertaking surveillance or going behind enemy lines) widely recognized as prisoners of war? What are our moral obligations to combatants, lawful and unlawful?

In other enthymemes, the conclusion of an argument may be unstated. In a debate over the question whether witnesses to a crime can be trusted, an arguer might state: “ They are friends of the accused, and no friends of the accused can be trusted.” Such claims clearly suggest that “ They cannot be trusted,” a claim that needs to be recognized as an implicit conclusion in an attempt to standardize and evaluate the argument.

As in the case of verbal arguments, implicit premises and conclusions play an important role in many visual and multimodal arguments. In EXAMPLE 9 (below), the title “Just Add Vodka” is superimposed over a gigantic bottle of vodka pouring its contents on the scene below. Outside the vodka splash, one sees a sleepy hamlet. Inside the vodka splash, it is transformed into a bustling cityscape which boasts skyscrapers and a nightlife with lights, people, nightclubs, bars, and restaurants.

The image in this advertisement is a visual metaphor which suggests that vodka can transform one’s life in the way that it transforms the scene in the advertisement, turning the (dreary) life one lives in a sleepy hamlet into the kind of exciting nightlife one finds in a cosmopolitan urban center. Insofar as this message is featured in an advertisement, we can plausibly understand this message as a proposed reason to buy the vodka in question.

We might standardize the argument this implies as follows.

EXAMPLE 9: (Visual) Premise : If you drink our vodka, it can transform your night life in the way it transforms the village in the photograph: into an exciting major city. Implicit Premise : A life of urban excitement is preferable to the quiet life of a village. Conclusion : You should “Just Add Vodka” to your life.

I have summarized the first premise in the argument in a way that refers to the image because it is difficult to fully capture the (almost magical) transformation the advertisement promises if we try to translate it into words. The implicit premise in the standardization is a key part of the argument because it is a key assumption made in the move from the (visually) explicit premise to the conclusion. So understood the argument can be analyzed and evaluated as an argument which has a questionable explicit and implicit premise (and an inference which is an instance of the fallacy “Affirming the Consequent”).

The attempt to identify implicit premises or conclusions in an argument can raise some significant theoretical questions when it is standardized. All arguments rely on many assumptions, raising the question whether and when and how they should be recognized. In cases in which it is clear that an unstated premise or conclusion needs to be recognized, it can often be interpreted in different ways.

One principle that many systems of informal logic use to choose between alternative interpretations of implicit argument components is the “Principle of Charity.” It favors an interpretation of an argument which makes it as credible as possible. In many situations, this can best be accomplished by attributing it a “logical minimum,” understood as the weakest implicit component needed to successfully connect the argument’s premises to its conclusion.

3.2 Key Component Tables and Diagrams

Standardizing arguments by listing their premises and conclusions is one way to delineate the content of an argument and prepare it for evaluation. But standardizing of this sort has a major shortcoming, for it does not distinguish between different kinds of inferences an argument may contain. One way to address the issues that this raises is by standardizing an argument with a table that catalogues an argument’s premises and conclusions (a Key Component or “KC” table) and combining it with an argument diagram (a “mapping”) that depicts the structure of its inferences.

In EXAMPLE 10 (below), I have used this method to standardize an argument that Kretzmann uses in his account of the medieval philosopher William of Sherwood, where he concludes that it is likely that William was a Master at the University of Paris. The first column of the KC table lists the premises and conclusion of the argument; the second assigns them their role as premise or conclusion; the third lists the source from which they are derived. The diagram that follows depicts the relationships between the argument’s key components, using arrows to indicate the inferences it includes.

This standardization shows that the Sherwood argument is supported by three “convergent” premises which provide three independent reasons that support the argument’s conclusion in different ways.

In other instances of argument, two or more “linked” premises combine to support a conclusion with a single inference. The argument “The murderer was very strong, so George cannot be the murderer.” is an enthymeme which assumes that “George is not very strong.” Using square brackets ([ ]) to indicate this implicit component, we can standardize the argument with the following KC table and diagram. In this case, the linked premises are indicated with a plus sign (“+”) that connects them in the diagram. It indicates that they support a conclusion (only) when combined.

The use of KC tables and diagrams is not limited to purely verbal arguments. When arguments have non-verbal key components, visual and multimodal premises and conclusions can be recognized in a key component table by ostension, or by reproducing them in some way. Once they are identified in a KC table, the inferences such arguments depend on can be mapped in standard ways.

The following table and diagram standardizes EXAMPLE 3, which contains an inference from three linked premises (two visual and one verbal) to the conclusion that there is water on the planet Mars.

KC table are not the only way to diagram the structure of real life arguments. Diagramming (mapping) has its own history, which incorporates many different ways of diagramming arguments. The usefulness of diagramming is already recognized in Whately 1826 and, in the early twentieth century, in Wigmore 1913, who develops a form of mapping (“evidence charts”) designed to portray and analyze complex chains of judicial reasoning.

The development of informal logic has kindled a renewed interest in different kinds of diagrams which are supported by the development of associated software (Rationale, Reason!Able, Araucaria, Athena, Compendium, Theseus) and online aids (Debate Mapper, TruthMapping.Com, Argunet, Agora).

3.3 Supplemented Diagrams

KC tables and diagrams prepare the way for argument evaluation by clarifying the internal structure and content of an argument. Other aspects of an argument that may need to be considered in argument assessment can be included in a “supplemented” diagram which adds an account of the context in which it is embedded. Three aspects of arguments merit note in this regard.

The first is the audience to which an argument is addressed. Most real life arguments are used in an attempt to convince some intended audience of some point of view. In view of this, successful arguments must be built with this in mind.

A convincing argument for the conclusion that the United Nations cannot be trusted must address different issues when it is directed at a Chinese, Norwegian, Kenyan, Israeli, Swiss, Palestinian, etc. audience. As rhetoric has emphasized since its beginnings, this means that successful arguers must construct their arguments in ways that recognize the beliefs, attitudes and values of their intended audience (and in this sense ‘speak’ to them). Tindale (1999, 2004, 2010) has imported this notion into informal logic, advocating an informal logic that incorporates an analysis of audience.

A second contextual factor relevant to the evaluation of an argument is the goal of the arguer. As Hitchcock 2002 points out, acts of arguing may make a declaration (“The evidence shows that you committed an assault, so I find you guilty as charged.”); command or make a request (“You were there, so you must tell us what happened.” “The children are shivering, so please close the door.”); make a promise (“I know it matters to you, so I promise to go tomorrow.”); express a sentiment (“What we did was inexcusable, so we beg your forgiveness”); and function in many other ways.

As Pinto and Gilbert have emphasized, this means that a successful argument (which is successful in the sense that it accomplishes the arguer’s purpose in presenting the argument), may produce a withholding of assent (or full assent) to some proposition, a particular attitude, an emotional state like fear or hope, or a certain kind of behavior (by, e.g., as when an argument demands that people take up arms against a foe or take action in support of social change). When one bargains, the goal of argument is not truth, but a bargain that serves one’s interests. As Hoffman 2016 notes, an argument may not aim to resolve disagreement, but to promote reflection and the raising of important questions.

The different goals arguer’s try to achieve via argument may make their success and legitimacy turn on norms and rules that add to (or subtract from) the traditional strictures that guide argument evaluation. In some circumstances, it makes ultimatums, exaggerations, threats and insults a permissible element of arguing. In sharp contrast, they are unacceptable in attempts to determine what is true from a scientific point of view, where such behaviors are instances of the fallacy ad baculum .

Walton 2007 accommodates the different goals associated with real life arguments by distinguishing between different kinds of dialogues in which arguing occurs. The rules for a particular kind of dialogue define what types of argumentative moves are allowed, what kinds of questions and responses are permitted, and what norms arguments must adhere to.

The seven basic types of dialogue he distinguishes can be summarized as follows.

In dialogues of inquiry, arguments are used as tools in an attempt to establish what is true. So understood, arguments must adhere to strict standards that determine what counts as evidence and counter-evidence for some point of view. In eristic dialogue, arguing is combat and the aim is to vanquish one’s opponent (and humiliate them, ideally by wowing one’s audience with one’s mental gymnastics). In doing so, sophistical tricks and fallacious reasoning are welcome if they serve this end.

Walton’s dialogue typology leaves room for more narrowly defined kinds of dialogue. Collective bargaining is a specific kind of negotiation dialogue which is governed by legal rules and well established practices that very precisely delineate what is and is not allowed in bargaining. Different subdialogues are associated with different norms, rules and practices.

A third contextual factor that may warrant comment in a supplemented diagram is the dialectical context in which an argument occurs. In his account of argument, Johnson 2000 distinguishes between the “illative” core of an argument and the argument’s “dialectical tier,” understanding the former as a “proto-argument” which consists of a set of premises offered in support of some conclusion. It is the kernel of an argument, but he considers it an argument in the full sense only if it fully engages the dialectical tier, considering alternative points of view, addressing objections to the conclusion it proposes.

The ultimate goal of informal logic is normative: an account of argument and systems of informal logic that can be used to determine when and whether real life arguments are strong or weak, good or bad, convincing or unconvincing. Standardizing prepares arguments for such assessment. This makes T in the BLAST definition — the tools and methods which are used to test argument strength — the most important defining element in a system of informal logic.

Almost all informal logics understand a good (strong) argument to be an argument with “acceptable” premises and a “valid” inference — i.e. a conclusion that follows from them. Hansen 2012 has argued that informal logic should follow classical logic, and not concern itself with the assessment of premise acceptability, but its engagement with real life arguments (and the desire to evaluate them in a fulsome way) has produced a field that includes this within argument evaluation. In this and in other regards, systems of informal logic adopt many different approaches to argument assessment.

4.1 AV Criteria

In classical logic, an argument is (deductively) valid if it is impossible for its premises to be true and its conclusion false. On this account, the ultimate aim of arguing is a “sound” argument: i.e. a valid argument with true premises.

Within informal logic, the simplest criterion for good arguments is an informal analogue of soundness. It understands a good argument to be an argument that justifies its conclusion by providing good (strong, credible, etc.) reasons for believing it. Within the argument, this implies premises which are “acceptable” and a conclusion that follows from them. We can summarize this basic criterion as T = { A , V }, where A is an account of premise acceptability, and V is an account of informal validity which determines when a conclusion follows from premises in a way that is approved of (deductively, inductively, conductively, and/or abductively, etc.).

Following Johnson and Blair (1977, 1994), many systems of informal logic adopt an ARS version of these AV criteria, making “Acceptability, Relevance and Sufficiency” the requirements for good argument. On this account, premises are acceptable when they are true or acceptable in some other way; relevant when they provide some (i.e. any) support for the conclusion of the argument; and sufficient when they provide enough support to warrant its acceptance — as likely, true, plausible, etc.

Premise Acceptability

Within informal arguments, premises may be acceptable in a variety of ways. In many circumstances, they are acceptable if they are likely true and unacceptable if likely false.

It is worth noting that this truth criterion can be expanded to apply to visual premises. An image functioning as a visual premise may, for example, be evaluated as a “true” or “false” depiction of what it represents. A photograph or an image may be unacceptable because it is untrustworthy or categorizes a situation in a misleading way. Photographs are often ‘doctored’ or in other ways designed to present things in ways that do not accurately reflect what is photographed. Other kinds of multimodal premises can be understood as likely true to the extent that they are a reliable basis for an inference.

In some kinds of dialogue — in the exchanges that characterize negotiation, bargaining, eristic, and persuasion — acceptable premises may not need to be true. In bargaining (“haggling”) a buyer may claim that “I will not give you a penny more than $300 for that lamp” as a premise in support of the conclusion that the seller should agree to a lower price. This counts as an acceptable premise even if it is an idle threat that the buyer will never carry out, for threats of this sort are an acceptable element of the arguments that take place in this kind of dialogue.

In other cases, informal logics use acceptability rather than truth as a criterion for judging premises in contexts in which it is difficult to judge premises as true or false. In such cases, acceptable premises may be plausible (or exploratory) hypotheses, claims that can only be said to be generally accepted or assumed, or ethical or aesthetic judgments which are not easily categorized as true or false.

In still other circumstances, truth may be required for acceptability, but only one of a number of conditions that must be satisfied. Even when a premise is true, it may be unacceptable because it violates the rules of interaction that govern the dialogue in which it is embedded. In a legal proceeding or a formal hearing, premises and arguments must not entertain premises that violate rules of procedure.

In situations in which arguments are attempts to convince a specific audience of a conclusion, an acceptable premise may need to be true, but also acceptable to the members of this audience. As Aristotle suggests in the Rhetoric , successful arguments may need to have premises that are in keeping with the pathos of an audience (and do so in a way that does not undermine the character — the ethos — of the arguer). As Gilbert 1997, 2014 has emphasized, there are many real life circumstances in which the emotional acceptability of a premise is required for argument success.

One of the first VR productions by the New York Times was “Kiya,” a production which recreated an incidence of domestic abuse in an attempt to provide support for the importance of attempts to address issues of domestic violence. Its producer describes it as an attempt to use “the immersive power of virtual reality: its ability to generate intense empathy on the part of the viewer; to wring from the audience the intense emotional connection that these stories deserve” (NYT, Jan 21, 2016).

Inference Validity

In its attempt to account for a broad range of real life arguing, informal logics have expanded traditional notions of premise acceptability. Something similar has happened in the case of inference validity. The end result is an expansion of both sides of the AV criteria for good argument.

In the case of inference validity, this expansion has been accomplished by treating deductive validity as one variant of validity, and by recognizing other “defeasible,” non-deductive ways in which premises may entail conclusions. Govier 1987 dubs the deductive/inductive distinction as “the great divide,” emphasizing the latter over the former. Sometimes informal logic systems understand inductive arguments narrowly, as inductive generalizations. Sometimes more broadly, as arguments which have premises that imply that a conclusion is (only) probable or plausible, leaving open the possibility that it is false.

“Conductive” arguments support their conclusions by accumulating non-decisive reasons in their favor. They are valid when they collect enough reasons to warrant their conclusions. In a particular case, different elements of evidence may suggest but not prove that someone charged with murder is guilty but make the conclusion likely if enough evidence of this sort is accumulated (a witness claims he pulled the trigger, the ballistics report shows that the bullet came from a gun he owned, he wrote an e-mail saying he would “get” the victim, etc.).

Strong “abductive” arguments are convincing instances of “inference to the best explanation” (see Harman 1965). They recognize some facts, point out that they are entailed by some hypothesis, and conclude that the hypothesis is true. Taken at face value, abductive arguments appear to be instances of the deductive fallacy “affirming the consequent,” but play an important role to play in medical, scientific and legal inquiry (see Walton 2004).

4.2 Fallacy Theory

AV criteria are in many ways an extension of the notion of good argument enshrined in classical logic. In the search for ways to deal with real life arguments, some informal logicians have moved in a different direction, reviving fallacy theory as an alternative. Hamblin 1970 has become a touchstone for moves in this direction.

Systems of informal logic that rely on fallacies test arguments by asking whether their proponents are guilty of fallacious reasoning. While there is no agreed-upon taxonomy of fallacies, many canonical fallacies have been emphasized in the analysis of informal arguments. They include formal fallacies like affirming the consequent and denying the antecedent; and informal fallacies like ad hominem (“against the person”), slippery slope, ad baculum (“appeal to threat or force”), ad misericordiam (“appeal to pity”), “hasty generalization,” and “two wrongs” reasoning (as in “two wrongs don’t make a right”). The systems of informal logic taught in textbooks often add specialized variants of the standard fallacies (“misleading vividness” designates the misuse of vivid anecdotal evidence in hasty generalizations, and so on.)

Woods and Walton 1982 and Hansen and Pinto 1995 contain detailed discussions of the definition, analysis and assessment of fallacies. In argumentation theory, van Eemeren and Grootendorst 1992 propose a “pragma-dialectical” account of fallacies which treats them as violations of rules that govern critical discussion -- dialogues that attempt to resolve a difference of opinion. Battersby and Bailin 2011 view fallacies as patterns of argument patterns “whose persuasive power greatly exceeds its probative [i.e. evidential] value,” making fallacies errors in reasoning that ordinary arguers are attracted to in view of their rhetorical appeal.

Some fallacies — e.g., equivocation and begging the question (i.e. circular reasoning) — highlight important issues that frequently interfere with real life arguing, but fallacy theory has been criticized when it is adopted as a general account of argument. The issues this raises include its unsystematic nature, disagreements about the definition and nature of specific fallacies, and the emphasis that fallacy theory places on faulty reasoning rather than good argument. Hitchcock (1995, 324) writes that the idea that we should teach reasoning by fallacies is “like saying that the best way to teach somebody to play tennis without making the common mistakes is to demonstrate these faults in action and get him to label and respond to them” (see Feldman 2009).

The theoretical issues raised by fallacy theory are compounded by instances of traditional fallacies which have a reasonable role to play in real life arguing. Appeals to pity and other appeals to emotion have, to take one example, a legitimate role to play in moral, political and aesthetic debate. The following examples highlight other circumstances in which arguments which fit the definition of a traditional fallacy cannot be so readily dismissed.

EXAMPLE 12: A remark from a Danish television debate over the question whether the Danish church should be separated from the Danish state (Jorgensen 1995, 369): “You should not listen to my opponent. He wants to sever the Danish church from the state for his own personal sake.” This seems a classic example of ad hominem , Kahane 1995 (p. 65), describing it as a fallacy that occurs when an arguer attacks “his opponent rather than his opponent’s evidence and arguments.” But this is an accusation of conflict of interest which cannot be dismissed out of hand. If there is reason to believe that an arguer favors a point of view because they have something to gain from it (say, the purchase of a company in which they own shares), this does raise questions about the extent to which their arguments should be entertained. EXAMPLE 13: Martin Luther King Jr., influenced by Gandhi, argued that one can justifiably break laws in a struggle for social justice. Such arguments play a central role in the civil rights movement. They cannot be summarily dismissed, though they appear to be clear cases of “two wrongs make a right.” EXAMPLE 14: The argument that “the attempt to use military force to put an end to terrorism is wrong because it will take us down a slippery slope that will end in improper interference in the affairs of independent states” cannot be dismissed because it is an instance of slippery slope reasoning. If it is true that some action will precipitate a chain of consequences that lead down an alleged slippery slope, this is a good reason to question it.

Examples of this sort have forced careful accounts of fallacies to make room for reasonable arguments which share the form of traditional fallacies.

In doing so, it is helpful to distinguish between fallacies which do and do not have non-fallacious instances. Equivocation, post hoc ergo propter hoc , non sequitor and hasty generalization are commonly classified as forms of argument that are inherently mistaken. In contrast, traditional fallacies like ad hominem , two wrongs reasoning, guilt by association, and appeal to pity are patterns of reasoning which can, when they are constructed in the right way, play a legitimate role within real life reasoning (and are, in view of this, sometimes treated as argument schemes rather than fallacies).

4.3 Natural Language Deductivism

Natural Language Deductivism (“NLD”) is an approach to informal reasoning that retains classical logic’s focus on deductive validity (see Groarke 1999, and Govier 1987 , who develops an initial account NLD, but ultimately favors a more radical break from classical logic). It suggests that we should interpret informal arguments as attempts to create deductively valid inferences which can be analyzed and assessed accordingly. In a deductivist system of informal logic, the V in the AV criteria for good arguments is this classical notion of validity.

NLD has frequently been rejected on the basis of the common, but mistaken, notion that deductively valid arguments must have certain conclusions — a misconception that seems founded on deductive validity’s historical connections to formal logic and mathematics (see Groarke 1999). It is true that ordinary arguing rarely satisfies the strict proof procedures they imply, but deductive validity is not restricted to this compass and there are many instances of ordinary argument which are clear examples of deductively valid argument.

In cases of deductive reasoning, the conclusion of an argument need not be certain, but only as certain as the premises, creating ample room for conclusions which are merely likely, plausible, or probable. EXAMPLE 15 is taken from a radio commentary on population growth.

EXAMPLE 15: The population of the world will grow from 6 to 9 billion in the next fifteen years so we will, if we are to provide sufficient food for everyone, need to find a way to provide food for an additional 3 billion people.

In this case, the premise of the argument is not certain, but reasonably thought to be true — because it was (in the commentary) backed by an extrapolation from well established population trends. The deductively valid inference based upon it makes it reasonable to judge the conclusion of the argument true as well, though it is not certain, as all predictions about population growth are, at best, plausible conjectures.

NLD’s plausibility as a general theory of argument turns on its account of arguments which are not prima facie deductive. Faced with arguments of this sort, it preserves the deductivist approach by attributing an implicit premise (essentially, a deductive warrant) to such arguments in a way that deductively connects an argument’s premises to its conclusion. Typically this is an “associated” conditional of the form “If P, then C” where P is the argument’s premises and C is its conclusion.

A 2015 blog by a professional dietician (Dr. Cristina Sutter) criticizes arguments that justify the claim that garcinia cambogia is a miracle weight loss pill by appealing to the authority of the popular television personality “Dr. Oz.” — instances of the reasoning “Dr. Oz Says It, So It Must Be True.” We can standardize the pattern of argument she criticizes as:

EXAMPLE 16: Premise : Dr. Oz says [that gracinia cambogia is a miracle weight loss pill]. Conclusion : This must be true.

If one judges only by the explicit premise in this argument (“Dr. Oz says...”), this is not a deductively valid argument, for it is obvious that the premise could be true and the conclusion false: Dr. Oz could say gracinia cambogia is a miracle weight loss pill and be wrong. That said, anyone using this argument must assume an associated conditional which can be understood as an implicit premise, allowing us to standardize the argument as:

Premise : Dr. Oz says [that gracinia cambogia is a miracle weight loss pill]. Implicit Premise : If Dr. Oz says this, it must be true. Conclusion : This must be true.

So understood, the Dr. Oz argument is deductively valid, but not sound, as it is a valid argument with a problematic implicit premise.

NLD deals with inductive generalizations in a similar way. Consider the following example from a conversation about French men.

EXAMPLE 17: “French men are fastidious about their appearance. I have worked with many and this was what I found.”

In this example, the move from the premise to the conclusion of the argument assumes that the sample of French men the arguer is familiar with are a representative sample of French men. If this is not likely, then the sample does not provide good reasons for concluding that French men are, in general, as fastidious if they are. If we recognize this assumption as an implicit premise when we standardize the argument, then the argument is deductively valid, for a representative sample is a subset of a population that accurately reflects the characteristics of the larger group.

In this way, NLD turns the uncertainty that characterizes inductive generalizations (and other forms of informal validity) into uncertainty that is connected to an associated conditional which warrants the move from an argument’s premises to its conclusion. This does not eliminate such uncertainty, but maintains it as a key consideration in argument evaluation, for it makes the status of this warrant an essential element of premise acceptability.

In favor of NLD, it has been argued that reconstruction of many arguments it proposes is a dialectically useful way to make explicit the key assumptions that arguments depend on, and frees us from the need to distinguish between different kinds of validity in ways that can be problematic when they are applied to real life arguing. The pragma-dialectical account of indirect speech acts (Eemeren and Grootendorst 2002, Groarke 1999) provides a way to reconstruct arguments as deductive arguments when NLD requires it, though Johnson 2000 and Godden 2004 argue that NLD is an artificial theory which forces informal arguments to adhere to an overly restrictive model of inference.

4.4 Argument Schemes

Argument (or “argumentation”) schemes are recurring patterns of reasoning. Once identified they can be used to evaluate an argument which is instance of a scheme, or as a template or recipe arguers can use when they construct an argument which is an instance of a scheme. Walton, Reed and Macagno 2008 provide a compendium of 96 schemes. Wageman has developed a Periodic Table of Arguments which provides a systematized account of basic schemes.

Rules of inference like modus ponens and modus tollens can be understood as deductive schemes. Other schemes commonly used in ordinary arguing include Argument by Sign, Argument by Analogy, Argument by Example, and Slippery Slope Reasoning. Dove 2016 and Groarke 2019 have shown how visual arguments with non-verbal visual elements may be instances of common schemes, and have identified some schemes which are inherently visual.

The most common approach to argument schemes combines a pattern of argument with a set of “critical questions” with which it is associated. The scheme Argument from Authority (“Appeal to Authority,” “Appeal to Expert Opinion”) and the critical questions it raises can be formalized as follows.

A is an authority in domain D. A says that T is true. T is within D. (Therefore) T is true.

Critical questions : 1. How credible is A? 2. Is A an authority in domain D? 3. What did A assert that implies T? 4. Is A someone who can be trusted? 5. Is T consistent with what other experts assert? 6. Is A’s assertion of T based on evidence?

Prakken 2010b understands argument schemes as inference rules, some critical questions ensuring the truth of an argument’s premises, others ensuring that the context of the inference is appropriate.

Another approach to schemes builds the answers to critical questions into a ‘full’ definition of the scheme, treating them as required premises in convincing instances of the scheme. Taking this approach, the scheme Argument from Authority can be defined as follows.

A is a credible authority in the domain D. A asserted X, which implies T. A can be trusted and T is within domain D. T is consistent with what other experts in domain D assert. A’s assertion of T is based on evidence. (Therefore) T is true.

The critical question approach to schemes suggests that a credible argument from authority must include acceptable premises of the form outlined in the definition of the scheme, and is valid if it is backed by answers to the scheme’s critical questions. Taking the ‘full’ approach, a valid argument from authority must include the premises that define it as as a good argument as explicit (or possibly implicit) premises. In both cases, the result is an account of premise acceptability and validity which is tailored to specifically apply to arguments from authority.

Different argument schemes are a refinement of general AV criteria, creating specific criteria which can be applied to different kinds of argument.

When a student essay claims that “we should not stockpile nuclear weapons” because Einstein told us that this would lead “to destruction even more terrible than the present destruction of life” (EXAMPLE 18), this is an appeal to authority which invokes Einstein as an authority. In order for it to be a convincing argument from authority, it would need to fully satisfy the conditions outlined in our definition of the scheme Argument from Authority .

The attempt to satisfy the requirements this implies produces a version of the argument which can be summarized as follows.

EXAMPLE 18 expanded to satisfy the criteria for a convincing argument from authority: 1. Einstein (A) is a credible authority on nuclear weapons (D). 2. Einstein (A) asserted that the stockpiling of nuclear weapons would precipitate “destruction even more terrible than the present destruction of life” (X), which implies that we should not stockpile nuclear weapons (T). 3. Einstein (A) can be trusted and questions about stockpiling nuclear weapons (T) are questions about nuclear weapons (D). 4. The claim that we should not stockpile nuclear weapons (T) is consistent with what other experts on nuclear weapons (D) assert. 5. Einstein’s (A’s) assertion that we should not stockpile nuclear weapons (T) is based on evidence. 6. (Therefore) We should not stockpile nuclear weapons (T).

This attempt to satisfy the conditions for a good instance of argument from authority fails because it produces a number of problematic premises. Other experts disagreed (and continue to disagree) with Einstein’s suggestion that it is a mistake to stockpile nuclear weapons. More fundamentally, the proposed argument is founded on too loose an account of nuclear weapons as a domain of expertise. Einstein is a renowned expert on nuclear physics but this it does not make him an expert on the social and political issues raised by nuclear weapons. This is something that a convincing version of the argument would have to establish.

In some ways, the scheme approach to argument assessment rectifies problems that arise in systems of informal logic that adopt the fallacy approach to argument evaluation. For traditional fallacies which have non-fallacious instances can be understood, not as fallacies, but as argumentation schemes which are legitimate forms of reasoning when they are properly employed. Fallacy definitions can be turned into scheme definitions by identifying a list of critical questions (or required premises) that specify what is required to make the arguments in question valid.

Ad hominem is a case in point, for there are many instances in which criticisms of an arguer (rather than their position) are a reasonable way to cast doubt on their views. We can specify when this is so by listing critical questions that determine whether this is so in a particular case of argument. The basic question that must be asked is whether there is a good reason why the arguer’s views should not be taken seriously — a question that subsumes the more specific questions whether they have repeatedly shown poor judgment, are biased, lack expertise in the area in question, or are for some other reason an arguer who should not be listened to.

Treated in this way, ad hominem is a legitimate scheme of argument, but there are (as there are in the case of all schemes) many situations in which ad hominem arguments are poor instances of the scheme. In this and many other cases, traditional fallacies can be regarded as deviations from an inherently correct scheme of reasoning.

4.5 Testing Systems

Most informal logics combine different ways of evaluating arguments. In this way, T in a particular system tends to be some combined set of tools that can be used in this endeavor. In most cases, T = { F , AV , AS }, where F is some list of fallacies, AV is some kind of AV criteria (say, the standard ARS criteria) and AS is some set of argumentation schemes.

But systems of informal logic can accommodate other, less common criteria for deciding whether arguments are good or bad, strong or weak. A system might, for example, incorporate criteria which are founded on a virtue-based approach to argument (see Virtues and Arguments) , on feminist principles, or on notions derived from rhetoric, theories of communication, or other cognate fields.

Informal logic’s attempt to understand argument as it occurs in a broad range of real life situations continues to evolve in a way that is influenced by the study of real life reasoning that takes place within the broader scope of argumentation theory.

One cognate field of note is Artificial Intelligence (AI). It relies on step by step accounts of informal reasoning in a wide array of contexts. Informal logics provide this in a general way that has influenced the attempt to model argumentation between agents in multi-agent systems which mimic or assist human reasoning. Computational models have been applied to large-scale collections (‘webs’) of inter-connected arguments, and to reasoning about medical decisions, legal issues, chemical properties and other complex systems. Automated argument assistance functions as a computational aid that can assist in the construction of an argument. Verheij 2014 provides an overview of the issues that this raises.

The development in the empirical study of real life reasoning is the study of argument “corpora” — large collections of argument drawn from natural language discourse. In an early study of this sort, Jorgenson, Kock and Rorbech 1991 analyzed a series of 37 one-hour televised debates from Danish public TV. The debates featured well-known public figures arguing for and against policy proposals. A representative audience of 100 voters voted before and after the debate. Their conclusions were compared with standard notions of “proper” and “valid” argumentation. Other studies consider corpora made up of large databases of selected written texts (see, e.g., Goodwin and Cortes 2010, and Mochales and Ieven 2009).

Argument mining is a subfield of data mining, or text mining (and computational linguistics). It uses software and algorithms that automatically process texts looking for argument structures — for premises, conclusions, argumentation schemes, and extended webs of argument. The texts studied include legal documents , on-line debates, product reviews, academic literature, user comments on proposed regulations, newspaper articles and court cases, as well as dialogical domains. ARG-tech , the Centre for Argument Technology, has played a central role in studies of this sort.

Other research relevant to informal logic has highlighted many ways in which the success of real life arguments depends on aspects of argumentation which are not well integrated into standard systems of informal logic. The latter include an arguer’s ability to draw attention to their argument (using “argument flags” that attract the attention of an intended audience); their personal credibility, ethos , or standing; or their ability to situate their argument within a broader context of debate or dialogue. The study of these and other pragmatic, social, dialectical, semiotic and rhetorical features of arguing will probably play a role in the continued development of informal logic.

The expansion of informal logic to account for an ever broader range of argument is evident in discussions of the use of narratives within argument. In Plato’s Republic , Glaucon uses the mythical story of “The Ring of Gyges” to prove that humans are inherently selfish. In this and many other situations, stories of various kinds (accounts of some historical event, biographies, fables, parables, morality plays, etc) are designed to provide support for some conclusion. It has often been said that a novel or some other work of fiction is an argument for socialism, freedom of expression, or some other value.

One can understand the argumentative use of narratives in a variety of ways: as rhetorical embellishment, as a form of argument by analogy, as implicit generalization (the characters in a story functioning as variables within such generalizations), or reasoning that requires the development of unique standards of argument assessment. According to Fisher 1987, argument itself is best understood as narrative. According to Nussbaum 1990, literature is a way to better understand, and argue about, complex moral situations. Within informal logic, Walton, Reed, and Macagno 2008 identify narrative-based schemes of argument while others continue to debate the role that narratives play in ordinary argument (see Govier and Ayer 2013, Olmos 2014, Plumer 2015).

Informal logic’s interest in real life argument has, from the start, been tied to its interest in the teaching of reasoning skills. In view of this, its interests overlap with fields and disciplines that study education and pedagogy — links manifest in its own influence on critical thinking and the philosophy of education (and movements like “Philosophy for Children”).

Some of the educational issues raised by informal logic are manifest in the development of critical thinking tests which attempt to measure argumentation skills. They are used to test the abilities of students or others and, in a self-reflective way, as an empirical way to test the success of attempts to teach informal reasoning.

Critical thinking (or, even more so, creative thinking) skills are not easily assessed using standardized tests which are designed for large scale use, and typically rely on multiple choice question and answers (see Sobocan 2021). In real life contexts, what counts as good arguing (and thinking) is open ended and unpredictable, dialectical, and influenced by pragmatic and contextual considerations which are difficult to incorporate within standard tests. The California Critical Thinking Test reflects the view of critical thinking elaborated in “The Delphi Report,” commissioned by the American Philosophical Association in 1987, a report that focuses on a narrow range of critical thinking skills which tends to oversimplify the competencies required for good informal argument.

Ennis 2013 provides a comprehensive proposal for dealing with the issues raised by critical thinking tests, and with other challenges raised by attempts to teach critical thinking.

The field of informal logic is a recent invention, but one that continues historical attempts to understand and teach others how to argue. In the Western philosophical tradition, it begins with the sophists’ fifth century boast that they could teach others how to be successful arguers. In Aristotle it is manifest in his systematic account of reasoning, which is expressly designed to teach others how to argue well. Within the history of philosophy, one finds numerous other attempts to formulate accounts of argument that can be used to explain, evaluate, and teach real life reasoning.

The practice of philosophy itself assumes (and frequently develops) an account of argument as it assembles evidence for different philosophical perspectives. Systems of informal logic assume, and often depend upon, the resulting views of reason, rationality and what counts as evidence and knowledge. The philosophical issues in play are tied to complex, unsettled epistemological questions about evidence and knowledge.

Mercier and Sperber (2011) argue that reasoning is a practice which has evolved from, and needs to be understood in terms of, the social practice of argumentation. Johnson 2000 pushes in the opposite direction, arguing that a comprehensive account of argument must be built upon a philosophical account of rationality. Goldman 1999 situates knowledge in the social interactions that take place within interpersonal exchange and knowledge institutions, emphasizing informal argument and the constraints which make it a valuable practice. A 2002 volume of Philosophica on Hilary Putnam’s philosophy suggests pragmatism as an epistemology that best fits informal logic as a discipline.

Some aspects of informal logic raise deep questions that have implications for logic and philosophy. One notable feature of informal logic as it is now practiced is a proliferation of different systems of informal logic which approach the analysis and evaluation of informal reasoning in different ways — employing fallacies, AV criteria, argumentation schemes, methods of formal analysis, and other models of good argument. One implication is a broadening of the conditions for argument felicity.

Other issues are raised by informal logic’s recognition that real life arguments frequently employ visual images, non-verbal sounds, and other non-verbal elements, challenging the traditional assumption that arguments are sets of sentences -- or the propositions (“the bearers of truth value”) that sentences refer to. However one understands visual and multimodal arguments, there is no easy way to reduce them to sets of sentences, for there is no precise way to translate what we see, hear, experience, etc. into words.

In its current state of development, informal logic’s connection to philosophy does not lie in its influence on key philosophical disciplines so much as their influence on it. In North America and elsewhere, informal logic is a field in which philosophers apply theories of argument (rationality, knowledge, etc.) to everyday argument. In keeping with this, philosophers continue to be the core contributors to informal logic; philosophy departments in colleges and universities continue to be the core departments that teach the courses that are its pedagogical focus.

The influence of informal logic courses on the enormous numbers of students that enrol in them require that we at least qualify Rescher’s remark (2005, p. 22) that: “The fact is that philosophy has little or no place in American popular (as opposed to academic) culture... philosophy nowadays makes virtually no impact on the wider culture of North America.” The extent to which informal logic’s attempt to broadly instill good reasoning habits within public education and public discussion and debate is difficult to judge.

Though informal logic addresses many issues relevant to core philosophical disciplines (most notably, epistemology and philosophy of mind; evident in the work of Goldman, Crosswhite, Thagard, and others), it has had limited impact on mainstream approaches to their subject matter. Woods 2000 has speculated on the reasons why. In part, informal logic’s position within philosophy reflects the broader fragmentation of philosophy within North America, Rescher 2005 (p. 4) writing that: “The most striking aspect of contemporary American philosophy is its fragmentation. The scale and complexity of the enterprise is such that if one seeks in contemporary American philosophy for a consensus on the problem agenda, let alone for agreement on the substantive issues, then one is predestined to look in vain.”

In this context, it can best be said that informal logic, like applied ethics, has become a standard offering that helps sustain philosophy departments in North America by contributing to what Rescher describes as “The rapid growth of ‘applied philosophy’ ...[that] is a striking structural feature of contemporary North American philosophy.” (p. 9). The goal of applied philosophy is philosophically informed and nuanced reasoning that addresses complex real life situations. Informal logic is one field which has made a valuable contribution to this goal.

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• Reed, C.A. and D.N. Walton, 2002. “ Applications of Argumentation Schemes ” in Hansen, H.V., C. W. Tindale, J.A. Blair, and R.H. Johnson (eds.), Argumentation and Its Applications: Proceedings of the 4th Conference of the Ontario Society for the Study of Argument , pp. 1–12, Windsor: Ontario Society for the Study of Argumentation.
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• Siegel, Harvey, 1988. Educating reason: rationality, critical thinking, and education , New York: Routledge.
• Shelley, Cameron, 2003. “Aspects of visual argument: A study of the ‘March of Progress’,” Informal Logic , 21(2): 92–112.
• Sobocan, Jan (ed.), 2021. Critical Thinking, Education and Assessment: Can Critical Thinking Be Tested? , 2nd ed. Windsor: Windsor Studies in Argument.
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• Walton, Douglas N., 2012. “Story similarity in arguments from analogy,” Informal Logic , 32: 190–221.
• –––, 2007. Dialog Theory for Critical Argumentation , Amsterdam: John Benjamins Publishing Company.
• –––, 2004. Abductive Reasoning , Tuscaloosa, University of Alabama Press.
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• –––, 1989. Informal Logic: A Handbook for Critical Argumentation , New York: Cambridge University Press.
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• Wilson, David Carl, 2020. A Guide to Good Reasoning: Cultivating Intellectual Virtues , 2nd edition, Twin Cities: University of Minesota Libraries.
• Woods, John and Douglas Walton 1982. Argument: The Logic of the Fallacies , Roronto: McGraw-Hill Ryerson.
• Woods, John, 2000. “How Philosophical is Informal Logic? ” Informal Logic , 20(2): 139–167.
• –––, 1995. “Appeal to Force,” in Hansen and Pinto 1995, pp. 240–250.
• Zenker, F. 2013. Bayesian Argumentation: The practical side of probability (Synthese Library 362), Dordrecht: Springer.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• AGORA-net: Argument mapping
• ARGDIAP: A Polish initiative dedicated to the issues of argumentation, dialogue and persuasion
• ARG-tech: Centre for Argument Technology
• Argument and Computation
• Centre for Argument Technology
• The listserv ARGTHRY: Argument Theory
• The Arguer’s Lexicon , A lexicon of the names of informal logicians and other argumentation theorists.
• Argumentation et Analyse du Discours
• Automated Argument Assistance
• Association for Informal Logic and Critical Thinking: AILACT
• Centre for Research in Reasoning, Argumentation & Rhetoric
• Cogency: Journal of Reasoning and Argumentation
• COMMA: Computational Models of Argument
• Fallacies , entry by Bradley Dowden in the Internet Encyclopedia of Philosophy , J. Fieser (ed.)
• Fallacies: The Nizkor Project
• Foundation for Critical Thinking
• Logical Fallacies
• Ontario Society for the Study of Argumentation (OSSA)
• The Periodic Table of Arguments
• Silva Rhetoricae: The Forest of Rhetoric
• Studies in Logic, Grammar and Rhetoric
• Virtues and Arguments: A Bibliography
• Windsor Studies in Argumentation

abduction | Aristotle, General Topics: rhetoric | children, philosophy for | fallacies | feminist philosophy, topics: perspectives on argumentation | logic: and games | logic: classical | logic: inductive | logical consequence | Port Royal Logic | science: and pseudo-science

Acknowledgments

The author would like to acknowledge and thank the members of ARGTHRY, Tony Blair, and the SEP reviewer for many corrections and suggestions.

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Educating Reason: Critical Thinking, Informal logic, and the Philosophy of Education

Informal Logic

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Contemporary anxieties around 'fake news', propaganda and echo chambers, often lead to a simplistic solution of the need to teach 'critical thinking' to students and to the public at large. Broadly this critical thinking means informal logic, drawn from Aristotle, but reduced to a sort of toolkit for detecting fallacies. While logic as mathematics is unproblematic, the uses of logic and reason within Critical Thinking are predominantly to scrutinise and doubt the beliefs of others, accusing them of being irrational, emotional or biased. This chapter conducts a Foucauldian 'archaeology' of Critical Thinking, focusing on how it produces knowledge by reducing language to premises and conclusions, and the vast array of logical errors it proposes. An analysis of the uses of reason and doubt in Descartes is offered, paralleled by Protestantism with Gnostic routes. Moving to the present, the chapter gives a detailed analysis of how contemporary Critical Thinking guides motivate ungenerous critique, heroises the critical thinker as exceptional and gives no knowledge beyond doubt. This interpretation is supported by Argyrou on thought as a gift relation and Horvath's notion of alchemy as deconstructive-atomising beliefs into a series of premises. Amid contemporary anxieties around fake-news, propaganda and ideological manipulation, many intellectuals, academics and politicians have called for better education in reasoning and critical thinking. Philosophers may propound complex ideas about knowledge, epistemology and discourse, but the trend towards a 'post-truth' world must be resisted, it seems. Of course, these trends are hardly new; the twentieth century is awash with critiques of political propaganda and debunking of commercial fantasies. What is perhaps noteworthy is that recent decades, Critical Thinking has become institutionalised within education, at university and beyond-even in the public sphere, where numerous websites offer free and for-profit courses in reasoning. A longer historical perspective reminds us that Socrates and Plato reacted against the rhetorical manipulation of truth by Sophists, again for political persuasion and simply to make money. Furthermore, Aristotle's formalisation of logic in the syllogism provided a definitive technique to test reason. So, in our present predicament, can we rely on reason to solve the present problems of the public sphere? While Aristotle's work has attracted legions of commentators over centuries, and philosophers of logic have amended his work, the principle of the syllogism is easily stated: Simple premises which share some of the same terms may build up into a conclusion: If A is B and all B's are C, then A = C. It has the simple elegance of mathematics, thoroughly

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Why Is Critical Thinking Important? A Survival Guide

Updated: December 7, 2023

Published: April 2, 2020

Why is critical thinking important? The decisions that you make affect your quality of life. And if you want to ensure that you live your best, most successful and happy life, you’re going to want to make conscious choices. That can be done with a simple thing known as critical thinking. Here’s how to improve your critical thinking skills and make decisions that you won’t regret.

What Is Critical Thinking?

You’ve surely heard of critical thinking, but you might not be entirely sure what it really means, and that’s because there are many definitions. For the most part, however, we think of critical thinking as the process of analyzing facts in order to form a judgment. Basically, it’s thinking about thinking.

How Has The Definition Evolved Over Time?

The first time critical thinking was documented is believed to be in the teachings of Socrates , recorded by Plato. But throughout history, the definition has changed.

Today it is best understood by philosophers and psychologists and it’s believed to be a highly complex concept. Some insightful modern-day critical thinking definitions include :

• “Reasonable, reflective thinking that is focused on deciding what to believe or do.”
• “Deciding what’s true and what you should do.”

The Importance Of Critical Thinking

Why is critical thinking important? Good question! Here are a few undeniable reasons why it’s crucial to have these skills.

1. Critical Thinking Is Universal

Critical thinking is a domain-general thinking skill. What does this mean? It means that no matter what path or profession you pursue, these skills will always be relevant and will always be beneficial to your success. They are not specific to any field.

2. Crucial For The Economy

Our future depends on technology, information, and innovation. Critical thinking is needed for our fast-growing economies, to solve problems as quickly and as effectively as possible.

3. Improves Language & Presentation Skills

In order to best express ourselves, we need to know how to think clearly and systematically — meaning practice critical thinking! Critical thinking also means knowing how to break down texts, and in turn, improve our ability to comprehend.

4. Promotes Creativity

By practicing critical thinking, we are allowing ourselves not only to solve problems but also to come up with new and creative ideas to do so. Critical thinking allows us to analyze these ideas and adjust them accordingly.

5. Important For Self-Reflection

Without critical thinking, how can we really live a meaningful life? We need this skill to self-reflect and justify our ways of life and opinions. Critical thinking provides us with the tools to evaluate ourselves in the way that we need to.

6. The Basis Of Science & Democracy

In order to have a democracy and to prove scientific facts, we need critical thinking in the world. Theories must be backed up with knowledge. In order for a society to effectively function, its citizens need to establish opinions about what’s right and wrong (by using critical thinking!).

Benefits Of Critical Thinking

We know that critical thinking is good for society as a whole, but what are some benefits of critical thinking on an individual level? Why is critical thinking important for us?

1. Key For Career Success

Critical thinking is crucial for many career paths. Not just for scientists, but lawyers , doctors, reporters, engineers , accountants, and analysts (among many others) all have to use critical thinking in their positions. In fact, according to the World Economic Forum, critical thinking is one of the most desirable skills to have in the workforce, as it helps analyze information, think outside the box, solve problems with innovative solutions, and plan systematically.

2. Better Decision Making

There’s no doubt about it — critical thinkers make the best choices. Critical thinking helps us deal with everyday problems as they come our way, and very often this thought process is even done subconsciously. It helps us think independently and trust our gut feeling.

3. Can Make You Happier!

While this often goes unnoticed, being in touch with yourself and having a deep understanding of why you think the way you think can really make you happier. Critical thinking can help you better understand yourself, and in turn, help you avoid any kind of negative or limiting beliefs, and focus more on your strengths. Being able to share your thoughts can increase your quality of life.

4. Form Well-Informed Opinions

There is no shortage of information coming at us from all angles. And that’s exactly why we need to use our critical thinking skills and decide for ourselves what to believe. Critical thinking allows us to ensure that our opinions are based on the facts, and help us sort through all that extra noise.

5. Better Citizens

One of the most inspiring critical thinking quotes is by former US president Thomas Jefferson: “An educated citizenry is a vital requisite for our survival as a free people.” What Jefferson is stressing to us here is that critical thinkers make better citizens, as they are able to see the entire picture without getting sucked into biases and propaganda.

6. Improves Relationships

While you may be convinced that being a critical thinker is bound to cause you problems in relationships, this really couldn’t be less true! Being a critical thinker can allow you to better understand the perspective of others, and can help you become more open-minded towards different views.

7. Promotes Curiosity

Critical thinkers are constantly curious about all kinds of things in life, and tend to have a wide range of interests. Critical thinking means constantly asking questions and wanting to know more, about why, what, who, where, when, and everything else that can help them make sense of a situation or concept, never taking anything at face value.

8. Allows For Creativity

Critical thinkers are also highly creative thinkers, and see themselves as limitless when it comes to possibilities. They are constantly looking to take things further, which is crucial in the workforce.

9. Enhances Problem Solving Skills

Those with critical thinking skills tend to solve problems as part of their natural instinct. Critical thinkers are patient and committed to solving the problem, similar to Albert Einstein, one of the best critical thinking examples, who said “It’s not that I’m so smart; it’s just that I stay with problems longer.” Critical thinkers’ enhanced problem-solving skills makes them better at their jobs and better at solving the world’s biggest problems. Like Einstein, they have the potential to literally change the world.

10. An Activity For The Mind

Just like our muscles, in order for them to be strong, our mind also needs to be exercised and challenged. It’s safe to say that critical thinking is almost like an activity for the mind — and it needs to be practiced. Critical thinking encourages the development of many crucial skills such as logical thinking, decision making, and open-mindness.

11. Creates Independence

When we think critically, we think on our own as we trust ourselves more. Critical thinking is key to creating independence, and encouraging students to make their own decisions and form their own opinions.

12. Crucial Life Skill

Critical thinking is crucial not just for learning, but for life overall! Education isn’t just a way to prepare ourselves for life, but it’s pretty much life itself. Learning is a lifelong process that we go through each and every day.

How to Think Critically

Now that you know the benefits of thinking critically, how do you actually do it?

How To Improve Your Critical Thinking

• Define Your Question: When it comes to critical thinking, it’s important to always keep your goal in mind. Know what you’re trying to achieve, and then figure out how to best get there.
• Gather Reliable Information: Make sure that you’re using sources you can trust — biases aside. That’s how a real critical thinker operates!
• Ask The Right Questions: We all know the importance of questions, but be sure that you’re asking the right questions that are going to get you to your answer.
• Look Short & Long Term: When coming up with solutions, think about both the short- and long-term consequences. Both of them are significant in the equation.
• Explore All Sides: There is never just one simple answer, and nothing is black or white. Explore all options and think outside of the box before you come to any conclusions.

How Is Critical Thinking Developed At School?

Critical thinking is developed in nearly everything we do. However, much of this important skill is encouraged to be practiced at school, and rightfully so! Critical thinking goes beyond just thinking clearly — it’s also about thinking for yourself.

When a teacher asks a question in class, students are given the chance to answer for themselves and think critically about what they learned and what they believe to be accurate. When students work in groups and are forced to engage in discussion, this is also a great chance to expand their thinking and use their critical thinking skills.

How Does Critical Thinking Apply To Your Career?

Once you’ve finished school and entered the workforce, your critical thinking journey only expands and grows from here!

Impress Your Employer

Employers value employees who are critical thinkers, ask questions, offer creative ideas, and are always ready to offer innovation against the competition. No matter what your position or role in a company may be, critical thinking will always give you the power to stand out and make a difference.

Careers That Require Critical Thinking

Some of many examples of careers that require critical thinking include:

• Human resources specialist
• Marketing associate
• Business analyst

Truth be told however, it’s probably harder to come up with a professional field that doesn’t require any critical thinking!

Photo by  Oladimeji Ajegbile  from  Pexels

What is someone with critical thinking skills capable of doing.

Someone with critical thinking skills is able to think rationally and clearly about what they should or not believe. They are capable of engaging in their own thoughts, and doing some reflection in order to come to a well-informed conclusion.

A critical thinker understands the connections between ideas, and is able to construct arguments based on facts, as well as find mistakes in reasoning.

The Process Of Critical Thinking

The process of critical thinking is highly systematic.

What Are Your Goals?

Critical thinking starts by defining your goals, and knowing what you are ultimately trying to achieve.

Once you know what you are trying to conclude, you can foresee your solution to the problem and play it out in your head from all perspectives.

What Does The Future Of Critical Thinking Hold?

The future of critical thinking is the equivalent of the future of jobs. In 2020, critical thinking was ranked as the 2nd top skill (following complex problem solving) by the World Economic Forum .

We are dealing with constant unprecedented changes, and what success is today, might not be considered success tomorrow — making critical thinking a key skill for the future workforce.

Why Is Critical Thinking So Important?

Why is critical thinking important? Critical thinking is more than just important! It’s one of the most crucial cognitive skills one can develop.

By practicing well-thought-out thinking, both your thoughts and decisions can make a positive change in your life, on both a professional and personal level. You can hugely improve your life by working on your critical thinking skills as often as you can.

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Student Development in Logical Reasoning: Results of an Intervention Guiding Students Through Different Modes of Visual and Formal Representation

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• Hugo Bronkhorst   ORCID: orcid.org/0000-0002-8181-1299 1 ,
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Due to growing interest in twenty-first-century skills, and critical thinking as a key element, logical reasoning is gaining increasing attention in mathematics curricula in secondary education. In this study, we report on an analysis of video recordings of student discussions in one class of seven students who were taught with a specially designed course in logical reasoning for non-science students (12th graders). During the course of 10 lessons, students worked on a diversity of logical reasoning tasks: both closed tasks where all premises were provided and everyday reasoning tasks with implicit premises. The structure of the course focused on linking different modes of representation (enactive, iconic, and symbolic), based on the model of concreteness fading (Fyfe et al., 2014 ). Results show that students easily link concrete situations to certain iconic referents, such as formal (letter) symbols, but need more practice for others, such as Venn and Euler diagrams. We also show that the link with the symbolic mode, i.e. an interpretation with more general and abstract models, is not that strong. This might be due to the limited time spent on further practice. However, in the transition from concrete to symbolic via the iconic mode, students may take a step back to a visual representation, which shows that working on such links is useful for all students. Overall, we conclude that the model of concreteness fading can support education in logical reasoning. One recommendation is to devote sufficient time to establishing links between different types of referents and representations.

En raison de l’intérêt grandissant porté aux compétences arrimées au 21 e  siècle ainsi que du rôle central qu’y joue la pensée critique, le raisonnement logique gagne sans cesse en importance dans le programme d’enseignement des mathématiques au secondaire. Dans cette étude, nous présentons l’analyse d’enregistrements vidéo de discussions qui ont eu lieu dans une classe de sept élèves à qui l’on a donné une formation spécialement élaborée en raisonnement logique et destinée aux élèves de 12 e  année qui ne sont pas en sciences. Pendant une période s’étendant sur 10 leçons, les élèves ont exécuté diverses tâches de raisonnement logique, incluant des tâches dites « fermées» où toute l’information nécessaire est donnée explicitement et des tâches de tous les jours faisant appel au raisonnement par inférence implicitement. La structure du cours a porté essentiellement sur l’établissement de liens entre différents modes de représentation (énactive, iconique, et symbolique), en se fondant sur le modèle de «  concreteness fading (l’atténuation du concret)» (Fyfe et al., 2014 ). Les résultats indiquent que les élèves établissent facilement des liens entre des situations concrètes et certains référents iconiques tels que des symboles (de lettres) formels, mais qu’ils ont besoin de s’entraîner davantage avec d’autres comme les diagrammes de Venn et d’Euler. Nous démontrons aussi que le lien qui s’établit en mode symbolique c’est-à-dire par la représentation de modèles plus génériques et abstraits n’est pas très marqué. Ceci peut être dû au manque d’entraînement. Toutefois, dans la transition qui s’effectue du concret au symbolique en mode iconique, les élèves peuvent faire un pas en arrière et choisir une représentation visuelle, ce qui démontre que de travailler sur ces liens s’avère utile pour tous les élèves. Notre conclusion générale est que le modèle de «  concreteness fading (l’atténuation du concret)» peut soutenir l’enseignement en ce qui concerne le raisonnement logique. Nous recommandons d’allouer suffisamment de temps à l’établissement de liens entre les différents types de référents et de représentations.

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Introduction

It is generally accepted that the development of twenty-first-century skills is essential for success in work and life, and this should be an important objective in all stages of education. One key element of twenty-first-century skills is critical thinking (Brookhart, 2010 ; P 21 , 2015 ; Vincent-Lancrin et al., 2019 ). In an earlier article, we stressed the importance of logical reasoning for the development of critical thinking skills (Bronkhorst et al., 2020a ). Liu et al. ( 2015 ) even claim that logical reasoning is the “core foundation” (p. 337) of critical thinking. However, one unresolved question is how students with relatively little experience in logical reasoning can be taught to reason logically in various situations and recognise logical fallacies. Based on research about effective instruction in mathematics education, we developed an intervention for non-science students with specific emphasis on the use of visual and formal representations in logical reasoning tasks, such as syllogism tasks, tasks with if–then statements, and argument analysis tasks. In previous work, we found that student development of logical reasoning was supported by the use of visual and formal representations. In this study, we explicitly focus on how students developed the ability to use these representations effectively over the course of the intervention.

Theoretical Background

First, we will elaborate on our definition of logical reasoning, which includes formal and informal reasoning. Formal reasoning is considered to occur within a system of predefined rules and symbols, based on unchanging premises. Valid conclusions are reached if the rules are followed, for example, rules of logic and mathematics (e.g. Schoenfeld, 1991 ; Teig & Scherer, 2016 ). Informal or everyday reasoning is often considered to be reasoning that is expressed in ordinary language and used to construct an argument where the reasoning and conclusions are context-dependent without strict validity (e.g. Bronkhorst et al., 2020a ; Johnson & Blair, 2006 ; Kuhn, 1991 ; Voss et al., 1991 ). Because both formal and informal reasoning are important to accomplish critical thinking, logical reasoning should not be considered synonymous with formal reasoning alone, but needs a broader definition. In this vein, Nunes ( 2012 ) defines logical reasoning as “a form of thinking in which premises and relations between premises are used in a rigorous manner to infer [emphasis added] conclusions that are entailed (or implied) by the premises and the relations” (p. 2066). The book, How People Learn II: Learners, Contexts, and Cultures (National Academies of Sciences, 2018 ), refers to inferential reasoning as “making logical connections between pieces of information in order to organize knowledge for understanding and to drawing conclusions through deductive reasoning, inductive reasoning, and abductive reasoning” (p. 93; based on: Seel, 2012 ). As we intend to emphasise the importance of making connections between information and using formal and informal reasoning, we define logical reasoning as “selecting and interpreting information from a given context, making connections, and verifying and drawing conclusions based on provided and interpreted information and the associated rules and processes” (Bronkhorst et al., 2020a , p. 1676), which we will use in this study.

Perhaps because of the influence of the twenty-first-century skills movement, mathematics curricula from around the world stipulate that apart from developing students’ formal logical reasoning applied within mathematics tasks, mathematics education should foster reasoning that can be applied beyond the classroom (e.g. cTWO, 2012 ; Liu et al., 2015 ; McChesney, 2017 ; National Council of Teachers of Mathematics, 2009 ). In the Netherlands, the domain of “logical reasoning” has recently been introduced into the mathematics curriculum for pre-university non-science students (College voor Toetsen en Examens, 2016 ).

Since we stress that logical reasoning is applied within a diversity of contexts and thus should be used in a variety of tasks in the classroom, we make a distinction between formal reasoning tasks and everyday reasoning tasks, following Galotti ( 1989 , p. 335). The key elements of formal reasoning tasks are that “all premises are provided, problems are self-contained, there is typically one correct answer, [and] it is typically unambiguous when the problem is solved” (Galotti, 1989 , p. 335). For everyday reasoning tasks, the key elements are that “some premises are implicit, and some are not supplied at all, problems are not self-contained, there are typically several possible answers that vary in quality, [and] it is often unclear whether the current ‘best’ solution is good enough” (Galotti, 1989 , p. 335). For the readability of this article, we will refer to formal reasoning tasks as closed tasks . Consider the conclusions in the following two examples as reasoning in closed syllogism tasks:

(1) All A are B . (2) All B are C . (So) All A are C.

(1) All humans are mammals . (2) All mammals are animals . (So) All humans are animals .

Although the examples are presented differently, with formal letter symbols in an abstract model (I) versus concrete objects (II), both conclusions follow logically from the given premises, they are valid, and they are conclusive. An example of an everyday reasoning task is the analysis of the argument in a newspaper article. In such tasks, not all premises are provided; therefore, the reader must make some implicit assumptions and review the most likely outcome.

Concreteness Fading

The model of “concreteness fading” (CF) provides a useful framework (Fyfe et al., 2014 ) to describe the phases of our intervention with a course in logical reasoning. The model is inspired by Bruner’s theory of instruction (Bruner, 1966 ), which distinguishes three stages, with students using different modes of representation that are applied in successive stages of skill learning. In the first stage of learning, the enactive mode, students rely on concrete knowledge and actions to achieve satisfactory outcomes. In the second stage, students start using iconic modes of representation, such as images or graphical representations. In the final stage, the symbolic mode, students use abstract representations, such as symbols and logical propositions used in reasoning with certain rules or laws. The strength of a representation may be different for each individual, depending on their understanding, but Bruner states that every problem situation can always be transformed in a recognisable way for the learner.

Bruner’s model has been translated into the Concrete Representational Abstract (CRA) instruction framework, commonly used in the USA (Butler et al., 2003 ), or the Concrete Pictorial Abstract (CPA) framework as adopted in, for example, Singapore (Kim, 2020 ). Although the description of the different modes in the CRA and CPA frameworks is similar to the stages in CF, we will use the terminology of CF because it explicitly focuses on consecutively establishing the links between the three stages. All stages are equally important, and by fading from the concrete information through the use of various representations, students gradually move to the iconic and symbolic stages in their development.

Fyfe et al. ( 2014 ) emphasise that spending sufficient time on making the connections between the stages should be the strength of the model. Although much research concerning this model focuses on elementary students (e.g. Fyfe et al., 2015 ) and the fact that many textbooks in middle and higher secondary school do not address the sequence correctly (Witzel et al., 2008 , p. 272), there are some studies that show positive effects of CF in comparison with other approaches within mathematics education at other levels (e.g. Kim, 2020 ; McNeil & Fyfe, 2012 ; Ottmar & Landy, 2017 ). The different stages of this model and the focus on successful transitions between the different stages guide the activities in our intervention.

In terms of the teaching of logical reasoning, the enactive mode refers to reasoning in concrete situations that stimulates learners to explore the situations in ordinary language. As soon as unrelated context is removed or formal symbols are introduced, a learner leaves the enactive mode and enters the iconic mode. Using schematic representations may help students to interiorise schemata that can be used in abstract reasoning (Chu et al., 2017 ). These representations are called graphic pictorial models in CF and should not be confused with concrete pictorial drawings (e.g. Hegarty & Kozhevnikov, 1999 ), which are concrete representations aimed at representing an authentic and complete image of a situation with unnecessary details (Chu et al., 2017 ). These are part of the enactive mode and are called “drawings” in this study. The symbolic mode is the most abstract level and refers to, for example, general rules of logic, such as modus ponens and modus tollens .

Following the development represented by the model, students move from the enactive mode to the iconic and, finally, to the symbolic mode. However, students should be capable of translating reasoning used in the symbolic mode back to the concrete world and vice versa. More specifically, while diagrams provide a means for students to apply the rules of logic to everyday situations, students should also learn to link these conclusions to the everyday situations and might even use representations from all three modes in their reasoning. This is visualised in Fig.  1 . The overlapping areas show the possible links (see also Tondevold, 2019 ).

Different modes of representation with links in all directions

Formal and Visual Representations

Prior research among university students indicates that teaching formal reasoning can be beneficial for the development of reasoning skills in general (e.g. Lehman et al., 1988 ; Stenning, 1996 ); however, Stenning ( 2002 ) also acknowledges that not all teaching in formal reasoning and representations is beneficial and that informal methods might be sufficient. Representations taught to students should capture relevant aspects of contexts and leave out irrelevant details to support their thinking (McKendree et al., 2002 ). Hegarty and Kozhevnikov ( 1999 ) conclude that instruction in visual representations “should encourage students to construct spatial representations of the relations between objects in a problem and discourage them from representing irrelevant pictorial details” (p. 688).

Research in secondary education suggests that the use of formal representations improves student reasoning and can be taught (Adey & Shayer, 1993 ; Van Aalten & De Waard, 2001 ). Based on Halpern ( 2014 ) and Van Gelder ( 2005 ), as well as on our own findings (Bronkhorst et al., 2018 , 2020a ), we conjecture that diagrams (such as Venn and Euler diagrams), scheme-based methods, and knowledge of formal logical rules will be highly beneficial for all sorts of reasoning tasks for our target group. The use of such representations is illustrated in Fig.  2 for the syllogism: “(1) All humans are mammals . (2) All mammals are animals . (So) All humans are animals .” On the left, the Euler diagram offers a visual representation of the context provided, while the diagram on the right is more general and even further formalised with the formulas on the right-hand side. The conclusion A ⇒ C can be verified by using the modus ponens (m.p.) rule.

Syllogism schematised on the left, more general on the right

Intervention

The intervention consisted of 10 50-min lessons on logical reasoning. Here, we provide an overview of the intervention with some task examples first, before showing how these lessons are linked to modes of representation in CF. In the design, the first two lessons were devoted to an exploration of reasoning in concrete tasks, mainly short newspaper articles, as an introduction. In the following lessons, students practised creating and working with visual and formal representations in small, mainly closed and meaningful tasks, with specific attention paid to links between the different modes of representation: first from enactive to iconic and later from iconic to symbolic. This was done with all sorts of syllogisms and several if–then claims, with specific attention paid to the students’ own solution methods. Recognising the importance of discourse in mathematics education, opportunities to discuss and justify their methods in pairs, in groups, and as a class were provided (Gravemeijer, 2020 ; Grouws & Cebulla, 2000 ; National Research Council, 1999 ). Figure  3 provides an example. On the left, the syllogism is stated in ordinary language (concrete version). In an earlier task, students were asked to find a general structure for this syllogism, after which letter symbols (first step iconic mode) were introduced. In this task, students were asked to use a visualisation for this syllogism (further exploration of representations in iconic mode) before Venn and Euler diagrams were introduced in the lesson materials. If–then statements, such as “If it rains, the street gets wet”, were used to make the connection between iconic and symbolic modes of representation. Using Euler diagrams (iconic), formal notations with logical symbols (∧, ∨, ⇒, and ¬) were explored to discover the rules of modus ponens ( A ⇒ B. A , so B ) and modus tollens ( A ⇒ B. ¬B , so ¬A ) (symbolic). During the final lessons, students were encouraged to apply and combine the representations of the different modes they had learned in a task such as that shown in Fig.  4 .

Visualising task for a syllogism

Analysis statement newspaper article (based on: Koelewijn, 2016 )

Figure  5 shows the structure of the intervention based on the three modes of representation. After two lessons of explorations in concrete situations, two lessons were aimed at establishing the link between enactive and iconic modes of representation (arrow 1). Subsequently, two lessons aimed at linking iconic and symbolic modes of representation (arrow 2) and enactive and symbolic modes of representation (arrow 3). Afterwards, two lessons offered students opportunities to use and link all three modes (area 4). The last two lessons consisted of further practice.

Structure of intervention

The intervention was developed via two iterative cycles (Van den Akker et al., 2013 ) in collaboration with a group of teachers. After a pilot study and evaluation, adjustments were made, mainly to provide students with sufficient time to develop their own solutions, for discussion in small groups or with the whole class, and for additional practice. During the sessions with the teachers, materials and implementation guidelines were discussed extensively. These guidelines were also provided in a teacher manual. In particular, attention to the links between the different modes of representation of CF and the importance of classroom discussions about the various representations were emphasised during the meetings.

Research Question

In an earlier experimental study, we found a significant increase in the use of visual and formal representations among students in the experimental group but not in the control group (Bronkhorst et al., 2020b ). In the experimental group, the use of Venn and Euler diagrams positively correlated with the scores on closed tasks. We also found that, in the post-test, students from the experimental group used Venn and Euler diagrams much more frequently than symbolic logical rules. In this article, the focus is on student development through the different modes of representation in the classroom. We focus on the way students developed effective use of visual and formal representations over the course of the intervention. In evaluating student development during the different parts of our intervention, our study was guided by the following research question: How do students use and apply visual and formal representations (iconic and symbolic) in logical reasoning tasks?

In this article, we will mainly use the analysis of video recordings from one group of students (12th graders) who were taught with the specially designed course in logical reasoning described in the introduction of this article. From the 10 lessons, the third through the seventh were videotaped by the first author of this article. These five lessons were selected because of the central focus on linking different modes of representations as important in CF.

Participants

The recorded group consisted of seven students from a school in the northern part of the Netherlands. The students were in their last year of pre-university education (12th graders): there were four boys (Adam, Daniel, Liam, and Owen) and three girls (Julia, Nora, and Riley). The small class size is common for this mathematics course because the course with logical reasoning is an elective for non-science students (College voor Toetsen en Examens, 2016 ). Their mathematics teacher has long-standing experience and has taught at this school for 32 years. All students and the teacher agreed to the video and audio recordings. An informed consent release was sent to all participating students and their parents, which was approved by the ethics committee of the authors’ university.

Data Collection

Five lessons of the experimental group were recorded. Classroom discussions were videotaped and interactions between students during work in pairs or groups of three were recorded with voice recorders. For some tasks, student worksheets were collected. After each lesson, the teacher filled out a logbook and rated statements concerning the implementation of the intervention (Likert scale 1–5).

The teacher’s logbook was used to verify whether the lessons were implemented according to plan. We used the video and audio recordings to analyse students’ statements and discussions. The recordings and corresponding transcripts were analysed in Dutch by the first two authors of this article. For this article, selected excerpts of these conversations have been translated into English.

Discussions among students and classroom discourse were analysed qualitatively in an interpretive way (e.g. Cohen et al., 2007 ) and categorised based on the different modes of representation of CF and the links between them, as represented in Figs. 1 and 5 . If students’ answers were concrete, with text in ordinary language or concrete pictorial drawings, it was categorised as reasoning in the enactive mode. Iconic modes of representation were identified by (1) the use or introduction of formal symbols as abstract referents, such as letter symbols, logical symbols, and arrows, but without manipulating them or applying general rules, or (2) the use of schematic diagrams and visual representations such as Venn and Euler diagrams. Figure  2 , for the example discussed above, shows iconic representations: (1) letter symbols A , B , and C to represent humans, mammals, and animals respectively and (2) Euler diagrams. If abstract referents were used in a model to discover structural patterns or to apply general formal rules, the students’ reasoning was categorised as symbolic. Examples are the abstract rules modus ponens and modus tollens , which are shown in the bottom right corner of our example in Fig.  2 .

According to the teacher’s reports in the logbooks, we concluded that the lessons were implemented according to our intentions. The video recordings confirmed that the teacher provided opportunities for the students to work in pairs or groups of three on the tasks (about 50% of the lesson time). The teacher reported high student participation and that different solutions were discussed in student groups and the classroom (about 25% of the lesson time). We observed much more discussion among students in the second half of the intervention. The different phases of the intervention will be described in detail below according to the sequential structure of the intervention, as shown in Fig.  5 .

From Enactive to Iconic Modes of Representation

Below, we describe two activities that aimed at establishing the link between the enactive and iconic modes of representation (see arrow 1 in Fig.  5 ).

Letter Symbols

After some explorations of the meaning of logical reasoning, students were introduced to syllogisms and explored the truth and validity of these short arguments. A typical example of these syllogism tasks was the following:

Premise 1: All humans are mortal.

Premise 2: Socrates is human.

Conclusion: Socrates is mortal.

In an open task, students were asked to find “a structure” for this syllogism individually and to compare their “structure” with others. However, Julia and Riley immediately started discussing this and introduced the symbols P and Q at the beginning of their conversation to abbreviate the premises, later using A and B as well.

[1] Julia: oh, do we have to do something with P , Q , at least that is all I can think of now

[2] Riley: yes, then it is P , Q

[3] Julia: Q , P

[4] Riley: P , so Q

[5] Julia: huh? Wait, why P , Q ?

[6] Riley: because, those are just the things they always use

[7] Julia: no, there are several forms, right?

[8] Riley: I can do that P and Q ...[inaudible]…, it doesn’t matter what you use

[9] Julia: no, I mean the form

[10] Riley: yes, but you might say A , P . A , so P . It doesn’t matter what you say, right? Or am I saying something stupid now?

[11] Julia: P , Q , P are humans then?

[12] Riley: yes, P is humans, and Q is mortal. He is human so he is mortal.

[13] Julia: ah, wow

[14] Riley: right? Or A , B , A , B , you know, you have to decide yourself.

From this transcript, we observe that Julia introduced the letter symbols P and Q (line [1]) and Riley agreed with this (line [2]), linking the concrete situation in the task to iconic representations. In the conversation, Riley made new reasoning steps (even numbered lines), while Julia asked questions or confirmed Riley’s reasoning (odd numbered lines). Riley understood that the letter symbols chosen were arbitrary (lines [8] and [14]) and that concrete meaning (here: humans for P and mortal for Q ) could be assigned to them (line [12]), which shows an initial understanding of the general form of a syllogism with letter symbols as an abstract model. Julia confirmed that she understood the link between the syllogism with letter symbols and the concrete example (line [13]).

Visual Representations

After further practice with letter symbols, students were asked to individually come up with their own visual representations of the syllogism about Socrates and then compare their ideas with their peers (see Fig.  3 ). The goal was that students would not only be able to generalise these syllogisms into a form with letter symbols, as in the previous task, but would also be able to use other forms of iconic representation such as Venn diagrams. Footnote 1

Nora and Daniel completed the first part of the task individually. Figures  6 and 7 show their answers. In the transcript below, Nora’s work is discussed (see Fig.  6 ). She literally tried to visualise the situation presented in the syllogism.

[1] Teacher: you have very different things, did you have a look at each other’s work?

[2] Nora: yes, then it is, I drew some dummies

[3] Daniel: also nice

[4] Nora: also nice. I drew both premises separately, so that those are very clear and I have derived the conclusion from there. So I quite literally translated the premises with the symbols into pictures.

[5] Teacher: yes, okay, […] but then you stay really close to the example, right?

[6] Nora: yes, true, shouldn’t I have done that?

[7] Teacher: the aim was actually, the way of reasoning, so, this is in general, such a syllogism, in fact for A you can take humans, but also other things, how would you visualise that?

[8] Nora: oh

[9] Daniel: yes, you can just do the same thing, but leave out the dummies, you can put

[10] Nora: A ’s there

[11] Daniel: just one A

[12] Nora: that’s basically what I

[13] Daniel: you just do the same instead of drawings

Nora’s visualisation for the Socrates syllogism

Daniel’s visualisation for the Socrates syllogism

Here, we observe that Nora made a drawing to represent the syllogism about Socrates. She visualised the meaning of the words literally in a pictorial drawing (see Fig.  6 ) and used arrows to schematise the implications, as she explained in line [4]. Apart from the arrows, the rest of her drawing was limited to the real situation described in the task and thus an enactive representation. The teacher tried to convince her to link her concrete model to more abstract referents (lines [5] and [7]). Nora thought that she could just replace the dummies by A' s (line [10]), but Daniel stated that one letter A for the whole set was enough (lines [9] and [11]). This suggests that Daniel tried to discover a more general structural pattern.

After this conversation, Nora made a second visualisation (see Fig.  8 ) and the teacher asked her to explain it, but Nora was not able to.

[14] Teacher: please explain it to me, A arrow B

[15] Nora: yes, that is let’s say, a fixed reasoning pattern with all A are B or A is B , C is A , so C is B

[16] Teacher: okay, but the information all or one, is that still important? Or can you just leave it out?

[17] Nora: I don’t really get it right now anymore, so I try something new, these are just variables that you can add or not

Nora’s second and third attempts to visualise the syllogism

The teacher wanted Nora to explain the meaning of the arrows (line [14]) and the difference between “all A ” and a single C (line [16]), because in her structure, both premises look the same. Nora only translated the conjugations of the verb “to be” into an implication arrow (line [15]), and she seemed confused by the meaning of the letter symbols as variables (line [17]). Although she said she was giving it another try (line [17]), she only wrote a question mark behind the 3 (Fig.  8 ).

Nora’s transformations from the concrete situation in this task to a visual representation started with an enactive representation (pictorial drawing), before she tried to link her drawing to an iconic representation. The following transcript shows how Daniel progressed from the use of letter symbols to the use of circles as a visualisation in another iconic representation (see Fig.  7 ).

[18] Daniel: well, I think mine is the most suitable, because it’s just really cool.

[19] Nora: but is it clear? If I have a look at it

[20] Daniel: isn’t it clear to you?

[21] Nora: no

[22] Daniel: why? you clearly see that all A are B , and all C are A , so all C are also B .

[23] Nora: mmm, okay, I understand where you are going to, but if I see all those circles, I wouldn’t say that

We observe that Daniel tried to convince Nora (lines [20] and [22]), but she did not accept Daniel’s visualisation (line [23]). Later, the teacher asked Daniel to further explain his diagram. Daniel: “Well, okay, you do have B , so you have, okay, all A are B , so everything from A is part of B , then you also have C , which is part of A and then, so C is always part of B .” Notable is the use of the phrase “is part of” instead of using “are” as in his earlier explanation (line [22]). This shows that he understood his Venn diagram in a general way, which might help him to make the link to symbolic representations.

Summary: from Enactive to Iconic Modes of Representation

These transcripts show that the tasks stimulated students to link concrete situations with iconic representations. We found that the students linked a concrete representation of a logical reasoning problem to a situation with letter symbols, but when asked for a visualisation, they had different interpretations. It was apparent that Nora knew that formal letter symbols could be used to represent a concrete model, but that she could not yet establish the exact links between the concrete and abstract referents. Daniel’s visualisation showed that students may come up with a Venn diagram as a representation for a concrete situation. Daniel easily changed his vocabulary to words that connected the Venn diagram to an abstract referent, while Nora only acknowledged his use of circles and, at that moment, clearly needed more guidance and practice to link concrete situations to an abstract pictorial model.

Towards Symbolic Modes of Representation

Below, we describe two tasks in which students were encouraged to take steps towards symbolic modes of representation. The first task concerned the relation between if–then statements and Venn diagrams, and aimed at linking iconic and symbolic modes of representation (arrow 2 in Fig.  5 ). The second task concerned similarities between if–then statements, and it intended to link enactive with symbolic representations (arrow 3 in Fig.  5 ).

Linking Iconic and Symbolic Modes of Representation

To explore the relation between if–then statements and a corresponding Venn diagram, students were provided with the following situation taken from a newspaper article about a court case.

A: The baby is poisoned.

B : The baby turns blue.

Statement: If the baby is poisoned, then the baby turns blue.

Students were asked to generalise the concretely formulated if–then statement and to provide a Venn diagram. Liam and Owen translated the statement into “If A , then B ” and discussed this expression with the teacher.

[1] Liam: I have a question, how do you put if-then in a Venn diagram?

[2] Teacher: yes, that’s a tricky one, isn’t it? Owen has something, what did you do?

[3] Owen: I put the B in A

[4] Teacher: the B in A

[5] Owen: if A then certainly B

[6] Teacher: ok, we will have a look at it if we are all done, but are you convinced?

[7] Liam: no, not yet

[8] Owen: me neither, but this seems the most logical to me

Owen put B in A (line [3] and Fig.  9 ), which is incorrect, and expressed that “if A , then certainly B ” (line [5]). The teacher did not agree or disagree but asked Liam if he was convinced by Owen’s explanation (line [6]). Both students expressed their doubts (lines [7] and [8]).

Owen’s Venn diagram

They did not discuss this further in this setting; however, the teacher started a classroom discussion about the connection between if–then statements and Owen’s Venn diagram (see Fig.  9 ), because he saw that other students had drawn similar diagrams.

[13] Teacher: yes, okay, but if A is true then B , so if you are in set A [points to A] then you are also in this set [points to B]

[14] Nora: we should have switched the order

[15] Adam: no, wait a minute, because if the baby is poisoned, then it will turn blue, or should I have done it the other way around, indeed?

[16] Teacher: if he is poisoned, then he will turn blue

[17] Nora: I think we should switch B and A

[18] Adam: no, we must change the order, indeed, ah, rubbish

Here, we observe that when the teacher pointed to the different areas of the Venn diagram (line [13]), this triggered Nora to exclaim that A and B should be switched (line [14]). This was not immediately clear to Adam, so he needed it translated back to the concrete example of the baby before he was convinced (lines [15] and [18]) about the correct positioning of the circles in the Venn diagram.

Linking Enactive and Symbolic Modes of Representation

The students were already introduced to valid and invalid conclusions in if–then statements before they were asked about the similarities in if–then statements. They were provided with the following two statements:

Statement 1 for a set of stones with pictures of animals on one side and astronomical objects on the other side: “If there is a moon on one side, then there is a fish on the other side.”

Statement 2: “If I rob the Dutch national bank, I will be rich.”

Both statements represent concrete scenarios. To make a judgement about their similarities, it would be useful to translate them into a symbolic expression, which Nora did quickly, clearly showing the structural pattern: “If you just translate this to regular symbols, then they both are if A then B .”

Adam and Liam experienced more difficulties understanding why these two if–then statements were similar and mainly reasoned with the concrete information, although Liam shortened the first statement to “Moon = Fish” in his notebook, not visualising the direction of the statement. Adam and Liam had the following conversation, which demonstrates that Adam did not agree with the equals sign.

[1] Liam: is it true, that the moon cannot be combined with another animal?

[2] Adam: I think so, if you say that if there is a moon on one side, then you have fish on the other side, and you say there is a butterfly, then there may still be a moon

[3] Liam: it is still possible, then any astronomical object is possible

[4] Adam: because moon means fish, but fish does not automatically mean moon

Line [4] shows that Adam did not accept the reversibility of the given statement, and thus that Fish on one side does not necessarily imply Moon on the other side. Near the end of their discussion, Liam concluded: “If this statement is true [refers to moon-fish statement], then this [Statement 2] is just like this one.” Later, during the classroom discussion, the teacher wrote the correct expression on the board using an implication arrow “Moon ⇒ Fish”, and he only indicated that the two statements were similar because they both were “just if–then statements” without elaborating on this or verifying the students’ understanding.

Summary: Towards Symbolic Modes of Representation

These transcripts show that the tasks stimulated the students to link referents from the iconic mode with abstract rules from the symbolic mode, but the link between if–then statements and a correct iconic visualisation was not made automatically. We saw that students put the consequent in the antecedent in their Venn diagrams. Only during the classroom discussion and after some guidance by the teacher did one of the students (Nora) recognise the invalid conditions in the diagram. Another student, Adam, needed a translation back to the concrete situation to verify the correctness of the diagram. Moreover, in the second task, Nora quickly used general rules to conclude that the concrete statements were similar, but not all students accepted this and did not use the general form A ⇒ B to derive conclusions for the concrete situations.

Linking Enactive, Iconic, and Symbolic Modes of Representation

In the last phase of the intervention, students were challenged to use their acquired knowledge and establish links between enactive, iconic, and symbolic modes of representation to verify their reasoning both in closed tasks and everyday reasoning tasks about newspaper articles (see number 4 in Fig.  5 ). In this section, we describe the students’ reasoning in a closed if–then task and a task presenting an argument from a newspaper article.

In the closed task, the following arguments were provided (based on: College voor Toetsen en Examens, 2017 ):

(I): “If you are strong, then you go to bed late. You are not strong, so you do not go to bed late.”

We can represent the statement in the first sentence with symbols as follows: S ⇒ L .

(II): (1) “If you are strong, then you go to bed late.”

(2) “If you are weak, then you do not go to bed late.”

First, students were asked to show that the second statement in argument (I) does not follow from the first statement and is in fact an incorrect conclusion. Riley and Nora discussed this and quickly switched to terminology connected to the symbolic mode.

[1] Riley: well, that is not S is not L , isn’t it?

[2] Nora: here it says if A then B , or if S then L

[3] Nora: but, look, you have, let’s say, two of those, this is not modus ponens, but modus tollens, so it should be not B, so not A

[4] Riley: yes indeed

[5] Nora: so it is a fallacy

[6] Riley: yes

[7] Nora: that’s like

[8] Riley: yes exactly

[9] Nora: even if you are not strong, you can still go to bed late

Here, we observe that Riley translated the second proposition and used the letter symbols S and L as provided in the task (line [1]). Nora showed that she could switch easily from Riley’s letter symbols to a general form with A and B (line [2]). Subsequently, Nora applied general rules to support her argument (line [3]) with modus tollens and thus showed why the order was wrong. With that information Nora easily translated that part to the concrete context again (line [9]) and Riley confirmed all her steps.

For the second subtask (II), students were asked if the conclusion, “If you are weak, then you are not strong,” is allowed on the basis of statements (1) and (2). Nora and Riley discussed this first in the concrete context and were not sure how to approach this argument, but then Nora heard another group using a Venn diagram and convinced Riley to use a Venn diagram as well.

[10] Nora: no wait, I want to draw a Venn diagram, I heard those guys doing that, I think that’s quite a good idea!

[11] Riley: oh yes

[12] Nora: look, it would be right! Because then you have if you are strong [A], you go to bed late [B]. If you are weak (so C), you do not go to bed late. Is now actually separate. Conclusion: so if you are weak, you are not strong. If you see it like this, it is possible. [see Fig. 10 ]

[13] Riley: wait a moment, B , A and then

[14] Nora: just say C is separated from that

[15] Riley: C is not B . If C is separated from it, yes then C is also not A. Yes it is.

[16] Nora: yes

Nora’s Venn diagram

Here, we observe that Nora quickly came to the right conclusion (line [12]) by using a Venn diagram, which is an iconic representation, and translated the conclusion back to the concrete context. She considered the space outside B as ¬ B , which is correct, and concluded that C lays in that area. Noteworthy is her use of general letter symbols A , B , and C , which could be used in an abstract model, although Nora and Riley did not reason with logical rules in an abstract model. Footnote 2

We do not have discussion data about the reasoning on the newspaper article tasks. However, five worksheets for one of the subtasks were available, where students had been asked to visualise, schematise, or create a diagram for a paragraph of a newspaper article. The full task is shown in Fig.  4 . Four of the five students used a Venn diagram, but none used letter symbols or logical symbols. One student tried to make a Venn diagram, but crossed it out and made an argument in ordinary language. From these answers, we conclude that they were able to use iconic representations for an everyday reasoning task but did not demonstrate any symbolic representations.

Summary: Linking Enactive, Iconic, and Symbolic Modes of Representation

Based on the closed subtasks, we found that Nora and Riley used abstract rules (symbolic mode) for statements with one step (argument (I) in the first task), but if they had to take two steps (argument (II) in the first task), they used alternatives, such as a Venn diagram (iconic representation), which was sufficient help in many of the tasks. During the classroom discussion, the teacher showed the solution using symbolic expressions, but did not verify whether the students had understood the steps. In the newspaper article task, the students did not use abstract models, but almost all of them used Venn diagrams, perhaps because of the implicit nature of the task, as we will discuss in the next section.

Conclusions and Discussion

In this article, we reported on student development of logical reasoning. The students were guided through the different stages of CF, working mainly in pairs or groups of three on logical reasoning problems. Our study addressed the research question: “How do students use and apply visual and formal representations (iconic and symbolic) in logical reasoning tasks?”.

Our main conclusion is that students were able to establish the link between enactive and iconic modes of representation. They rapidly resorted to the use of letter symbols in concrete tasks. Most often they chose general letter symbols ( A , B , C or P , Q , R ), even before they were introduced in the teaching materials or by the teacher. In later phases, most students directly introduced letter symbols to start their reasoning. Although this was stimulated during the lessons in logical reasoning, this might also be explained by the fact that students are used to doing this for other mathematics topics.

We also showed that before the students were introduced to the use of Venn and Euler diagrams for logical reasoning, some created a correct generalised diagram as a visual representation. However, for other students, more practice or guidance was needed before they saw the merits of Venn diagrams and started using them. Nevertheless, we conclude that after the intervention, the students had all added Venn and Euler diagrams to their reasoning toolboxes. Moreover, we saw that students sometimes preferred a Venn diagram over formal logical rules. This might be explained by the concreteness of the tasks, but may also indicate that the link with symbolic modes of representation is less well developed, as more time was spent on iconic representations than on formal logical rules in the intervention. Additionally, the way and speed in which students made the transition from enactive and iconic representations to symbolic representations differed between students. We saw that taking a step back to the iconic mode can help scaffold their reasoning. In summary, we conclude that visual representations play a major role in solving logical reasoning problems.

However, the use of symbolic modes of representation was not completely absent. We observed that some students used modus tollens , although only in one-step reasoning. The students experienced difficulties when they had to apply more than one rule in a task. This is consistent with the results of the post-test from our earlier study (Bronkhorst et al., 2020b ), in which students rarely used general, abstract expressions and logical rules. One possible explanation for using iconic representations instead of symbolic ones in everyday reasoning tasks is that students consider the former as more useful or appropriate for the tasks. It is possible that the implicit nature of the everyday reasoning tasks caused confusion or doubt among the students (Galotti, 1989 ), which results in the choice of a visual rather than an abstract model to gain a better understanding or overview. More practice time and tasks might lead to a better understanding and acceptance of other representations and support the relevance of abstract tools for everyday reasoning problems (Witzel et al., 2008 , p. 275).

Our results reveal the importance of the interplay between the different modes of representation, as shown in Figs. 1 and 5 . Over time, the link between enactive and iconic modes of representation became stronger and, as mentioned above, this might also be the case for the link with symbolic modes of representation if students are given more time and practice.

Our study was limited to a specific target group of non-science students, for whom correct logical reasoning has societal relevance. In general, our target group was not strong in mathematics and did not like using abstract rules, and this would probably also apply to logical reasoning problems. Febriana et al. ( 2019 ) showed that for elementary school students, whether they reached the iconic or symbolic stage depended on their mathematical ability. We cannot verify this for our target group, but it is recommended that further research compares groups with different mathematical abilities and adjusts the teaching of logical reasoning for both groups.

Our analysis of the interactions between the students indicated that conversations often led them to another representation or better understanding. Although the role of the teacher was not a separate object of research in this study, the way he guided the pair and group work was important for students’ conversations. As we showed, he did this by encouraging students to explain their solutions and to elaborate on them. In classroom discussions, he often tried to show the connection between the different representations, but did not always verify whether the students had understood the symbolic modes of representation. Perhaps more explicit attention to abstract reasoning would result in better understanding and use of the general rules.

Thus far, we have only discussed links and transfer between the different modes of representation. Another interesting question would be whether the students’ use of representations transfers to other contexts. Although we used meaningful everyday reasoning tasks, such as newspaper articles, the students still fulfil tasks within the mathematics classroom and expect that there should be one correct solution, as is common in mathematics exercises (e.g. Jäder et al., 2017 ). While we showed CF’s contribution to student development of logical reasoning in the context of mathematics education, we need more research on the transfer of their logical reasoning abilities to daily life contexts, as an important indicator of their development of twenty-first-century skills (Liu et al., 2015 ).

Recommendations

In the “ Introduction ” section, we stressed the importance of logical reasoning for twenty-first-century learning (P21, 2015 ). CF appears to be a useful framework (Fyfe et al., 2014 ) to develop teaching materials that work gradually from concrete reasoning to more abstraction. The combined approach of students working on their own solutions and discussing them in small groups and at the classroom level shows promise as a way to strengthen the links between the different modes of representation. However, as not all kinds of representations were internalised by students in this study, we not only recommend further research but also that teachers provide more practice time for tasks that improve the link with the symbolic mode. Finally, while the participants in our research were part of a specific group of non-science students, we recommend the teaching of logical reasoning skills to other secondary education students as well, as the fostering of such skills should be a major goal for all students.

We will use the word “Venn” for all Venn and Euler diagrams in the “ Results ” section, because the lesson materials and the students used the term Venn for both.

Using the letter symbols, they could have verified with logical rules in an abstract model that the conclusion is allowed in two steps: C ⇒ ¬B (statement 2), ¬ B⇒ ¬A ( modus tollens , statement 1), so: C⇒ ¬A .

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This work is part of the research programme Doctoral Grant for Teachers with project number 023.007.043, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).

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Bronkhorst, H., Roorda, G., Suhre, C. et al. Student Development in Logical Reasoning: Results of an Intervention Guiding Students Through Different Modes of Visual and Formal Representation. Can. J. Sci. Math. Techn. Educ. 21 , 378–399 (2021). https://doi.org/10.1007/s42330-021-00148-4

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Parents' Guide

Introduction, critical thinking development: ages 5 to 9.

Critical thinking must be built from a solid foundation. Although children aged five to nine are not yet ready to take on complicated reasoning or formulate detailed arguments, parents can still help their children lay a foundation for critical thinking. 

In order to develop high-level critical thinking skills later in life, five- to nine-year-old children must first make progress along four different tracks. This includes developing basic reasoning skills and interests, building self-esteem, learning emotional management skills, and internalizing social norms that value critical thinking. The following sections will discuss the importance of these foundational aspects of critical thinking and offer parents guidance in how to support their young children’s development. 

1. Logic and Critical Thinking

Critical thinking is different from logical thinking. logical thinking is like math: it involves formal reasoning skills that can only be learned later in life. in contrast, critical thinking builds on everyday reasoning. so parents should guide their children’s critical thinking development from a very young age..

Formal logic is an important part of critical thinking, but ultimately critical thinking involves habits and skills going far beyond the domain of logic. Children are able to develop their critical faculties not from logical analysis, but everyday reasoning.

There are three main factors to keep in mind in differentiating logic from the everyday reasoning that underpins critical thinking.

First, logic is not a natural human trait. If logic were natural, we wouldn’t have to learn how to reason, and math wouldn’t be considered so difficult in school. The natural reasoning displayed by children is often founded on sensory experiences and marred by the cognitive biases discussed in the introduction. Consider this example. Someone says: “If it rains, I’ll take my umbrella with me.” And then a moment later adds: “It’s not raining.” What may we conclude? The vast majority of people — including both adults and children old enough to understand the question — will conclude that the person will not take an umbrella. In context, that is a reasonable conclusion to draw. 

Logic is not natural to humans and can only be acquired through learning.

But from a purely logical perspective, it does not follow. The fact that if it does rain, the speaker will take an umbrella implies nothing, strictly speaking, about what will happen in the case that it is not raining. Logic, the cognitive capacity for formal and reliable deduction, is not natural to humans. We can only acquire it through learning—and only at an age when the cognitive system and brain development allow for such learning (between ages 12 and 15).

Second, although logic is not natural, it can be taught with varying degrees of success, according to personality, cognitive profile, and so on. Multiple developmental psychology studies since Piaget have shown that our cognitive system can only become proficient in logical analysis later on, and with the correct training.

Third, if parents train children from ages five to nine to make more or less complex logical deductions, no deep knowledge is acquired. At a young age, the cognitive system does not yet have the capacity to discern logical invariables (i.e., the ability to reproduce a line of reasoning in a variable context). 

This is why we only explain mathematical principles to children when they are 13 to 14 years old. But again parents can encourage the basics of critical thinking at an early age by promoting social factors like self-esteem. 

Logic and Brain Development

Complex reasoning predominantly takes place in the prefrontal cortex and areas of the brain devoted to language. Language development is, of course, closely linked to explicit learning, as well as to implicit stimulation.

But reasoning requires more than just language skills. The prefrontal cortex carries out what are known as executive functions. It controls concentration, planning, decision-making, and many other functions. These allow us to break down complex tasks into a series of simpler tasks. Reasoning requires a strategy that breaks things down. The prefrontal lobe is a cerebral zone that only matures neurologically after the age of 20. 

Logic is neither natural nor easy. Its development requires a comfortable handling of language and the capacity for problem-solving in the prefrontal cortex. Where are we now? Where do we want to go? How can we get there?

Metacognition​

2. everyday reasoning, although their logical reasoning skills are undeveloped, young children can argue and express opinions. parents should encourage them. even though a child’s argument will tend to be based on emotion, the practice can help build a critical perspective and confidence..

Despite the fact that young children may not be able to grasp logical concepts, they still employ everyday forms of reasoning in both their use of language and in problem-solving and decision-making. It is from out of these capacities that critical thinking can begin to develop at this age.

As is readily apparent, communication via language is not logical. Natural language does not conform to a formal logical structure. It is contextual, whether we are talking about comprehension or expression. If someone says: “If I had a knife, I would cut my steak,” most people would understand that having a knife makes it possible to cut the steak. However, in formal logic, the sentence means that if I had a knife, I would be obliged to cut the steak. Logical language is systematic and obligatory. But a child learns to speak and to understand in a pragmatic and contextual, not logical, fashion.

Certain communication problems result from an overly rigid logical rigor, as in the case of people with Asperger’s syndrome, a type of high-functioning autism. Paradoxically, human communication only works because it is not a purely logical linguistic system. This is one of the reasons why automated translation between languages has been a thorn in the side of artificial intelligence experts since the 1970s.

Logical Proof and Factual Proof

Most real-life problems that we have been grappling with since infancy cannot be formally resolved by logical deduction .

Decision-making is based on a complex mix of different elements:

the cognitive processing of a situation and/or argument

intervention, conscious or unconscious, from our memory of similar past experiences, our preferences, and our personality in the broad sense

our emotions

This is how a child can choose between two toys or how an adult chooses between buying and renting an apartment. People with ultra-logical cognitive tendencies won’t have enough factors for their reasoning to work with, and may be incapable of making a decision—and therefore, incapable of taking action. Neurological studies, since those undertaken by Antonio Damasio in the 1990s, have shown us that decision-making processes and emotional processes are intimately linked , from both neurophysiological and behavioral perspectives.

Pure logic, besides often producing unfortunate results in the real world, can be a hindrance in a highly complicated universe where decisions require managing multiple factors. This is the main reason why artificial intelligence is only now starting to see results, despite the fact that information technology has been in use since the 1940s.

Computer engineers have needed to overcome their grounding in logical, mathematical, and hypothetical deduction, and to incorporate developments in cognitive science and neurology. Algorithms now operate more like children. That is to say, they make random decisions, analyze and memorize the outcomes in order to progress, and then correct themselves by discerning both the invariables and the contextual variables. This is called deep learning. 

Children cannot rely too heavily on logic, but they are still able to express opinions based on their experiences, intuitions, and emotions.

This is also how children between five and nine years old operate. They solve many problems and make many choices, without being able to demonstrate (in the purest sense of the word) why their conclusions and choices were correct.

Between the ages of five and nine, therefore, children cannot rely too heavily on logic. However, they are still able to to express opinions based on their experiences, intuitions, and emotions. To do this, they need to practice, have good self-esteem, and feel esteemed by others in order to believe they have the right, the desire, and the energy to put their critical thinking to use. In other words, they need to exist as a thinking and acting subject whose capacities are recognized by others.

At this age, children are able to argue based on things they have experienced and knowledge they have acquired at school or at home, from books, television, or the internet, or by talking with their friends. They are also able to argue with their “heart.” They assume that their emotions are arguments themselves. 

For example, a child might consider that we shouldn’t eat meat because innocent animals shouldn’t have to die. The child’s empathy is the crux of their argument and the strength of their insistence will often be proportional to that of their emotions.

Case in Point

We show children from this age group a drawing of a rectangular flask tipped at an angle, and we ask them: “If I fill this flask roughly halfway, could you draw the water line on the flask?” 

What would be the result? Most children will draw a line perpendicular to the flask’s longitudinal axis. Yet, since this axis does not run vertically but is at an angle, the line the child draws is not horizontal relative to the ground, as it should be. 

Children err here because their minds are referentially anchored to the flask, just as astronomers for many millennia fixated on the idea of the earth, and later the sun, as a reference point—before realizing that the universe does not have an absolute reference point.

Even if we explain the error to children—and they say they understand—many will, shortly afterwards, make the same mistake again. Their cognitive system is not mature enough to incorporate the logic behind reference and relativity. The example shows how logical thinking is not natural. It requires a learned ability to step back and remove oneself from immediate engagement with a particular situation. 

3. Preparing Kids to Think Critically

Parents or guardians can foster critical thinking skills in children from an early age. First, it’s important to understand the basics of how children learn to think and how a child’s mind differs from that of an adult. Critical thinking in their early years prepares children for life’s challenges and allows them to live a productive life.

How to teach critical thinking to your child

Here are four ways you can support your child’s early cognitive development and put them on the path to becoming critical thinkers. Teaching critical thinking may seem daunting, but having a primer on the particular needs of a child can help you better approach this important task.

1. Encourage children not to see everything as centered only on them by involving them in discussions on an array of topics, including current affairs.

Contrary to popular belief, from the age of five—and sometimes even earlier—children like to be involved in discussions, provided they are not drowned in technical vocabulary or formal logic . They also need to feel that adults are interested in what they are saying and that they are being listened to. Adults need to learn to step away from the role of educator and engage children at their level.

It is highly important for the development of critical faculties that children see their thoughts on the world are accepted. By taking those thoughts seriously, we are taking our children seriously and accepting them.

For example, ask five-year-old children whether Santa Claus exists and how they know. Listen to their arguments: they saw Santa at the mall; they know their Christmas presents must come from somewhere. Contradicting them or breaking down their worldview would be a grave mistake. It would fly in the face of our knowledge about cognitive development, and it would disregard their emotional need for this belief. Paradoxically, we need to let children formulate their own ideas and worldviews, namely through dreaming and imagination. In this way, they will grow happy and confident enough, in time and at their own pace, to move on to more mature ideas.

2. Value the content of what children say.

With encouragement, children will want to express their thoughts increasingly often, quite simply because they find it pleasurable. A certain structure in our brains, the amygdala, memorizes emotions linked to situations we experience. We are predisposed to pursue experiences and situations which induce pleasure, be it sensory or psychological. If a child puts energy into reflection in order to convince us that aliens exist, and we then dismantle their arguments and dreams, we will be inhibiting their desire to participate in this type of discussion again.

For children aged five to nine, the pleasure of thinking something through, of expressing and discussing their thoughts, of feeling language to be a source of joy, are all of far greater importance than argumentative rigor or logical reasoning .

Children debate and give their opinions. This stimulates their brain, which creates a whole host of connections, which, in turn, improve their abilities and their cognitive and emotional performance. The pleasure of discussion, of having someone listen to your ideas, releases a “flood” of neurotransmitters that promote cerebral development. An atmosphere of kindness and benevolence in which the child feels heard produces neural connections and develops various kinds of intelligence. As the child learns through debate, putting effort into reflective thought and into verbal and bodily expression, the brain evolves and invests in the future. This results from cognitive stimulation paired with  joie de vivre  that comes from being heard by others and receiving their undivided attention.

Parents should not hold back from bringing children into discussions and debates.

3. gradually, the ability to argue with pertinence, on both familiar topics of reflection or debate and new ones, will increase..

Numerous recent studies show that doing well in school results more so from pleasure and the development of self-esteem than heavy exposure to graded exercises, which can create anxiety and belittle children. Children are vulnerable and quickly internalize the labels others place on them.

In short, parents should not hold back from bringing children into discussions and debates, keeping to the principles outlined above. Also, be sure to respond to their desire to start discussions within their frame of reference and be sure to take them seriously.

4. Gradually, with time, pleasure, learning, and cognitive and emotional development, it will be possible to encourage children to argue without pressuring them through open-ended questions.

From the age of eight, children can start learning about  metacognition and the adoption of alternative points of view. They should also be trained at this time to understand the difference between an opinion, an argument, and a piece of evidence.

An opinion is the expression of an idea that is not, in and of itself, true or false. Children are empowered to express their opinions early on by all the preliminary work on building up self-esteem. “I think they should close down all the schools, so we can be on holiday all the time” is an opinion. A child of five can easily express such an opinion.

An argument is an attempt to convince others by offering information and reasoning. A child of eight might argue: “If we close down all the schools, we can get up later. Then we’ll have more energy to learn things better at home.”

Evidence are the facts we use to try to prove a point in an argument. Evidence can be highly powerful but it rarely amounts to conclusive proof. When an unambiguous proof is presented, alternative opinions evaporate, provided that one can cognitively and emotionally assimilate the perspective of the person presenting the proof. Something can be proven in two ways. On the one hand, it can be proven through formal reasoning—attainable from the age of nine upwards in real-life situations and, later on, in l more abstract situations. On the other hand, it can be established through factual demonstration. If a child claims that “you can scare away a mean dog by running after it,” proof can be given through demonstration. This leaves no need for argument.

From ages eight to nine, children can come to differentiate and prioritize opinion, argument, and evidence in what they say and hear, provided that their own flawed arguments at age five to six were met with respect and tolerance. This is vital for developing children’s self-esteem and respect for others. It enables them to take pleasure in argument and increases their desire to express themselves more persuasively.

Critical thinking exercises for kids

Hunting—for or against? For a debate like this one, with considerable social implications, focus on these concepts:

1. Teach children to distinguish between:

An opinion : I am against hunting…

An argument : … because it entails animal suffering and human deaths.

 Hunting significantly increases the production of stress hormones (such as hydrocortisone) in hunted animals.

There are around thousands of hunting accidents each year.

2. Teach children to adopt a counter-argument for practice:

An opinion : I am in favor of hunting…

An argument : … because it allows us to control the size of animal populations.

Evidence : Wild boar populations are high and cause a great deal of damage to farmland.

New Perspectives and Overcoming Biases

4. the importance of self-esteem, children need self-esteem to think themselves worthy of expressing their opinions. parents can strengthen their children’s self-esteem by encouraging them to try new things, stimulating their curiosity, and showing pride in their accomplishments., understanding the importance of self-esteem, the foundation of critical thinking.

Before children can learn to analyze and criticize complicated material or controversial opinions, they need to have a strong sense of themselves. Their capacity to question external sources of information depends on feelings of self-worth and security.

The terms “self-confidence” and “self-esteem” are often used interchangeably. There is, however, a difference between the two, even if they are related. Before we can have high self-esteem, we must first have self-confidence. The feeling of confidence is a result of a belief in our ability to succeed. 

Self-esteem rests on our conscious self-worth, despite our foibles and failures. It’s knowing how to recognize our strengths and our limitations and, therefore, having a realistic outlook on ourselves.

Self-esteem requires an ability to recognize our strengths and weaknesses, and to accept them as they are.

For example, children can have high self-esteem even if they know that they struggle with math. Self-esteem can also vary depending on context. Children in school can have high social self-esteem, but a lower academic self-esteem.

Self-esteem requires an ability to recognize our strengths and weaknesses, and to accept them as they are. Children must learn to understand that they have value, even if they can’t do everything perfectly.

Self-esteem starts developing in childhood. Very young children adopt a style of behavior that reflects their self-image. From the age of five, healthy self-esteem is particularly important when it comes to dealing with the numerous challenges they face. Children must, among other things, gradually become more independent, and learn how to read, write, and do mental arithmetic. This period is key, and children need self-confidence as well. More than anywhere else, it is in the family home that children develop the foundations for self-esteem.

Children with high self-esteem:

have an accurate conception of who they are and neither over- nor underestimate their abilities;

make choices;

express their needs, feelings, ideas, and preferences;

are optimistic about the future;

dare to take risks and accept mistakes;

keep up their motivation to learn and to progress;

maintain healthy relationships with others;

trust their own thoughts and trust others.

As parents, developing our own self-esteem enhances the development of our children’s self-esteem, as their identity is closely entwined with our own. Our children learn a great deal by imitating us. Modeling self-esteem can therefore be a great help to them. Here are some examples of what we can do:

Be openly proud of our accomplishments, even those which seem minor to us.

Engage in activities just for fun (and not for competitive reasons).

Don’t pay too much heed to other people’s opinions about us.

Don’t belittle ourselves: if we’ve made an error or if we aren’t so good at a certain task, explain to children that we are going to start again and learn to do it better.

At mealtimes, prompt everyone around the table to say something they did well that day.

On a big sheet of paper, write down the names of family members; then, write down next to everyone’s name some of their strengths.

5. Promoting Self-Esteem

To promote healthy self-esteem in children, parents must strike a balance between discipline and encouragement., the most important thing of all in the development of young children’s self-esteem is our unconditional love for them..

Children must feel and understand that our love will never be dependent on their actions, their successes, or their failures. It is this state of mind that allows them to embrace the unknown and to continue to progress despite the inevitable failures that come along with learning new skills.

Developing Self-Esteem

But be careful not to let unconditional love prevent the imposition of authority or limits. Instead of developing their self-esteem, the absence of limits promotes the feeling in children that they can do no wrong and renders them incapable of dealing with frustration. It is necessary to establish limits and to be firm (without being judgmental). The desired result is only reached if effort and respect are taken seriously.

Self-esteem means loving ourselves for who we are, for our strengths and our weaknesses, and it is based on having been loved this way since birth.

Advice: How to promote the development of a child’s self-esteem

As parents, we have a big influence on our children, particularly when they are young. Here are some ways to help build up children’s self-esteem:

Praise children’s efforts and successes. Note that effort is always more important than results. 

Don’t hesitate to reiterate to children that error and failure are not the same thing. Show them that you’re proud of them, even when they make mistakes. Reflect with them on how to do better next time.

Let children complete household chores; give them a few responsibilities they can handle. They will feel useful and proud.

Show children that we love them for who they are, unconditionally, and not for what they do or how they look.

Let children express their emotions and inner thoughts.

Assist children in finding out who they are. Help them to recognize what they like and where their strengths lie.

Encourage them to make decisions. For example, let them choose their own outfits.

Invite them to address common challenges (according to their abilities and age).

Pitfalls to avoid

Avoid being overprotective. Not only does this prevent children from learning, it also sends them a negative message: that they are incapable and unworthy of trust.

Don’t criticize them incessantly. If we’re always making negative comments about our children, and if we show ourselves to be unsatisfied with their work or behavior even when they’re doing their best, they will get disheartened. 

If children don’t act appropriately, stress that it is their behavior, rather than their personality, that must change. For example, it is better to explain that an action they may have done is mean, rather than that they are themselves mean.

Always be respectful towards children. Never belittle them. What we say to our children has a great impact on their self-image. 

Show them we’re interested in what they’re doing. Don’t ignore them. We are still at the center of their universe. 

Don’t compare them to their siblings or to other children their age. (“Your four-year-old sister can do it!”) Highlight how they are progressing without comparing them to anyone else.

Risk-Taking

6. the role of emotions, emotions are an important part of children’s cognitive development, but if emotions become overwhelming they can be counterproductive. parents should help their children learn how to express their feelings calmly and prevent emotions from becoming a distraction., understanding the role of emotions  in the development of critical thinking.

Young children may develop skills in language and argument, and benefit from a level of self-esteem allowing them to stand their ground and explore the unknown. Nonetheless, the development of their critical faculties will still be limited if they haven’t learned how to manage their emotions.

Emotions appear in a part of the brain called the limbic system , which is very old in terms of human evolution. This system develops automatically at a very early stage. But very quickly, children experience the need to rein in the spontaneous and unrestricted expression of their emotions. These emotions are, of course, closely connected to basic relations to others (and initially most often to one’s parents) and to cultural norms. 

The prefrontal lobe contains the greatest number of neural networks that simultaneously regulate the scope of conscious emotions and their expression in verbal and non-verbal language, as well as in behavior. From the age of five or six, children start their first year of primary school, where they are forced to sit for hours on end each day. They must also listen to a curriculum designed more around societal needs and expectations, rather than around the desires and emotions of children. Frontal lobe development enables the inhibition of urges and the management of emotions , two prerequisites for intellectual learning and for feelings of belonging in family and society.

The ability to manage emotions has a two-fold constructive impact on the development of children’s critical faculties. First, it enables children to override their emotions, so they may focus their attention and concentrate. This is essential for both cognitive development in general and their argumentative, logical, and critical skills.

Management of emotions also allows us to feel settled and to convince and influence others when we speak. Paradoxically, children learn that, by managing their emotions (which is initially experienced as repression), they can have an impact on their peers, make themselves understood, and even be emulated. The pleasure they derive from this reinforces the balance between spontaneity and control, and both pleasure in self-expression and respect for others will increase. Self-esteem will therefore progress, also allowing the child to assert his or her will. 

Development of the critical faculties will benefit from a heightened level of self-esteem. But it’s important to remember that this is a balancing act.

If family or social pressures excessively inhibit emotional expression, feelings of uniqueness and self-worth are compromised. In this case, even with otherwise normal (and even excellent) cognitive development, children’s critical faculties can be impeded. A child won’t truly become an individual and the development of his or her critical faculties will therefore be stunted. Such a child is like a mere cell, rather than a whole organ. This lack of individuality is found in the social conventions and education systems established by totalitarian regimes. Highly intelligent, cultured, logical people can, under such regimes, remain devoid of critical thinking skills.

Emotion is the psychological motor of cognition. But in high and uncontrolled doses, emotion can override cognition.

Conversely, if children’s emotions and expressions of emotion are badly managed or not curtailed at all, they will come to see themselves as almost omnipotent. The consequent behavior will be mistaken for high self-esteem . In reality, cognitive and intellectual development will be dampened due to a lower attention span caused by poor emotional management. Logical and argumentative skills will be less developed and what may appear to be “critical” thinking will, in fact, be nothing more than a systematic, unthinking opposition to everything. 

Critical thinking without cognitive and intellectual development does not truly exist. Real, constructive critical thinking requires listening, attention, concentration, and the organization of one’s thoughts. The development of these faculties itself requires good emotional management, which must intensify from around the age of five or six, in order to strengthen learning skills and social life. Above all, parents should not try to snuff out a child’s emotions. Emotions are what give children vital energy, the desire to learn, and the strength to exercise self-control. Emotion is the psychological motor of cognition. But in high and uncontrolled doses, emotion can override cognition.

7. Managing Emotions

Parents should not ignore or simply silence their children when they act out or are overcome with emotion. they should work with them on strategies for coping and discuss how they can more calmly and productively express their emotions., how to help our children to control their emotions.

Our emotions are a part of who we are: we have to learn to manage and accept them. In order to help children manage their emotions, we must set limits (for example, by forbidding them to waste food or lie). However, setting limits on their behavior does not mean setting limits on their feelings.

We cannot stop children from getting angry even if they are forbidden from acting on that anger rather we can coach children in controlling their reactions. Sending them to their rooms to calm down will not prevent them from being upset and frustrated. On the contrary, by conveying to them the idea that they must face their emotions alone, we encourage them to repress their feelings. When children repress their emotions, they can no longer manage them consciously, which means they are liable to resurface at any moment.

An angry child is not a bad person, but a hurt person. When children lose control over their emotions, it is because they are overwhelmed.

These outbursts, when our children seem to have totally lost control of themselves, can frighten us as parents. Indeed, if children habitually repress their emotions, they become unable to express them verbally and rage takes over.

Failing to acknowledge children’s emotions can prevent them from learning to exercise self-control.

Advice: How do children learn to manage their emotions?

Children learn from us. When we yell, they learn to yell. When we speak respectfully, they learn to speak respectfully. Likewise, every time we manage to control our emotions in front of our children, they learn how to regulate their own emotions.

To help children manage their emotions, we should explicitly explain how to do so and discuss it with them.

Even older children need to feel a connection with their parents to manage their emotions. When we notice our children having difficulties controlling their emotions, it is important to reconnect with them. When children feel cared for and important, they become more cooperative and their feelings of joy cancel out bad behavioral traits.

The best way to help children become autonomous is to trust them and to entrust them with tasks and little challenges.

An angry child is not a bad person, but a hurt person. When children lose control over their emotions, it is because they are overwhelmed. Controlling their emotions is beyond their capacities at that particular moment in time and emotional control is something that they’ll build gradually as they mature.

If we continue treating them with compassion, our children will feel safe enough to express their emotions. If we help them to cry and let out their emotions, these feelings of being overwhelmed will go away, along with their anger and aggression.

Is it important to teach children specific language for expressing emotions?

Of course it is! But don’t try to force children to voice their emotions. Instead, focus on accepting their emotions. This will teach them that:

There is nothing wrong with emotions—they enrich human life.

Even if we can’t control everything in life, we can still choose how we react and respond. 

When we are comfortable with our emotions, we feel them deeply, and then they pass. This gives us the sensation of letting go and of releasing tension.

If we actively teach these lessons—and continue to work on resolving our own emotions—we will be happy to find that our children will learn to manage their feelings. It will eventually become second nature to them.

Emotional Management

8. critical thinking and social life, critical thinking is a positive social norm, but it requires the support of background knowledge and genuine reasoning skills. without them, critical thinking can become an illusion..

Parents should balance their encouragement of children’s argumentative skills and self-expression with an emphasis on intellectual rigor.

Taking account of social norms and peer groups

No child grows up in a vacuum. As they develop, children internalize many of the norms and ways of thinking that are dominant in their families, social lives, schools, and society more broadly. Parents should be aware of the positive and negative influences these different spheres can have on their children. They should know what they can do to expose their children to norms that will foster healthy and independent thinking.

It seems that the right, even the responsibility, to think for oneself and to exercise one’s critical faculties has become increasingly tied to notions of dignity and individuality. More and more we see factors that have historically determined who has the “right” to be critical—age, origin, gender, level of general knowledge, or other implicit hierarchies—fade in importance. 

Thus, it is becoming more and more common for students (with disconcerting self-assurance) to correct their teachers on aspects of history or other issues that are matters of fact. This raises some important questions, notably regarding the role of the educator, the goals of education, and the relationships between generations. 

Our society encourages critical thinking from a very early age. We have insisted on the fact that, for young children, although intellectual rigor is difficult to attain, it is crucial to develop self-esteem and self-affirmation. But we have also seen that from around the age of eight, it is necessary to move towards teaching them basic reasoning skills.

The risk of making the “right to critical thinking” a social norm from a young age is that we lower intellectual standards. If the encouragement of children to think critically is not paired with intellectual progress in other areas, critical thinking is rendered a mere simulation of free thought and expression. This is as true for children as it is for teenagers or adults.

The entire population may feel truly free and have high self-esteem. However, if the intellectual rigor that comes with arguing, debating, and reasoning, is missing from children’s intellectual and social education, the people will be easily manipulated. Giving our children the freedom to exercise their critical faculties must be paired with the demand for intellectual rigor and linguistic mastery, without which “critical thinking” would offer the mere illusion of liberty.

Striking a balance:

For parents today, it is a matter of striking a balance between fostering critical thought from an early age, in spite of gaps in knowledge and logic, and developing our children’s cognitive faculties and knowledge base. Without these faculties of listening, attention, comprehension, expression, argument, and deduction, critical thinking is an illusion, a pseudo-democratic farce. This  can lead to a society plagued by ignorance and vulnerable to barbarism.

On the other hand, we cannot simply slip back into old social conventions whereby children were told to simply keep quiet and learn their lessons passively. The only thing this approach ensures is that the child won’t become a troublemaker.

What is needed is an approach that harmonize advances in philosophy and psychology, which consider children as fully fledged individuals, on the one hand, with an understanding of the intellectual immaturity of this child, on the other.

Disagreeing in a civilized manner, in the end, allows us to agree on what matters most.

With the help of an affectionate, attentive, but also sometimes restrictive and guiding parent—who is at once intellectually stimulating, indulgent, and patient with the child’s needs—early development of self-affirmation and critical thinking becomes compatible with growing intellectual aptitude.

This intellectual aptitude is crucial to a healthy social life as well. People lacking this intellectual maturity cannot even disagree with each other productively; they lack the ability to discuss subjects worthy of critical interest, as well as the social and cognitive skills of listening, argument, and logical deduction. Disagreeing in a civilized manner, in the end, allows us to agree on what matters most.

Consider this discussion between two eight year olds.

 – “I saw a show on TV yesterday that proved that aliens really exist. Tons of people have seen them, and they’ve found marks left by flying saucers in the desert!”

– “But there’s no real evidence. Those clues and eyewitness accounts weren’t very specific. Different witnesses described the aliens in very different ways—some said they were little green men, while others said they were big with glowing eyes. And the marks from UFOs could have been formed by strong winds.”

– “Oh, so you think you’re smarter than the scientists on TV, is that it?”

One child declares that a TV show they saw proves the existence of aliens. He or she takes it for granted that what we see on TV is true. The second is educated into a norm that calls claims into question and demands evidence. The first child doesn’t understand the second, because, to him or her, seeing it on TV is proof enough. From this point onward, the discussion can only go in circles. In this case, different social or family norms are incompatible.

Independent Thinking

Case study 1, metacognition.

Already at a young age children can begin to gain perspective on how they reason.  One good way to help them foster this metacognition is by pointing out the variety of different methods available for solving a particular problem. By, for example, seeing the multiple different methods available for solving a math problem, children can begin to think about their own thought processes and evaluate various cognitive strategies. This will gradually open up the world of reasoning to them. They will begin to pay more attention to how they solve problems or complete tasks involving reasoning, instead of focusing only on answering correctly or completing the task. 

How do children calculate 6 x 3, for example? 

There are several ways:

They could add 6 + 6 + 6;

They could recall that 6 x 2 = 12, then add six more to get 18;

They could simply memorize and recall the answer: 18;

They could draw a grid of 6 by 3 units and then count how many boxes are in the grid.

Or they could use one of various other techniques…

Our culture values accurate and precise results but tends to pay little attention to the route taken to arrive at those results. Yet, if children are aware of their train of thought, they will be in a better position to master the technique—to perfect it to the point where they may even decide to switch to another technique if they need to increase their speed, for example. That is why it is important to help children understand the method they are using to the point that they can explain it themselves.

In helping their children with schoolwork or other projects involving reasoning, parents should ask them to explain themselves, make explicit the steps they’re taking to solve a particular problem, and discuss the advantages and disadvantages of their method and alternative methods. The result will be a much deeper understanding not only of the particular task at hand, but also of the practice of reasoning itself.

Case Study 2

Logical proof and factual proof​​.

At this stage, we can begin to introduce rudimentary logical concepts and distinctions. In everyday conversation, children have already begun using what we might call “natural logic.” They may, for example, get in arguments, like the one below, in which they draw conclusions based on premises. When children present these types of arguments, parents can intervene to teach basic logical concepts and ask children how a given conclusion might be proven or disproven. 

One distinction appropriate to teach at this age is that between logical proof (proof that draws logical conclusions from certain premises) and factual proof (proof that uses actual facts to prove or disprove a given statement). The following anecdote provides the opportunity for such a lesson.

William and Eve, two children walking their dog in the park, are having a conversation about Labradors:

— “There are two kinds of Labradors—black and golden,” declares William.

— “That’s not true; there are also chocolate Labradors,” replies Eve. “My friend Adam has one.”

— “Well, his dog must not be a Labrador then,” William says.

How might we interpret this conversation?

In terms of logical proof, if Labradors are either black or golden, Adam’s chocolate “Labrador” cannot be a Labrador. That is a logically formulated proof. The reasoning is valid. It is the basic premise, William’s initial declaration that there are only two kinds of Labradors, that is false. It is, therefore, possible for William to draw a false conclusion even though his logic is technically correct.

In terms of factual proof, if we can prove that the chocolate-colored dog has two Labrador parents, we can factually prove that William’s premise is wrong: there are at least three types of Labrador.

There are many opportunities like this one to begin to make explicit the logical steps involved in everyday conversations with your children and to show them that they are already using logic, even if they may not know it. This serves to get them thinking about their own thinking, and it makes the topics of logic and reasoning less intimidating.

Case Study 3

What is bias.

A bias is a simply a preconceived and unreasoned opinion. Often biases are formed due to upbringing, larger societal biases, or particular subjective experiences. They exist in many forms and can persist into adulthood unless a child builds a firm foundation in critical thinking and reasoning.

How to overcome bias

The following anecdotes demonstrate how parents can use everyday events to help their children better understand and relate to perspectives outside their own. In order to think critically, children must be able to imaginatively and empathetically put themselves outside their own experiences and perspectives. Children thereby begin to come to terms with the limitations their own upbringings and backgrounds necessarily impose on them. 

This is a vital part of metacognition since it allows children to see themselves, their attitudes, and their views as if from the outside. They become better at overcoming biases, prejudices, and errors in thinking. This process also enables them to entertain the perspectives of others and thereby engage in argument and debate in the future with more charity and nuance. Finally, it encourages them to seek out new experiences and perspectives and to develop intellectual curiosity.

In this first anecdote, a child learns to broaden her horizons through an interaction with another child whose experience is different from her own. In the second, a child learns that his attitude toward particular objects can depend strongly on the context in which they are experienced. 

Overcoming Bias Example 1: Fear of Dogs

Jane is eight years old and lives in a small village. Her parents own several animals, including two Labradors. 

Jane’s cousin Max is nine and a half and lives in central Paris.

Max is always happy to visit Jane, and they play together outside, dreaming up adventures and climbing trees. But he is terribly afraid of Jane’s big dogs; whenever they come near him, he screams at the top of his lungs and runs indoors to hide. Jane finds this funny, calling her cousin a “fraidy cat” and devising ploys to lure Max close to the dogs.

Jane does not realize that, unlike her, Max is not used to having animals in his daily environment. She interprets his attitude exclusively from the viewpoint of her own experience.

What would you do if you were Jane’s parents?

At the dinner table, Jane’s mom asks her to stop teasing Max and explains that he is not used to animals because he lives in different circumstances than she does.

She asks Max to tell them what it is like living in the city. Max talks about his daily life and, notably, how he takes the metro by himself to school in the mornings, two stations from home.

The blood drains from Jane’s face: “You take the metro all by yourself? I could never do that, I’d be much too scared of getting lost.”

Her mom says to her: “You see, Jane, you fell into a trap—thinking that your cousin was just like you. We are all different. You need to remind yourself of that in the future because it’s easy for you to forget!”

This focused discussion has given Jane the opportunity to overcome her own egocentrism by realizing that she and Max inhabit different worlds. She, therefore, realizes that even though Max is scared of dogs (whereas she is not), he is capable of things that intimidate her, like taking the metro alone. This allows her to re-examine her way of reasoning through a “meta” example of her own ideas about the world, eventually leading her to change her attitude toward her cousin.

As parents, we should look for and take advantage of opportunities to open up our children to new perspectives, especially with respect to unexamined biases they may have against peers or outsiders. They will gradually learn to identify and guard against the tendency we all have to generalize recklessly from our own limited experience. Moreover, they will develop the capacity to see things from other perspectives and interests outside their own narrow sphere.

Overcoming Bias Example 2: Fear of Nettles

Josh has recently been on a field trip with his class. Before a hike, the teacher warns the students to steer clear of the nettle plants in the area  These “stinging nettles” can cause a nasty itching and burning rash. 

A few days later, at dinner, Josh finds that his parents have prepared a nettle soup . Boiling water makes the nettles safe to touch and eat.  But he refuses to eat it, since his experience tells them to keep nettles as far away from his body as possible— especially his mouth.

Josh vehemently refuses to try the soup at first and insists on having a frozen pizza instead. But his parents are firm with him and show him that the soup poses no danger by eating it themselves. Finally, Josh relents and tries the soup. He finds that it causes him no harm, and, much to his surprise, he actually enjoys it.

Children who do not know that nettles are safe to eat formulate their prejudice against the soup based solely on their experience, which is limited to the nettle’s irritant qualities. These kinds of learning experiences can be good moments for parents to point out to their children how they may falsely generalize their own limited experiences and how those experiences can produce unwarranted biases. These prejudices may stop them from trying out new things that may very well enrich their lives. 

Case Study 4

Developing self esteem.

Climbing Esther and Ali, both five years old, are at a playground, looking at a climbing wall designed for five to 10 year olds.

Esther goes over to the wall, looks at it, and touches the climbing holds. She starts climbing, pulling herself up with her arms and putting her feet on the lower holds to relieve her arms.

When she is about six feet up the wall, Esther stops.

“Go on, Esther — you’re almost there! Come on, just one more push. You can do it!” calls out her father from the bench he is sitting on.

Esther looks at the top of the wall. She wants to make it all the way up, but her hands hurt from clutching the climbing holds. She lets go and lands on the soft covering of the playground.

“Oh—you almost made it,” her father calls out.

Ali’s father goes over to his son: “Do you want to try? Grab onto these with your hands, and then put your feet on the ones at the bottom. Then you move your hands up more, and then your feet—hands and feet… Go slowly; it’ll be tricky to start with. Check where the holds are before you start climbing.”

Ali goes to the foot of the wall and grabs the holds to see what they feel like. He starts climbing, following his father’s advice.

Ali climbs slowly. He is about halfway up the wall, far below where Esther reached. He asks to get down, and his father takes him in his arms and puts him on the ground.

 “Great job, son! That was really good for a first try! I’m proud of you. That wall isn’t easy—it’s for children up to 10.”

In these two examples of the same situation, what is the impact of each parent’s behavior on the child’s self-esteem? What will each child remember from their first try at climbing?

Esther will probably be left with a sense of failure, thinking that she disappointed her father because she didn’t reach the top of the wall on her first try. She may not be willing to try again in the future, and she may hesitate to take on other new challenges. Even though he didn’t reach as high as Esther, Ali’s first climbing experience will likely be gratifying to him. His efforts have been recognized and encouraged by his father. He may be motivated now to make new efforts in the future, both in climbing and in other challenging new activities.

Case Study 5

Risk taking.

An important part of supporting the development of critical thinking skills at this age is encouraging children to take risks. Parents should beware of being hypercritical when their children make mistakes. They should also be proactive in exposing their children to new and potentially challenging situations. Finally, they should encourage their children to put themselves at risk in these situations, especially when it comes to putting forward arguments or answering questions. When they are (inevitably) wrong, children should be encouraged and supported rather than criticized. Being wrong should not become a source of shame for the child, but an opportunity to learn and grow. Consider the following anecdote.

Eight-year-old classmates Laura and Adam sit next to each other in a theater. Some 60 children, including Laura and Adam’s class, are on a field trip to see a historical reenactment. 

Before the curtain rises, the activity leader presenting the show asks the children: ″Who can tell me the name of the Roman emperor who conquered Gaul?”

Adam, who happens to be an avid reader of a cartoon about history, knows the answer immediately (Julius Caesar) and wants desperately to say it—but is afraid of making a mistake in front of everyone and, as a result, remains silent.

Laura hesitates. Several names spring to mind as she thinks back to what she learned in history class: Nero, Caligula, etc. Finally, a few seconds later, no longer able to restrain herself, she blurts out, “Julius Caesar!”

The activity leader congratulates her and then gets the show started.

In this situation, we see two different attitudes toward the risk of being wrong:

Adam would rather keep quiet than risk giving a wrong answer. We can deduce from this that Adam associates mistakes with something negative that could earn him disapproval or lead to him being mocked—even punished. He has thus pressured himself into thinking that only perfection is acceptable and has therefore reduced his ability to try things out.

Laura, on the other hand, would rather risk being wrong than remain silent. We can deduce from this that she does not feel shame about making mistakes; in any case, her desire to try and the excitement of taking risks outweigh the drawbacks of being wrong.

We learn through trial and erro r , which is necessary for the development of the ability to reason. Risk-taking and trial and error are vital.

Children’s environments, and notably their parents’ attitudes regarding mistakes, are determining factors in how they approach risk-taking and in whether they allow themselves to make mistakes.

Case Study 6

In addition to acquiring perspective on their own experiences and their own reasoning, children should, at this age,  begin to acquire perspective on their own emotions and to learn strategies for managing their emotions.  Without these management skills, children will be continually overwhelmed by their emotions and allow them to compromise their reasoning. The anecdote below can be used as a model to help parents guide their children in learning to express and manage their emotions, and to think clearly in spite of strong emotional reactions.

Seven-year-old Eddie is on vacation by the sea with his parents, who suggest that they all go out and take a boat to a nearby island for a few hours. They can visit the lighthouse there.

Eddie, who is busy playing with his figurines, refuses to get ready for the trip as his parents have asked.

“I haven’t finished playing! I want to stay here,” he exclaims.

“You can play with your figurines at home whenever you want, Eddy, but this boat trip is special. It’s something we can only do on vacation,” argues his mother. “Come on now, hurry up and put your shoes on, and then go and get your bag. Take a jacket as well, please—it can be cold out at sea.”

Eddie’s parents are all ready, and he still has not budged. He carries on playing with his back to them.

“That’s enough now, Eddy. Get up and get ready so we can leave,” orders his father, raising his voice slightly.

Without looking at them, Eddy bursts into tears.

“I don’t want to go on a boat! I’m scared of falling in the ocean! And what if the boat sinks? There are sharks out there! Plus I get scared of swimming if I can’t touch the bottom—if the water is too deep for me,” he says with a quavering voice.

“Oh, Eddy, why didn’t you say so before? I didn’t realize you were worried about the boat. I didn’t even think of that. But you know what? It’s normal to be scared the first time. And the ocean is daunting, that’s for sure. Listen, I’ll tell you what: let’s look at the shipping forecast together. I checked it earlier and it’s going to be a really nice day, with a very calm sea. As for swimming offshore, that’s out of the question! We’ll go swimming at our usual beach when we get back later this afternoon. And we’ll all be wearing life jackets on the boat, so there’s no way you can drown! Are you less worried now?”

“Yes… But I don’t want you to think I’m a wimp…”

“Being scared is nothing to be ashamed of! It’s a normal feeling which helps to protect us from danger. You should always say if you’re scared. I can’t always guess how you’re feeling—you’ve got to tell me!”

In this scenario, after a bit of hesitation, Eddie was able to express his fears. His parents accepted this emotion and drew on it to reassure him with clear, objective facts, helping him to understand the unfamiliar circumstances. This way he could feel completely safe on the boat.

If Eddie had not expressed his fears—because he was afraid of his parents being judgmental, angry, or perhaps even making fun of him—the situation could have taken one of the following turns:

Eddie could have categorically refused to go on the trip, and his parents would either have had to force him to come, or drop the plan entirely.

Eddie could have obeyed them without saying anything, but the trip would have been ruined by his anxiety.

Although dealing with and expressing emotions may seem far afield from critical thinking, it is a vital precondition of critical and independent thinking that children have the confidence to recognize and acknowledge their emotions. Otherwise, children will be unable to set their emotions aside in order to  consider complicated questions or scenarios in a clear and unbiased way.

Case Study 7

What is independent thinking.

What does independent thinking mean? Independent thinking is when an individual forms their own thoughts rather than just going along with what others are thinking. They apply their personal experiences, knowledge, and observations to form a personal viewpoint.

Independent thinking vs critical thinking

We can think independently without thinking critically, but we can’t think critically without thinking independently. That is, independent thinking is a precondition of critical thinking. In order to begin assessing information and making judgments objectively, we must first prevent ourselves from being unduly influenced by our peers’ views.

Example of independent thinking

In certain scenarios, children’s developing perspectives on their own beliefs, reasoning, and emotions can combine in the analysis of a challenging source of information.  The wealth of media to which children are exposed today can be overwhelming, but these media can also provide opportunities for learning and practicing the skills of critical analysis. Parents can help guide their children in these situations by prompting them with questions and asking children to make their beliefs and reasoning explicit. At this young age, preparation for independent and critical thinking need not interfere with the fantasy life of the child, as the example below shows. 

Six-year-old Tom has just written a letter to Santa Claus. Now he is watching television, flipping between channels until a show about Christmas catches his attention.

The TV presenter explains that nowadays children do not believe in Santa Claus the way they used to. Christmas has been totally commercialized. What’s more, red only became the color of Christmas due to the branding of the Coca-Cola company. 

First part of the program: “What do those concerned say?” A journalist standing outside a school asks several children their opinion. The children interviewed say that their parents have told them about Santa Claus, but that he does not really exist, at least no more than witches and ghosts do. They say that they know exactly what they are going to get for Christmas and how much it will cost. Their little brothers or sisters may still believe in Santa, but they themselves are not babies anymore. Regardless of whether they’re “naughty or nice,” they know there will always be gifts for them under the tree.

Second part of the program: “Santa Claus: salesman.” Images in the background show check-out lines in toy stores, parents with shopping carts full to the brim, others taking photos of the shelves on their phones. We see Santa Clauses of all shapes and sizes in shopping malls, day care centers, in the street, and even sitting in donkey-drawn carriages. A narrator provides statistics on the average amount spent by families on gifts, as well as the percentage of gifts purchased in-store versus online.

Finally, the presenter comes back on the screen and concludes with, “Christmas has lost its magic!” before going to a commercial break.

Tom’s father came into the room while the show was on air and has seen part of it. He can tell that his son is both confused and unsettled.

“Why do you believe in Santa Claus, Tom? What are your reasons?”

“Because he’s come every year since I was little. And because he comes at nighttime. Who else could come in the middle of the night? Because he always drinks the hot chocolate we leave him under the tree, and he eats the cookies. Because I’ve seen him more than once, near the Christmas tree at school and in stores. Because no one else could make toys for every kid and deliver them all.”

“Yes, those are very good reasons to believe in him, Tom. And what about at school? Do you talk about Santa with the other kids?”

“The big kids say the same thing as the people on the TV: that he doesn’t exist and that their parents made him up. When I told them there was no way presents could just appear under the tree overnight, they said I was a baby. I don’t talk about Santa anymore because of that.”

“I think you’re right to assert yourself and say what you really think. There’s what they say on TV, what your friends say, and then there’s your own opinion. And it’s important for you to say what you think and defend your point of view. It’s important to listen to other people too, of course, because no one is right all the time. But having your own ideas and expressing them is really important all through your life.”

What would you have done if you were Tom’s father?

Would it have been better to admit the truth about Santa Claus to Tom and contradict his beliefs and imagination? If Tom’s dad had done that, what value would his son have placed on his own reasoning? Would he have dared to defend his opinion in the future? 

During this conversation, the father chose to give weight to Tom’s arguments by giving credit to them and praising the way he expressed his personal thoughts. He did not state his own opinion on the matter, but instead focused the discussion on dealing with clashing points of view and on arguing. He hopes that Tom will now see the value in his own arguments, even if they go against what was said on the television show. Now, the next time he finds himself in a similar situation, Tom will probably be confident enough to express his own opinion on the information he receives.

The repetition of situations such as this should allow Tom’s critical thinking skills to develop. They will reinforce and strengthen his self-esteem and build his confidence in his ability to develop his own thoughts.

This situation may seem counter-intuitive. We usually associate the development of critical thinking with questioning certain beliefs, in this case the belief in the existence of Santa Claus. 

This viewpoint, though, projects our own adult understanding onto Tom. Children of his age should instead be encouraged to express themselves, to be creative in their arguments, and to believe in the value of their own points of view—rather than in the truths that are thrust on them by adults, media, or their friends.

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