logic and critical thinking symbols

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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Symbolic Logic

Symbolic logic is a system that takes sentences apart and shows the connections between their pieces using symbols. By doing this, you can find out whether the sentence is set up in a way that makes logical sense . The beauty of symbolic logic is that it turns arguments into almost a puzzle that you can piece together. It’s like translating a sentence into a secret code where each symbol has a specific meaning.

Now, to get really clear about this: Imagine you have a statement like “If I study, then I will pass the test.” Symbolic logic would take this and turn it into something like “S → P,” where each letter and the arrow in between have special meanings. The first letter represents the “if” part (studying), the arrow symbolizes the “then,” and the second letter represents the outcome (passing the test). Symbolic logic helps us step back and look at the structure of our thinking without getting tangled up in the actual words we use.

How to Guide

Let’s break down symbolic logic into steps. First up are the symbols. Think of them as a secret alphabet where ‘∧’ stands for ‘and’, ‘∨’ for ‘or’, and so on. Then you’ll translate regular sentences into these symbols, kind of like learning a new language. Now, the fun begins! With the rules of logic, you can move these symbols around, swap them, or combine them in different ways to explore the argument . It’s a little like following a recipe where the rules are your ingredients and steps. The best part? You end up with a clear idea of whether the argument holds up to scrutiny.

• Propositional Logic: Think of this as the basic form of symbolic logic. It cares about whether whole sentences are true or false, but doesn’t worry about the smaller pieces within those sentences.
• Predicate Logic: This digs deeper than propositional logic. It looks into the subjects of sentences and what’s being said about them, giving us a more detailed picture.
• Modal Logic: Modal logic adds flavor to the mix by considering possibilities. It plays with concepts like what could happen or what has to happen.

Examples of Symbolic Logic

• ‘P → Q’ translates to ‘If P is true, then Q is true.’ For example, “If it is raining (P), then the ground is wet (Q).” This shows a cause-and-effect relationship neatly packed into a simple code.
• ‘P ∧ Q’ says both P and Q are true. With “I have a cat (P)” and “I have a dog (Q)”, ‘P ∧ Q’ combines these to mean “I have a cat and a dog.” It’s a logic sandwich where both fillings need to be present.
• ‘P ∨ Q’ hints that P, Q, or both could be true. With “I’ll have soup (P)” and “I’ll have salad (Q)” as options, ‘P ∨ Q’ tells you that lunch is definitely going to include soup, salad, or a combo meal.

Why is it important?

Symbolic logic is a podium where our thoughts and arguments can stand to be inspected. It dissects our statements, revealing their structure in stark clarity. It’s as vital as a compass for a sailor, guiding us through a sea of potentially confusing dialogues. When we’re debating life’s big questions or when we want to be sure we’re on solid ground with our reasoning, symbolic logic shines as a beacon of clarity.

Consider this: if you’re discussing something complex, like a peace treaty or a legal contract, symbolic logic allows you to zoom out and see the larger picture beyond the jumble of words. It helps avoid misunderstandings, maybe even conflicts. It brings out any hidden loopholes or flaws in thinking that might trip us up. For the average person, it means sharper critical thinking skills and stronger arguments, whether you’re debating which movie to watch or thinking about bigger life choices.

The roots of symbolic logic stretch way back to thinkers like Aristotle , but it wasn’t until folks like George Boole and Gottlob Frege stepped up in the 1800s that it truly got its wings. These pioneers crafted symbolic logic into the precise, finely tuned tool that it is today, comparable to a mathematician’s trusty set of formulas and equations.

Controversies

The big head-scratcher with symbolic logic is whether it captures everything about how we communicate. Think about the colors of a sunset or the feeling of a first kiss – they might not fit neatly into symbols. Critics caution that symbolic logic is brilliant but not the only show in town. It should play nice with the other ways we understand conversations and arguments.

Other Important Facts

Symbolic logic’s influence isn’t just limited to philosophy. It’s a jack-of-all-trades, finding its way into computer science to steer algorithms and into mathematics to underpin proofs. Even in philosophy’s diverse landscape, from ethics to metaphysics , it’s a versatile tool for disentangling some exceptionally knotty problems.

Related Topics

Think of the following as cousins to symbolic logic, each bringing their own twist to understanding reality:

• Mathematical Logic: It’s the nuts and bolts of logic applied to math, proving theorems, and building the foundations of mathematics.
• Set Theory: Dives into collections of objects, thinking about them as single units and exploring the relations between them.
• Computational Logic: The brainpower behind computers, transforming our logic into a language machines can understand and act on.

In summary, symbolic logic is an intellectual Swiss Army knife, carving out clarity from the potentially murky world of arguments and ideas. It’s not just about symbols; it’s about seeing the hidden structure in our thoughts, like x-ray vision for the mind. Whether it’s mapping out the logic of a scientific discovery or navigating the ethics of right and wrong, symbolic logic helps us get there without getting lost. It might not have all the answers, but it sure points us in the right direction.

Logical Symbols

Compound statements.

Conjunction

Disjunction

Implication

Equivalence

Exercises in Propositional Logic

Pursuing Truth: A Guide to Critical Thinking

Chapter 4 propositional logic.

Categorical logic is a great way to analyze arguments, but only certain kinds of arguments. It is limited to arguments that have only two premises and the four kinds of categorical sentences. This means that certain common arguments that are obviously valid will not even be well-formed arguments in categorical logic. Here is an example:

• I will either go out for dinner tonight or go out for breakfast tomorrow.
• I won’t go out for dinner tonight.
• I will go out for breakfast tomorrow.

None of these sentences fit any of the four categorical schemes. So, we need a new logic, called propositional logic. The good news is that it is fairly simple.

4.1 Simple and Complex Sentences

The fundamental logical unit in categorical logic was a category, or class of things. The fundamental logical unit in propositional logic is a statement, or proposition 5 Simple statements are statements that contain no other statement as a part. Here are some examples:

• Oklahoma Baptist University is in Shawnee, Oklahoma.
• Barack Obama was succeeded as President of the US by Donald Trump.
• It is 33 degrees outside.

Simple sentences are symbolized by uppercase letters. Just pick a letter that makes sense, given the sentence to be symbolized, that way you can more easily remember which letter means which sentence.

Complex sentences have at least one sentence as a component. There are five types in propositional logic:

• Conjunctions
• Disjunctions
• Conditionals
• Biconditionals

4.1.1 Negations

Negations are “not” sentences. They assert that something is not the case. For example, the negation of the simple sentence “Oklahoma Baptist University is in Shawnee, Oklahoma” is “Oklahoma Baptist University is not in Shawnee, Oklahoma.” In general, a simple way to form a negation is to just place the phrase “It is not the case that” before the sentence to be negated.

A negation is symbolized by placing this symbol ‘ \(\neg\) ’ before the sentence-letter. The symbol looks like a dash with a little tail on its right side. If \(\textrm{D}\) = ‘It is 33 degrees outside,’ then \(\neg \textrm{D}\) = ‘It is not 33 degrees outside.’ The negation symbol is used to translate these English phrases:

• it is not the case that
• it is not true that
• it is false that

A negation is true whenever the negated sentence is false. If it is true that it is not 33 degrees outside, then it must be false that it is 33 degrees outside. if it is false that Tulsa is the capital of Oklahoma, then it is true that Tulsa is not the capital of Oklahoma.

When translating, try to keep the simple sentences positive in meaning. Note the warning on page 24, about the example of affirming and denying. Denying is not simply the negation of affirming.

4.2 Conjunction

Negations are “and” sentences. They put two sentences, called conjuncts, together and claim that they are both true. We’ll use the ampersand (&) to signify a negation. Other common symbols are a dot and an upside down wedge. The English words that are translated with the ampersand include:

• nevertheless

For example, we would translate the sentence ‘It is raining today and my sunroof is open’ as ‘ \(\textrm{R} \& \textrm{O}\) .’

4.3 Disjunction

A disjunction is an “or” sentence. It claims that at least one of two sentences, called disjuncts, is true. For example, if I say that either I will go to the movies this weekend or I will stay home and grade critical thinking homework, then I have told the truth provided that I do one or both of those things. If I do neither, though, then my claim was false.

We use this symbol, called a “vel,” for disjunctions: \(\vee\) . The vel is used to translate - or - eitheror - unless

4.4 Conditional

The conditional is a common type of sentence. It claims that something is true, if something else is also. Examples of conditionals are

• “If Sarah makes an A on the final, then she will get an A for the course.”
• “Your car will last many years, provided you perform the required maintenance.”
• “You can light that match only if it is not wet.”

We can translate those sentences with an arrow like this:

• \(F \rightarrow C\)
• \(M \rightarrow L\)
• \(L \rightarrow \neg W\)

The arrow translates many English words and phrases, including

• provided that
• is a sufficient condition for
• is a necessary condition for
• on the condition that

One big difference between conditionals and other sentences, like conjunctions and disjunctions, is that order matters. Notice that there is no logical difference between the following two sentences:

• Albany is the capital of New York and Austin is the capital of Texas.
• Austin is the capital of Texas and Albany is the capital of New York.

They essentially assert exactly the same thing, that both of those conjuncts are true. So, changing order of the conjuncts or disjuncts does not change the meaning of the sentence, and if meaning doesn’t change, then true value doesn’t change.

That’s not true of conditionals. Note the difference between these two sentences:

• If you drew a diamond, then you drew a red card.
• If you drew a red card, then you drew a diamond.

The first sentence must be true. if you drew a diamond, then that guarantees that it’s a red card. The second sentence, though, could be false. Your drawing a red card doesn’t guarantee that you drew a diamond, you could have drawn a heart instead. So, we need to be able to specify which sentence goes before the arrow and which sentence goes after. The sentence before the arrow is called the antecedent, and the sentence after the arrow is called the consequent.

Look at those three examples again:

The antecedent for the first sentence is “Sarah makes an A on the final.” The consequent is “She will get an A for the course.” Note that the if and the then are not parts of the antecedent and consequent.

In the second sentence, the antecdent is “You perform the required maintenance.” The consequent is “Your car will last many years.” This tells us that the antecedent won’t always come first in the English sentence.

The third sentence is tricky. The antecedent is “You can light that match.” Why? The explanation involves something called necessary and sufficient conditions.

4.4.1 Necessary and Sufficient Conditions

A sufficient condition is something that is enough to guarantee the truth of something else. For example, getting a 95 on an exam is sufficient for making an A, assuming that exam is worth 100 points. A necessary condition is something that must be true in order for something else to be true. Making a 95 on an exam is not necessary for making an A—a 94 would have still been an A. Taking the exam is necessary for making an A, though. You can’t make an A if you don’t take the exam, or, in other words, you can make an a only if you enroll in the course.

Here are some important rules to keep in mind:

• ‘If’ introduces antecedents, but Only if introduces consequents.
• If A is a sufficient condition for B, then \(A \rightarrow B\) .
• If A is a necessary condition for B, then \(B \rightarrow A\) .

4.5 Biconditional

We won’t spend much time on biconditionals. There are times when something is both a necessary and a sufficient condition for something else. For example, making at least a 90 and getting an A (assuming a standard scale, no curve, and no rounding up). If you make at least a 90, then you will get an A. If you got an A, then you made at least a 90. We can use a double arrow to translate a biconditional, like this:

• \(N \rightarrow A\)

For biconditionals, as for conjunctions and disjunctions, order doesn’t matter.

Here are some English phrases that signify biconditionals:

• it and only if
• when and only when
• just in case
• is a necessary and sufficient condition for

4.6 Translations

Propositional logic is language. Like other languages, it has a syntax and a semantics. The syntax of a language includes the basic symbols of the language plus rules for putting together proper statements in the language. To use propositional logic, we need to know how to translate English sentences into the language of propositional logic. We start with our sentence letters, which represent simple English sentences. Let’s use three borrowed from an elementary school reader:

We then build complex sentences using the sentence letters and our five logical operators, like this:

We can make even more complex sentences, but we will soon run into a problem. Consider this example:

\[ T \mathbin{\&} J \rightarrow S\]

We don’t know this means. It could be either one of the following:

• Tom hit the ball, and if Jane caught the ball, then Spot chased it.
• If Tom hit the ball and Jane caught it, then Spot chased it.

The first sentence is a conjunction, \(T\) is the first conjunct and \(M \rightarrow S\) is the second conjunct. The second sentence, though, is a conditional, \(T \mathbin{\&}M\) is the antecdent, and \(S\) is the consequent. Our two interpretations are not equivalent, so we need a way to clear up the ambiguity. We can do this with parentheses. Our first sentence becomes:

\[ T \mathbin{\&} (J \rightarrow S) \]

The second sentence is:

\[ (T \mathbin{\&} J) \rightarrow S\]

If we need higher level parentheses, we can use brackets and braces. For instance, this is a perfectly good formula in propositional logic:

\[ [(P \mathbin{\&} Q) \vee R] \rightarrow \{[(\neg P \leftrightarrow Q) \mathbin{\&} S] \vee \neg P\} \] 6

Every sentence in propositional logic is one of six types:

• Conjunction
• Disjunction
• Conditional
• Biconditional

What type of sentence it is will be determined by its main logical operator. Sentences can have several logical operators, but they will always have one, and only one, main operator. Here are some general rules for finding the main operator in a symbolized formula of propositional logic:

• If a sentence has only one logical operator, then that is the main operator.
• If a sentence has more than one logical operator, then the main operator is the one outside the parentheses.
• If a sentence has two logical operators outside the parentheses, then the main operator is not the negation.

Here are some examples:

Informally, we use ‘proposition’ and ‘statement’ interchangeably. Strictly speaking, the proposition is the content, or meaning, that the statement expresses. When different sentences in different languages mean the same thing, it is because they express the same proposition. ↩︎

It may be a good formula in propositional logic, but that doesn’t mean it would be a good English sentence. ↩︎

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!

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

Certificate Final Exam

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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.

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Saylor Direct Credit

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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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Critical Thinking

Critical thinking is a widely accepted educational goal. Its definition is contested, but the competing definitions can be understood as differing conceptions of the same basic concept: careful thinking directed to a goal. Conceptions differ with respect to the scope of such thinking, the type of goal, the criteria and norms for thinking carefully, and the thinking components on which they focus. Its adoption as an educational goal has been recommended on the basis of respect for students’ autonomy and preparing students for success in life and for democratic citizenship. “Critical thinkers” have the dispositions and abilities that lead them to think critically when appropriate. The abilities can be identified directly; the dispositions indirectly, by considering what factors contribute to or impede exercise of the abilities. Standardized tests have been developed to assess the degree to which a person possesses such dispositions and abilities. Educational intervention has been shown experimentally to improve them, particularly when it includes dialogue, anchored instruction, and mentoring. Controversies have arisen over the generalizability of critical thinking across domains, over alleged bias in critical thinking theories and instruction, and over the relationship of critical thinking to other types of thinking.

2.1 Dewey’s Three Main Examples

2.2 dewey’s other examples, 2.3 further examples, 2.4 non-examples, 3. the definition of critical thinking, 4. its value, 5. the process of thinking critically, 6. components of the process, 7. contributory dispositions and abilities, 8.1 initiating dispositions, 8.2 internal dispositions, 9. critical thinking abilities, 10. required knowledge, 11. educational methods, 12.1 the generalizability of critical thinking, 12.2 bias in critical thinking theory and pedagogy, 12.3 relationship of critical thinking to other types of thinking, other internet resources, related entries.

Use of the term ‘critical thinking’ to describe an educational goal goes back to the American philosopher John Dewey (1910), who more commonly called it ‘reflective thinking’. He defined it as

active, persistent and careful consideration of any belief or supposed form of knowledge in the light of the grounds that support it, and the further conclusions to which it tends. (Dewey 1910: 6; 1933: 9)

and identified a habit of such consideration with a scientific attitude of mind. His lengthy quotations of Francis Bacon, John Locke, and John Stuart Mill indicate that he was not the first person to propose development of a scientific attitude of mind as an educational goal.

In the 1930s, many of the schools that participated in the Eight-Year Study of the Progressive Education Association (Aikin 1942) adopted critical thinking as an educational goal, for whose achievement the study’s Evaluation Staff developed tests (Smith, Tyler, & Evaluation Staff 1942). Glaser (1941) showed experimentally that it was possible to improve the critical thinking of high school students. Bloom’s influential taxonomy of cognitive educational objectives (Bloom et al. 1956) incorporated critical thinking abilities. Ennis (1962) proposed 12 aspects of critical thinking as a basis for research on the teaching and evaluation of critical thinking ability.

Since 1980, an annual international conference in California on critical thinking and educational reform has attracted tens of thousands of educators from all levels of education and from many parts of the world. Also since 1980, the state university system in California has required all undergraduate students to take a critical thinking course. Since 1983, the Association for Informal Logic and Critical Thinking has sponsored sessions in conjunction with the divisional meetings of the American Philosophical Association (APA). In 1987, the APA’s Committee on Pre-College Philosophy commissioned a consensus statement on critical thinking for purposes of educational assessment and instruction (Facione 1990a). Researchers have developed standardized tests of critical thinking abilities and dispositions; for details, see the Supplement on Assessment . Educational jurisdictions around the world now include critical thinking in guidelines for curriculum and assessment.

For details on this history, see the Supplement on History .

2. Examples and Non-Examples

Before considering the definition of critical thinking, it will be helpful to have in mind some examples of critical thinking, as well as some examples of kinds of thinking that would apparently not count as critical thinking.

Dewey (1910: 68–71; 1933: 91–94) takes as paradigms of reflective thinking three class papers of students in which they describe their thinking. The examples range from the everyday to the scientific.

Transit : “The other day, when I was down town on 16th Street, a clock caught my eye. I saw that the hands pointed to 12:20. This suggested that I had an engagement at 124th Street, at one o’clock. I reasoned that as it had taken me an hour to come down on a surface car, I should probably be twenty minutes late if I returned the same way. I might save twenty minutes by a subway express. But was there a station near? If not, I might lose more than twenty minutes in looking for one. Then I thought of the elevated, and I saw there was such a line within two blocks. But where was the station? If it were several blocks above or below the street I was on, I should lose time instead of gaining it. My mind went back to the subway express as quicker than the elevated; furthermore, I remembered that it went nearer than the elevated to the part of 124th Street I wished to reach, so that time would be saved at the end of the journey. I concluded in favor of the subway, and reached my destination by one o’clock.” (Dewey 1910: 68–69; 1933: 91–92)

Ferryboat : “Projecting nearly horizontally from the upper deck of the ferryboat on which I daily cross the river is a long white pole, having a gilded ball at its tip. It suggested a flagpole when I first saw it; its color, shape, and gilded ball agreed with this idea, and these reasons seemed to justify me in this belief. But soon difficulties presented themselves. The pole was nearly horizontal, an unusual position for a flagpole; in the next place, there was no pulley, ring, or cord by which to attach a flag; finally, there were elsewhere on the boat two vertical staffs from which flags were occasionally flown. It seemed probable that the pole was not there for flag-flying.

“I then tried to imagine all possible purposes of the pole, and to consider for which of these it was best suited: (a) Possibly it was an ornament. But as all the ferryboats and even the tugboats carried poles, this hypothesis was rejected. (b) Possibly it was the terminal of a wireless telegraph. But the same considerations made this improbable. Besides, the more natural place for such a terminal would be the highest part of the boat, on top of the pilot house. (c) Its purpose might be to point out the direction in which the boat is moving.

“In support of this conclusion, I discovered that the pole was lower than the pilot house, so that the steersman could easily see it. Moreover, the tip was enough higher than the base, so that, from the pilot’s position, it must appear to project far out in front of the boat. Moreover, the pilot being near the front of the boat, he would need some such guide as to its direction. Tugboats would also need poles for such a purpose. This hypothesis was so much more probable than the others that I accepted it. I formed the conclusion that the pole was set up for the purpose of showing the pilot the direction in which the boat pointed, to enable him to steer correctly.” (Dewey 1910: 69–70; 1933: 92–93)

Bubbles : “In washing tumblers in hot soapsuds and placing them mouth downward on a plate, bubbles appeared on the outside of the mouth of the tumblers and then went inside. Why? The presence of bubbles suggests air, which I note must come from inside the tumbler. I see that the soapy water on the plate prevents escape of the air save as it may be caught in bubbles. But why should air leave the tumbler? There was no substance entering to force it out. It must have expanded. It expands by increase of heat, or by decrease of pressure, or both. Could the air have become heated after the tumbler was taken from the hot suds? Clearly not the air that was already entangled in the water. If heated air was the cause, cold air must have entered in transferring the tumblers from the suds to the plate. I test to see if this supposition is true by taking several more tumblers out. Some I shake so as to make sure of entrapping cold air in them. Some I take out holding mouth downward in order to prevent cold air from entering. Bubbles appear on the outside of every one of the former and on none of the latter. I must be right in my inference. Air from the outside must have been expanded by the heat of the tumbler, which explains the appearance of the bubbles on the outside. But why do they then go inside? Cold contracts. The tumbler cooled and also the air inside it. Tension was removed, and hence bubbles appeared inside. To be sure of this, I test by placing a cup of ice on the tumbler while the bubbles are still forming outside. They soon reverse” (Dewey 1910: 70–71; 1933: 93–94).

Dewey (1910, 1933) sprinkles his book with other examples of critical thinking. We will refer to the following.

Weather : A man on a walk notices that it has suddenly become cool, thinks that it is probably going to rain, looks up and sees a dark cloud obscuring the sun, and quickens his steps (1910: 6–10; 1933: 9–13).

Disorder : A man finds his rooms on his return to them in disorder with his belongings thrown about, thinks at first of burglary as an explanation, then thinks of mischievous children as being an alternative explanation, then looks to see whether valuables are missing, and discovers that they are (1910: 82–83; 1933: 166–168).

Typhoid : A physician diagnosing a patient whose conspicuous symptoms suggest typhoid avoids drawing a conclusion until more data are gathered by questioning the patient and by making tests (1910: 85–86; 1933: 170).

Blur : A moving blur catches our eye in the distance, we ask ourselves whether it is a cloud of whirling dust or a tree moving its branches or a man signaling to us, we think of other traits that should be found on each of those possibilities, and we look and see if those traits are found (1910: 102, 108; 1933: 121, 133).

Suction pump : In thinking about the suction pump, the scientist first notes that it will draw water only to a maximum height of 33 feet at sea level and to a lesser maximum height at higher elevations, selects for attention the differing atmospheric pressure at these elevations, sets up experiments in which the air is removed from a vessel containing water (when suction no longer works) and in which the weight of air at various levels is calculated, compares the results of reasoning about the height to which a given weight of air will allow a suction pump to raise water with the observed maximum height at different elevations, and finally assimilates the suction pump to such apparently different phenomena as the siphon and the rising of a balloon (1910: 150–153; 1933: 195–198).

Diamond : A passenger in a car driving in a diamond lane reserved for vehicles with at least one passenger notices that the diamond marks on the pavement are far apart in some places and close together in others. Why? The driver suggests that the reason may be that the diamond marks are not needed where there is a solid double line separating the diamond lane from the adjoining lane, but are needed when there is a dotted single line permitting crossing into the diamond lane. Further observation confirms that the diamonds are close together when a dotted line separates the diamond lane from its neighbour, but otherwise far apart.

Rash : A woman suddenly develops a very itchy red rash on her throat and upper chest. She recently noticed a mark on the back of her right hand, but was not sure whether the mark was a rash or a scrape. She lies down in bed and thinks about what might be causing the rash and what to do about it. About two weeks before, she began taking blood pressure medication that contained a sulfa drug, and the pharmacist had warned her, in view of a previous allergic reaction to a medication containing a sulfa drug, to be on the alert for an allergic reaction; however, she had been taking the medication for two weeks with no such effect. The day before, she began using a new cream on her neck and upper chest; against the new cream as the cause was mark on the back of her hand, which had not been exposed to the cream. She began taking probiotics about a month before. She also recently started new eye drops, but she supposed that manufacturers of eye drops would be careful not to include allergy-causing components in the medication. The rash might be a heat rash, since she recently was sweating profusely from her upper body. Since she is about to go away on a short vacation, where she would not have access to her usual physician, she decides to keep taking the probiotics and using the new eye drops but to discontinue the blood pressure medication and to switch back to the old cream for her neck and upper chest. She forms a plan to consult her regular physician on her return about the blood pressure medication.

Candidate : Although Dewey included no examples of thinking directed at appraising the arguments of others, such thinking has come to be considered a kind of critical thinking. We find an example of such thinking in the performance task on the Collegiate Learning Assessment (CLA+), which its sponsoring organization describes as

a performance-based assessment that provides a measure of an institution’s contribution to the development of critical-thinking and written communication skills of its students. (Council for Aid to Education 2017)

A sample task posted on its website requires the test-taker to write a report for public distribution evaluating a fictional candidate’s policy proposals and their supporting arguments, using supplied background documents, with a recommendation on whether to endorse the candidate.

Immediate acceptance of an idea that suggests itself as a solution to a problem (e.g., a possible explanation of an event or phenomenon, an action that seems likely to produce a desired result) is “uncritical thinking, the minimum of reflection” (Dewey 1910: 13). On-going suspension of judgment in the light of doubt about a possible solution is not critical thinking (Dewey 1910: 108). Critique driven by a dogmatically held political or religious ideology is not critical thinking; thus Paulo Freire (1968 [1970]) is using the term (e.g., at 1970: 71, 81, 100, 146) in a more politically freighted sense that includes not only reflection but also revolutionary action against oppression. Derivation of a conclusion from given data using an algorithm is not critical thinking.

What is critical thinking? There are many definitions. Ennis (2016) lists 14 philosophically oriented scholarly definitions and three dictionary definitions. Following Rawls (1971), who distinguished his conception of justice from a utilitarian conception but regarded them as rival conceptions of the same concept, Ennis maintains that the 17 definitions are different conceptions of the same concept. Rawls articulated the shared concept of justice as

a characteristic set of principles for assigning basic rights and duties and for determining… the proper distribution of the benefits and burdens of social cooperation. (Rawls 1971: 5)

Bailin et al. (1999b) claim that, if one considers what sorts of thinking an educator would take not to be critical thinking and what sorts to be critical thinking, one can conclude that educators typically understand critical thinking to have at least three features.

• It is done for the purpose of making up one’s mind about what to believe or do.
• The person engaging in the thinking is trying to fulfill standards of adequacy and accuracy appropriate to the thinking.
• The thinking fulfills the relevant standards to some threshold level.

One could sum up the core concept that involves these three features by saying that critical thinking is careful goal-directed thinking. This core concept seems to apply to all the examples of critical thinking described in the previous section. As for the non-examples, their exclusion depends on construing careful thinking as excluding jumping immediately to conclusions, suspending judgment no matter how strong the evidence, reasoning from an unquestioned ideological or religious perspective, and routinely using an algorithm to answer a question.

If the core of critical thinking is careful goal-directed thinking, conceptions of it can vary according to its presumed scope, its presumed goal, one’s criteria and threshold for being careful, and the thinking component on which one focuses. As to its scope, some conceptions (e.g., Dewey 1910, 1933) restrict it to constructive thinking on the basis of one’s own observations and experiments, others (e.g., Ennis 1962; Fisher & Scriven 1997; Johnson 1992) to appraisal of the products of such thinking. Ennis (1991) and Bailin et al. (1999b) take it to cover both construction and appraisal. As to its goal, some conceptions restrict it to forming a judgment (Dewey 1910, 1933; Lipman 1987; Facione 1990a). Others allow for actions as well as beliefs as the end point of a process of critical thinking (Ennis 1991; Bailin et al. 1999b). As to the criteria and threshold for being careful, definitions vary in the term used to indicate that critical thinking satisfies certain norms: “intellectually disciplined” (Scriven & Paul 1987), “reasonable” (Ennis 1991), “skillful” (Lipman 1987), “skilled” (Fisher & Scriven 1997), “careful” (Bailin & Battersby 2009). Some definitions specify these norms, referring variously to “consideration of any belief or supposed form of knowledge in the light of the grounds that support it and the further conclusions to which it tends” (Dewey 1910, 1933); “the methods of logical inquiry and reasoning” (Glaser 1941); “conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication” (Scriven & Paul 1987); the requirement that “it is sensitive to context, relies on criteria, and is self-correcting” (Lipman 1987); “evidential, conceptual, methodological, criteriological, or contextual considerations” (Facione 1990a); and “plus-minus considerations of the product in terms of appropriate standards (or criteria)” (Johnson 1992). Stanovich and Stanovich (2010) propose to ground the concept of critical thinking in the concept of rationality, which they understand as combining epistemic rationality (fitting one’s beliefs to the world) and instrumental rationality (optimizing goal fulfillment); a critical thinker, in their view, is someone with “a propensity to override suboptimal responses from the autonomous mind” (2010: 227). These variant specifications of norms for critical thinking are not necessarily incompatible with one another, and in any case presuppose the core notion of thinking carefully. As to the thinking component singled out, some definitions focus on suspension of judgment during the thinking (Dewey 1910; McPeck 1981), others on inquiry while judgment is suspended (Bailin & Battersby 2009, 2021), others on the resulting judgment (Facione 1990a), and still others on responsiveness to reasons (Siegel 1988). Kuhn (2019) takes critical thinking to be more a dialogic practice of advancing and responding to arguments than an individual ability.

In educational contexts, a definition of critical thinking is a “programmatic definition” (Scheffler 1960: 19). It expresses a practical program for achieving an educational goal. For this purpose, a one-sentence formulaic definition is much less useful than articulation of a critical thinking process, with criteria and standards for the kinds of thinking that the process may involve. The real educational goal is recognition, adoption and implementation by students of those criteria and standards. That adoption and implementation in turn consists in acquiring the knowledge, abilities and dispositions of a critical thinker.

Conceptions of critical thinking generally do not include moral integrity as part of the concept. Dewey, for example, took critical thinking to be the ultimate intellectual goal of education, but distinguished it from the development of social cooperation among school children, which he took to be the central moral goal. Ennis (1996, 2011) added to his previous list of critical thinking dispositions a group of dispositions to care about the dignity and worth of every person, which he described as a “correlative” (1996) disposition without which critical thinking would be less valuable and perhaps harmful. An educational program that aimed at developing critical thinking but not the correlative disposition to care about the dignity and worth of every person, he asserted, “would be deficient and perhaps dangerous” (Ennis 1996: 172).

Dewey thought that education for reflective thinking would be of value to both the individual and society; recognition in educational practice of the kinship to the scientific attitude of children’s native curiosity, fertile imagination and love of experimental inquiry “would make for individual happiness and the reduction of social waste” (Dewey 1910: iii). Schools participating in the Eight-Year Study took development of the habit of reflective thinking and skill in solving problems as a means to leading young people to understand, appreciate and live the democratic way of life characteristic of the United States (Aikin 1942: 17–18, 81). Harvey Siegel (1988: 55–61) has offered four considerations in support of adopting critical thinking as an educational ideal. (1) Respect for persons requires that schools and teachers honour students’ demands for reasons and explanations, deal with students honestly, and recognize the need to confront students’ independent judgment; these requirements concern the manner in which teachers treat students. (2) Education has the task of preparing children to be successful adults, a task that requires development of their self-sufficiency. (3) Education should initiate children into the rational traditions in such fields as history, science and mathematics. (4) Education should prepare children to become democratic citizens, which requires reasoned procedures and critical talents and attitudes. To supplement these considerations, Siegel (1988: 62–90) responds to two objections: the ideology objection that adoption of any educational ideal requires a prior ideological commitment and the indoctrination objection that cultivation of critical thinking cannot escape being a form of indoctrination.

Despite the diversity of our 11 examples, one can recognize a common pattern. Dewey analyzed it as consisting of five phases:

• suggestions , in which the mind leaps forward to a possible solution;
• an intellectualization of the difficulty or perplexity into a problem to be solved, a question for which the answer must be sought;
• the use of one suggestion after another as a leading idea, or hypothesis , to initiate and guide observation and other operations in collection of factual material;
• the mental elaboration of the idea or supposition as an idea or supposition ( reasoning , in the sense on which reasoning is a part, not the whole, of inference); and
• testing the hypothesis by overt or imaginative action. (Dewey 1933: 106–107; italics in original)

The process of reflective thinking consisting of these phases would be preceded by a perplexed, troubled or confused situation and followed by a cleared-up, unified, resolved situation (Dewey 1933: 106). The term ‘phases’ replaced the term ‘steps’ (Dewey 1910: 72), thus removing the earlier suggestion of an invariant sequence. Variants of the above analysis appeared in (Dewey 1916: 177) and (Dewey 1938: 101–119).

The variant formulations indicate the difficulty of giving a single logical analysis of such a varied process. The process of critical thinking may have a spiral pattern, with the problem being redefined in the light of obstacles to solving it as originally formulated. For example, the person in Transit might have concluded that getting to the appointment at the scheduled time was impossible and have reformulated the problem as that of rescheduling the appointment for a mutually convenient time. Further, defining a problem does not always follow after or lead immediately to an idea of a suggested solution. Nor should it do so, as Dewey himself recognized in describing the physician in Typhoid as avoiding any strong preference for this or that conclusion before getting further information (Dewey 1910: 85; 1933: 170). People with a hypothesis in mind, even one to which they have a very weak commitment, have a so-called “confirmation bias” (Nickerson 1998): they are likely to pay attention to evidence that confirms the hypothesis and to ignore evidence that counts against it or for some competing hypothesis. Detectives, intelligence agencies, and investigators of airplane accidents are well advised to gather relevant evidence systematically and to postpone even tentative adoption of an explanatory hypothesis until the collected evidence rules out with the appropriate degree of certainty all but one explanation. Dewey’s analysis of the critical thinking process can be faulted as well for requiring acceptance or rejection of a possible solution to a defined problem, with no allowance for deciding in the light of the available evidence to suspend judgment. Further, given the great variety of kinds of problems for which reflection is appropriate, there is likely to be variation in its component events. Perhaps the best way to conceptualize the critical thinking process is as a checklist whose component events can occur in a variety of orders, selectively, and more than once. These component events might include (1) noticing a difficulty, (2) defining the problem, (3) dividing the problem into manageable sub-problems, (4) formulating a variety of possible solutions to the problem or sub-problem, (5) determining what evidence is relevant to deciding among possible solutions to the problem or sub-problem, (6) devising a plan of systematic observation or experiment that will uncover the relevant evidence, (7) carrying out the plan of systematic observation or experimentation, (8) noting the results of the systematic observation or experiment, (9) gathering relevant testimony and information from others, (10) judging the credibility of testimony and information gathered from others, (11) drawing conclusions from gathered evidence and accepted testimony, and (12) accepting a solution that the evidence adequately supports (cf. Hitchcock 2017: 485).

Checklist conceptions of the process of critical thinking are open to the objection that they are too mechanical and procedural to fit the multi-dimensional and emotionally charged issues for which critical thinking is urgently needed (Paul 1984). For such issues, a more dialectical process is advocated, in which competing relevant world views are identified, their implications explored, and some sort of creative synthesis attempted.

If one considers the critical thinking process illustrated by the 11 examples, one can identify distinct kinds of mental acts and mental states that form part of it. To distinguish, label and briefly characterize these components is a useful preliminary to identifying abilities, skills, dispositions, attitudes, habits and the like that contribute causally to thinking critically. Identifying such abilities and habits is in turn a useful preliminary to setting educational goals. Setting the goals is in its turn a useful preliminary to designing strategies for helping learners to achieve the goals and to designing ways of measuring the extent to which learners have done so. Such measures provide both feedback to learners on their achievement and a basis for experimental research on the effectiveness of various strategies for educating people to think critically. Let us begin, then, by distinguishing the kinds of mental acts and mental events that can occur in a critical thinking process.

• Observing : One notices something in one’s immediate environment (sudden cooling of temperature in Weather , bubbles forming outside a glass and then going inside in Bubbles , a moving blur in the distance in Blur , a rash in Rash ). Or one notes the results of an experiment or systematic observation (valuables missing in Disorder , no suction without air pressure in Suction pump )
• Feeling : One feels puzzled or uncertain about something (how to get to an appointment on time in Transit , why the diamonds vary in spacing in Diamond ). One wants to resolve this perplexity. One feels satisfaction once one has worked out an answer (to take the subway express in Transit , diamonds closer when needed as a warning in Diamond ).
• Wondering : One formulates a question to be addressed (why bubbles form outside a tumbler taken from hot water in Bubbles , how suction pumps work in Suction pump , what caused the rash in Rash ).
• Imagining : One thinks of possible answers (bus or subway or elevated in Transit , flagpole or ornament or wireless communication aid or direction indicator in Ferryboat , allergic reaction or heat rash in Rash ).
• Inferring : One works out what would be the case if a possible answer were assumed (valuables missing if there has been a burglary in Disorder , earlier start to the rash if it is an allergic reaction to a sulfa drug in Rash ). Or one draws a conclusion once sufficient relevant evidence is gathered (take the subway in Transit , burglary in Disorder , discontinue blood pressure medication and new cream in Rash ).
• Knowledge : One uses stored knowledge of the subject-matter to generate possible answers or to infer what would be expected on the assumption of a particular answer (knowledge of a city’s public transit system in Transit , of the requirements for a flagpole in Ferryboat , of Boyle’s law in Bubbles , of allergic reactions in Rash ).
• Experimenting : One designs and carries out an experiment or a systematic observation to find out whether the results deduced from a possible answer will occur (looking at the location of the flagpole in relation to the pilot’s position in Ferryboat , putting an ice cube on top of a tumbler taken from hot water in Bubbles , measuring the height to which a suction pump will draw water at different elevations in Suction pump , noticing the spacing of diamonds when movement to or from a diamond lane is allowed in Diamond ).
• Consulting : One finds a source of information, gets the information from the source, and makes a judgment on whether to accept it. None of our 11 examples include searching for sources of information. In this respect they are unrepresentative, since most people nowadays have almost instant access to information relevant to answering any question, including many of those illustrated by the examples. However, Candidate includes the activities of extracting information from sources and evaluating its credibility.
• Identifying and analyzing arguments : One notices an argument and works out its structure and content as a preliminary to evaluating its strength. This activity is central to Candidate . It is an important part of a critical thinking process in which one surveys arguments for various positions on an issue.
• Judging : One makes a judgment on the basis of accumulated evidence and reasoning, such as the judgment in Ferryboat that the purpose of the pole is to provide direction to the pilot.
• Deciding : One makes a decision on what to do or on what policy to adopt, as in the decision in Transit to take the subway.

By definition, a person who does something voluntarily is both willing and able to do that thing at that time. Both the willingness and the ability contribute causally to the person’s action, in the sense that the voluntary action would not occur if either (or both) of these were lacking. For example, suppose that one is standing with one’s arms at one’s sides and one voluntarily lifts one’s right arm to an extended horizontal position. One would not do so if one were unable to lift one’s arm, if for example one’s right side was paralyzed as the result of a stroke. Nor would one do so if one were unwilling to lift one’s arm, if for example one were participating in a street demonstration at which a white supremacist was urging the crowd to lift their right arm in a Nazi salute and one were unwilling to express support in this way for the racist Nazi ideology. The same analysis applies to a voluntary mental process of thinking critically. It requires both willingness and ability to think critically, including willingness and ability to perform each of the mental acts that compose the process and to coordinate those acts in a sequence that is directed at resolving the initiating perplexity.

Consider willingness first. We can identify causal contributors to willingness to think critically by considering factors that would cause a person who was able to think critically about an issue nevertheless not to do so (Hamby 2014). For each factor, the opposite condition thus contributes causally to willingness to think critically on a particular occasion. For example, people who habitually jump to conclusions without considering alternatives will not think critically about issues that arise, even if they have the required abilities. The contrary condition of willingness to suspend judgment is thus a causal contributor to thinking critically.

Now consider ability. In contrast to the ability to move one’s arm, which can be completely absent because a stroke has left the arm paralyzed, the ability to think critically is a developed ability, whose absence is not a complete absence of ability to think but absence of ability to think well. We can identify the ability to think well directly, in terms of the norms and standards for good thinking. In general, to be able do well the thinking activities that can be components of a critical thinking process, one needs to know the concepts and principles that characterize their good performance, to recognize in particular cases that the concepts and principles apply, and to apply them. The knowledge, recognition and application may be procedural rather than declarative. It may be domain-specific rather than widely applicable, and in either case may need subject-matter knowledge, sometimes of a deep kind.

Reflections of the sort illustrated by the previous two paragraphs have led scholars to identify the knowledge, abilities and dispositions of a “critical thinker”, i.e., someone who thinks critically whenever it is appropriate to do so. We turn now to these three types of causal contributors to thinking critically. We start with dispositions, since arguably these are the most powerful contributors to being a critical thinker, can be fostered at an early stage of a child’s development, and are susceptible to general improvement (Glaser 1941: 175)

8. Critical Thinking Dispositions

Educational researchers use the term ‘dispositions’ broadly for the habits of mind and attitudes that contribute causally to being a critical thinker. Some writers (e.g., Paul & Elder 2006; Hamby 2014; Bailin & Battersby 2016a) propose to use the term ‘virtues’ for this dimension of a critical thinker. The virtues in question, although they are virtues of character, concern the person’s ways of thinking rather than the person’s ways of behaving towards others. They are not moral virtues but intellectual virtues, of the sort articulated by Zagzebski (1996) and discussed by Turri, Alfano, and Greco (2017).

On a realistic conception, thinking dispositions or intellectual virtues are real properties of thinkers. They are general tendencies, propensities, or inclinations to think in particular ways in particular circumstances, and can be genuinely explanatory (Siegel 1999). Sceptics argue that there is no evidence for a specific mental basis for the habits of mind that contribute to thinking critically, and that it is pedagogically misleading to posit such a basis (Bailin et al. 1999a). Whatever their status, critical thinking dispositions need motivation for their initial formation in a child—motivation that may be external or internal. As children develop, the force of habit will gradually become important in sustaining the disposition (Nieto & Valenzuela 2012). Mere force of habit, however, is unlikely to sustain critical thinking dispositions. Critical thinkers must value and enjoy using their knowledge and abilities to think things through for themselves. They must be committed to, and lovers of, inquiry.

A person may have a critical thinking disposition with respect to only some kinds of issues. For example, one could be open-minded about scientific issues but not about religious issues. Similarly, one could be confident in one’s ability to reason about the theological implications of the existence of evil in the world but not in one’s ability to reason about the best design for a guided ballistic missile.

Facione (1990a: 25) divides “affective dispositions” of critical thinking into approaches to life and living in general and approaches to specific issues, questions or problems. Adapting this distinction, one can usefully divide critical thinking dispositions into initiating dispositions (those that contribute causally to starting to think critically about an issue) and internal dispositions (those that contribute causally to doing a good job of thinking critically once one has started). The two categories are not mutually exclusive. For example, open-mindedness, in the sense of willingness to consider alternative points of view to one’s own, is both an initiating and an internal disposition.

Using the strategy of considering factors that would block people with the ability to think critically from doing so, we can identify as initiating dispositions for thinking critically attentiveness, a habit of inquiry, self-confidence, courage, open-mindedness, willingness to suspend judgment, trust in reason, wanting evidence for one’s beliefs, and seeking the truth. We consider briefly what each of these dispositions amounts to, in each case citing sources that acknowledge them.

• Attentiveness : One will not think critically if one fails to recognize an issue that needs to be thought through. For example, the pedestrian in Weather would not have looked up if he had not noticed that the air was suddenly cooler. To be a critical thinker, then, one needs to be habitually attentive to one’s surroundings, noticing not only what one senses but also sources of perplexity in messages received and in one’s own beliefs and attitudes (Facione 1990a: 25; Facione, Facione, & Giancarlo 2001).
• Habit of inquiry : Inquiry is effortful, and one needs an internal push to engage in it. For example, the student in Bubbles could easily have stopped at idle wondering about the cause of the bubbles rather than reasoning to a hypothesis, then designing and executing an experiment to test it. Thus willingness to think critically needs mental energy and initiative. What can supply that energy? Love of inquiry, or perhaps just a habit of inquiry. Hamby (2015) has argued that willingness to inquire is the central critical thinking virtue, one that encompasses all the others. It is recognized as a critical thinking disposition by Dewey (1910: 29; 1933: 35), Glaser (1941: 5), Ennis (1987: 12; 1991: 8), Facione (1990a: 25), Bailin et al. (1999b: 294), Halpern (1998: 452), and Facione, Facione, & Giancarlo (2001).
• Self-confidence : Lack of confidence in one’s abilities can block critical thinking. For example, if the woman in Rash lacked confidence in her ability to figure things out for herself, she might just have assumed that the rash on her chest was the allergic reaction to her medication against which the pharmacist had warned her. Thus willingness to think critically requires confidence in one’s ability to inquire (Facione 1990a: 25; Facione, Facione, & Giancarlo 2001).
• Courage : Fear of thinking for oneself can stop one from doing it. Thus willingness to think critically requires intellectual courage (Paul & Elder 2006: 16).
• Open-mindedness : A dogmatic attitude will impede thinking critically. For example, a person who adheres rigidly to a “pro-choice” position on the issue of the legal status of induced abortion is likely to be unwilling to consider seriously the issue of when in its development an unborn child acquires a moral right to life. Thus willingness to think critically requires open-mindedness, in the sense of a willingness to examine questions to which one already accepts an answer but which further evidence or reasoning might cause one to answer differently (Dewey 1933; Facione 1990a; Ennis 1991; Bailin et al. 1999b; Halpern 1998, Facione, Facione, & Giancarlo 2001). Paul (1981) emphasizes open-mindedness about alternative world-views, and recommends a dialectical approach to integrating such views as central to what he calls “strong sense” critical thinking. In three studies, Haran, Ritov, & Mellers (2013) found that actively open-minded thinking, including “the tendency to weigh new evidence against a favored belief, to spend sufficient time on a problem before giving up, and to consider carefully the opinions of others in forming one’s own”, led study participants to acquire information and thus to make accurate estimations.
• Willingness to suspend judgment : Premature closure on an initial solution will block critical thinking. Thus willingness to think critically requires a willingness to suspend judgment while alternatives are explored (Facione 1990a; Ennis 1991; Halpern 1998).
• Trust in reason : Since distrust in the processes of reasoned inquiry will dissuade one from engaging in it, trust in them is an initiating critical thinking disposition (Facione 1990a, 25; Bailin et al. 1999b: 294; Facione, Facione, & Giancarlo 2001; Paul & Elder 2006). In reaction to an allegedly exclusive emphasis on reason in critical thinking theory and pedagogy, Thayer-Bacon (2000) argues that intuition, imagination, and emotion have important roles to play in an adequate conception of critical thinking that she calls “constructive thinking”. From her point of view, critical thinking requires trust not only in reason but also in intuition, imagination, and emotion.
• Seeking the truth : If one does not care about the truth but is content to stick with one’s initial bias on an issue, then one will not think critically about it. Seeking the truth is thus an initiating critical thinking disposition (Bailin et al. 1999b: 294; Facione, Facione, & Giancarlo 2001). A disposition to seek the truth is implicit in more specific critical thinking dispositions, such as trying to be well-informed, considering seriously points of view other than one’s own, looking for alternatives, suspending judgment when the evidence is insufficient, and adopting a position when the evidence supporting it is sufficient.

Some of the initiating dispositions, such as open-mindedness and willingness to suspend judgment, are also internal critical thinking dispositions, in the sense of mental habits or attitudes that contribute causally to doing a good job of critical thinking once one starts the process. But there are many other internal critical thinking dispositions. Some of them are parasitic on one’s conception of good thinking. For example, it is constitutive of good thinking about an issue to formulate the issue clearly and to maintain focus on it. For this purpose, one needs not only the corresponding ability but also the corresponding disposition. Ennis (1991: 8) describes it as the disposition “to determine and maintain focus on the conclusion or question”, Facione (1990a: 25) as “clarity in stating the question or concern”. Other internal dispositions are motivators to continue or adjust the critical thinking process, such as willingness to persist in a complex task and willingness to abandon nonproductive strategies in an attempt to self-correct (Halpern 1998: 452). For a list of identified internal critical thinking dispositions, see the Supplement on Internal Critical Thinking Dispositions .

Some theorists postulate skills, i.e., acquired abilities, as operative in critical thinking. It is not obvious, however, that a good mental act is the exercise of a generic acquired skill. Inferring an expected time of arrival, as in Transit , has some generic components but also uses non-generic subject-matter knowledge. Bailin et al. (1999a) argue against viewing critical thinking skills as generic and discrete, on the ground that skilled performance at a critical thinking task cannot be separated from knowledge of concepts and from domain-specific principles of good thinking. Talk of skills, they concede, is unproblematic if it means merely that a person with critical thinking skills is capable of intelligent performance.

Despite such scepticism, theorists of critical thinking have listed as general contributors to critical thinking what they variously call abilities (Glaser 1941; Ennis 1962, 1991), skills (Facione 1990a; Halpern 1998) or competencies (Fisher & Scriven 1997). Amalgamating these lists would produce a confusing and chaotic cornucopia of more than 50 possible educational objectives, with only partial overlap among them. It makes sense instead to try to understand the reasons for the multiplicity and diversity, and to make a selection according to one’s own reasons for singling out abilities to be developed in a critical thinking curriculum. Two reasons for diversity among lists of critical thinking abilities are the underlying conception of critical thinking and the envisaged educational level. Appraisal-only conceptions, for example, involve a different suite of abilities than constructive-only conceptions. Some lists, such as those in (Glaser 1941), are put forward as educational objectives for secondary school students, whereas others are proposed as objectives for college students (e.g., Facione 1990a).

The abilities described in the remaining paragraphs of this section emerge from reflection on the general abilities needed to do well the thinking activities identified in section 6 as components of the critical thinking process described in section 5 . The derivation of each collection of abilities is accompanied by citation of sources that list such abilities and of standardized tests that claim to test them.

Observational abilities : Careful and accurate observation sometimes requires specialist expertise and practice, as in the case of observing birds and observing accident scenes. However, there are general abilities of noticing what one’s senses are picking up from one’s environment and of being able to articulate clearly and accurately to oneself and others what one has observed. It helps in exercising them to be able to recognize and take into account factors that make one’s observation less trustworthy, such as prior framing of the situation, inadequate time, deficient senses, poor observation conditions, and the like. It helps as well to be skilled at taking steps to make one’s observation more trustworthy, such as moving closer to get a better look, measuring something three times and taking the average, and checking what one thinks one is observing with someone else who is in a good position to observe it. It also helps to be skilled at recognizing respects in which one’s report of one’s observation involves inference rather than direct observation, so that one can then consider whether the inference is justified. These abilities come into play as well when one thinks about whether and with what degree of confidence to accept an observation report, for example in the study of history or in a criminal investigation or in assessing news reports. Observational abilities show up in some lists of critical thinking abilities (Ennis 1962: 90; Facione 1990a: 16; Ennis 1991: 9). There are items testing a person’s ability to judge the credibility of observation reports in the Cornell Critical Thinking Tests, Levels X and Z (Ennis & Millman 1971; Ennis, Millman, & Tomko 1985, 2005). Norris and King (1983, 1985, 1990a, 1990b) is a test of ability to appraise observation reports.

Emotional abilities : The emotions that drive a critical thinking process are perplexity or puzzlement, a wish to resolve it, and satisfaction at achieving the desired resolution. Children experience these emotions at an early age, without being trained to do so. Education that takes critical thinking as a goal needs only to channel these emotions and to make sure not to stifle them. Collaborative critical thinking benefits from ability to recognize one’s own and others’ emotional commitments and reactions.

Questioning abilities : A critical thinking process needs transformation of an inchoate sense of perplexity into a clear question. Formulating a question well requires not building in questionable assumptions, not prejudging the issue, and using language that in context is unambiguous and precise enough (Ennis 1962: 97; 1991: 9).

Imaginative abilities : Thinking directed at finding the correct causal explanation of a general phenomenon or particular event requires an ability to imagine possible explanations. Thinking about what policy or plan of action to adopt requires generation of options and consideration of possible consequences of each option. Domain knowledge is required for such creative activity, but a general ability to imagine alternatives is helpful and can be nurtured so as to become easier, quicker, more extensive, and deeper (Dewey 1910: 34–39; 1933: 40–47). Facione (1990a) and Halpern (1998) include the ability to imagine alternatives as a critical thinking ability.

Inferential abilities : The ability to draw conclusions from given information, and to recognize with what degree of certainty one’s own or others’ conclusions follow, is universally recognized as a general critical thinking ability. All 11 examples in section 2 of this article include inferences, some from hypotheses or options (as in Transit , Ferryboat and Disorder ), others from something observed (as in Weather and Rash ). None of these inferences is formally valid. Rather, they are licensed by general, sometimes qualified substantive rules of inference (Toulmin 1958) that rest on domain knowledge—that a bus trip takes about the same time in each direction, that the terminal of a wireless telegraph would be located on the highest possible place, that sudden cooling is often followed by rain, that an allergic reaction to a sulfa drug generally shows up soon after one starts taking it. It is a matter of controversy to what extent the specialized ability to deduce conclusions from premisses using formal rules of inference is needed for critical thinking. Dewey (1933) locates logical forms in setting out the products of reflection rather than in the process of reflection. Ennis (1981a), on the other hand, maintains that a liberally-educated person should have the following abilities: to translate natural-language statements into statements using the standard logical operators, to use appropriately the language of necessary and sufficient conditions, to deal with argument forms and arguments containing symbols, to determine whether in virtue of an argument’s form its conclusion follows necessarily from its premisses, to reason with logically complex propositions, and to apply the rules and procedures of deductive logic. Inferential abilities are recognized as critical thinking abilities by Glaser (1941: 6), Facione (1990a: 9), Ennis (1991: 9), Fisher & Scriven (1997: 99, 111), and Halpern (1998: 452). Items testing inferential abilities constitute two of the five subtests of the Watson Glaser Critical Thinking Appraisal (Watson & Glaser 1980a, 1980b, 1994), two of the four sections in the Cornell Critical Thinking Test Level X (Ennis & Millman 1971; Ennis, Millman, & Tomko 1985, 2005), three of the seven sections in the Cornell Critical Thinking Test Level Z (Ennis & Millman 1971; Ennis, Millman, & Tomko 1985, 2005), 11 of the 34 items on Forms A and B of the California Critical Thinking Skills Test (Facione 1990b, 1992), and a high but variable proportion of the 25 selected-response questions in the Collegiate Learning Assessment (Council for Aid to Education 2017).

Experimenting abilities : Knowing how to design and execute an experiment is important not just in scientific research but also in everyday life, as in Rash . Dewey devoted a whole chapter of his How We Think (1910: 145–156; 1933: 190–202) to the superiority of experimentation over observation in advancing knowledge. Experimenting abilities come into play at one remove in appraising reports of scientific studies. Skill in designing and executing experiments includes the acknowledged abilities to appraise evidence (Glaser 1941: 6), to carry out experiments and to apply appropriate statistical inference techniques (Facione 1990a: 9), to judge inductions to an explanatory hypothesis (Ennis 1991: 9), and to recognize the need for an adequately large sample size (Halpern 1998). The Cornell Critical Thinking Test Level Z (Ennis & Millman 1971; Ennis, Millman, & Tomko 1985, 2005) includes four items (out of 52) on experimental design. The Collegiate Learning Assessment (Council for Aid to Education 2017) makes room for appraisal of study design in both its performance task and its selected-response questions.

Consulting abilities : Skill at consulting sources of information comes into play when one seeks information to help resolve a problem, as in Candidate . Ability to find and appraise information includes ability to gather and marshal pertinent information (Glaser 1941: 6), to judge whether a statement made by an alleged authority is acceptable (Ennis 1962: 84), to plan a search for desired information (Facione 1990a: 9), and to judge the credibility of a source (Ennis 1991: 9). Ability to judge the credibility of statements is tested by 24 items (out of 76) in the Cornell Critical Thinking Test Level X (Ennis & Millman 1971; Ennis, Millman, & Tomko 1985, 2005) and by four items (out of 52) in the Cornell Critical Thinking Test Level Z (Ennis & Millman 1971; Ennis, Millman, & Tomko 1985, 2005). The College Learning Assessment’s performance task requires evaluation of whether information in documents is credible or unreliable (Council for Aid to Education 2017).

Argument analysis abilities : The ability to identify and analyze arguments contributes to the process of surveying arguments on an issue in order to form one’s own reasoned judgment, as in Candidate . The ability to detect and analyze arguments is recognized as a critical thinking skill by Facione (1990a: 7–8), Ennis (1991: 9) and Halpern (1998). Five items (out of 34) on the California Critical Thinking Skills Test (Facione 1990b, 1992) test skill at argument analysis. The College Learning Assessment (Council for Aid to Education 2017) incorporates argument analysis in its selected-response tests of critical reading and evaluation and of critiquing an argument.

Judging skills and deciding skills : Skill at judging and deciding is skill at recognizing what judgment or decision the available evidence and argument supports, and with what degree of confidence. It is thus a component of the inferential skills already discussed.

Lists and tests of critical thinking abilities often include two more abilities: identifying assumptions and constructing and evaluating definitions.

In addition to dispositions and abilities, critical thinking needs knowledge: of critical thinking concepts, of critical thinking principles, and of the subject-matter of the thinking.

We can derive a short list of concepts whose understanding contributes to critical thinking from the critical thinking abilities described in the preceding section. Observational abilities require an understanding of the difference between observation and inference. Questioning abilities require an understanding of the concepts of ambiguity and vagueness. Inferential abilities require an understanding of the difference between conclusive and defeasible inference (traditionally, between deduction and induction), as well as of the difference between necessary and sufficient conditions. Experimenting abilities require an understanding of the concepts of hypothesis, null hypothesis, assumption and prediction, as well as of the concept of statistical significance and of its difference from importance. They also require an understanding of the difference between an experiment and an observational study, and in particular of the difference between a randomized controlled trial, a prospective correlational study and a retrospective (case-control) study. Argument analysis abilities require an understanding of the concepts of argument, premiss, assumption, conclusion and counter-consideration. Additional critical thinking concepts are proposed by Bailin et al. (1999b: 293), Fisher & Scriven (1997: 105–106), Black (2012), and Blair (2021).

According to Glaser (1941: 25), ability to think critically requires knowledge of the methods of logical inquiry and reasoning. If we review the list of abilities in the preceding section, however, we can see that some of them can be acquired and exercised merely through practice, possibly guided in an educational setting, followed by feedback. Searching intelligently for a causal explanation of some phenomenon or event requires that one consider a full range of possible causal contributors, but it seems more important that one implements this principle in one’s practice than that one is able to articulate it. What is important is “operational knowledge” of the standards and principles of good thinking (Bailin et al. 1999b: 291–293). But the development of such critical thinking abilities as designing an experiment or constructing an operational definition can benefit from learning their underlying theory. Further, explicit knowledge of quirks of human thinking seems useful as a cautionary guide. Human memory is not just fallible about details, as people learn from their own experiences of misremembering, but is so malleable that a detailed, clear and vivid recollection of an event can be a total fabrication (Loftus 2017). People seek or interpret evidence in ways that are partial to their existing beliefs and expectations, often unconscious of their “confirmation bias” (Nickerson 1998). Not only are people subject to this and other cognitive biases (Kahneman 2011), of which they are typically unaware, but it may be counter-productive for one to make oneself aware of them and try consciously to counteract them or to counteract social biases such as racial or sexual stereotypes (Kenyon & Beaulac 2014). It is helpful to be aware of these facts and of the superior effectiveness of blocking the operation of biases—for example, by making an immediate record of one’s observations, refraining from forming a preliminary explanatory hypothesis, blind refereeing, double-blind randomized trials, and blind grading of students’ work. It is also helpful to be aware of the prevalence of “noise” (unwanted unsystematic variability of judgments), of how to detect noise (through a noise audit), and of how to reduce noise: make accuracy the goal, think statistically, break a process of arriving at a judgment into independent tasks, resist premature intuitions, in a group get independent judgments first, favour comparative judgments and scales (Kahneman, Sibony, & Sunstein 2021). It is helpful as well to be aware of the concept of “bounded rationality” in decision-making and of the related distinction between “satisficing” and optimizing (Simon 1956; Gigerenzer 2001).

Critical thinking about an issue requires substantive knowledge of the domain to which the issue belongs. Critical thinking abilities are not a magic elixir that can be applied to any issue whatever by somebody who has no knowledge of the facts relevant to exploring that issue. For example, the student in Bubbles needed to know that gases do not penetrate solid objects like a glass, that air expands when heated, that the volume of an enclosed gas varies directly with its temperature and inversely with its pressure, and that hot objects will spontaneously cool down to the ambient temperature of their surroundings unless kept hot by insulation or a source of heat. Critical thinkers thus need a rich fund of subject-matter knowledge relevant to the variety of situations they encounter. This fact is recognized in the inclusion among critical thinking dispositions of a concern to become and remain generally well informed.

Experimental educational interventions, with control groups, have shown that education can improve critical thinking skills and dispositions, as measured by standardized tests. For information about these tests, see the Supplement on Assessment .

What educational methods are most effective at developing the dispositions, abilities and knowledge of a critical thinker? In a comprehensive meta-analysis of experimental and quasi-experimental studies of strategies for teaching students to think critically, Abrami et al. (2015) found that dialogue, anchored instruction, and mentoring each increased the effectiveness of the educational intervention, and that they were most effective when combined. They also found that in these studies a combination of separate instruction in critical thinking with subject-matter instruction in which students are encouraged to think critically was more effective than either by itself. However, the difference was not statistically significant; that is, it might have arisen by chance.

Most of these studies lack the longitudinal follow-up required to determine whether the observed differential improvements in critical thinking abilities or dispositions continue over time, for example until high school or college graduation. For details on studies of methods of developing critical thinking skills and dispositions, see the Supplement on Educational Methods .

12. Controversies

Scholars have denied the generalizability of critical thinking abilities across subject domains, have alleged bias in critical thinking theory and pedagogy, and have investigated the relationship of critical thinking to other kinds of thinking.

McPeck (1981) attacked the thinking skills movement of the 1970s, including the critical thinking movement. He argued that there are no general thinking skills, since thinking is always thinking about some subject-matter. It is futile, he claimed, for schools and colleges to teach thinking as if it were a separate subject. Rather, teachers should lead their pupils to become autonomous thinkers by teaching school subjects in a way that brings out their cognitive structure and that encourages and rewards discussion and argument. As some of his critics (e.g., Paul 1985; Siegel 1985) pointed out, McPeck’s central argument needs elaboration, since it has obvious counter-examples in writing and speaking, for which (up to a certain level of complexity) there are teachable general abilities even though they are always about some subject-matter. To make his argument convincing, McPeck needs to explain how thinking differs from writing and speaking in a way that does not permit useful abstraction of its components from the subject-matters with which it deals. He has not done so. Nevertheless, his position that the dispositions and abilities of a critical thinker are best developed in the context of subject-matter instruction is shared by many theorists of critical thinking, including Dewey (1910, 1933), Glaser (1941), Passmore (1980), Weinstein (1990), Bailin et al. (1999b), and Willingham (2019).

McPeck’s challenge prompted reflection on the extent to which critical thinking is subject-specific. McPeck argued for a strong subject-specificity thesis, according to which it is a conceptual truth that all critical thinking abilities are specific to a subject. (He did not however extend his subject-specificity thesis to critical thinking dispositions. In particular, he took the disposition to suspend judgment in situations of cognitive dissonance to be a general disposition.) Conceptual subject-specificity is subject to obvious counter-examples, such as the general ability to recognize confusion of necessary and sufficient conditions. A more modest thesis, also endorsed by McPeck, is epistemological subject-specificity, according to which the norms of good thinking vary from one field to another. Epistemological subject-specificity clearly holds to a certain extent; for example, the principles in accordance with which one solves a differential equation are quite different from the principles in accordance with which one determines whether a painting is a genuine Picasso. But the thesis suffers, as Ennis (1989) points out, from vagueness of the concept of a field or subject and from the obvious existence of inter-field principles, however broadly the concept of a field is construed. For example, the principles of hypothetico-deductive reasoning hold for all the varied fields in which such reasoning occurs. A third kind of subject-specificity is empirical subject-specificity, according to which as a matter of empirically observable fact a person with the abilities and dispositions of a critical thinker in one area of investigation will not necessarily have them in another area of investigation.

The thesis of empirical subject-specificity raises the general problem of transfer. If critical thinking abilities and dispositions have to be developed independently in each school subject, how are they of any use in dealing with the problems of everyday life and the political and social issues of contemporary society, most of which do not fit into the framework of a traditional school subject? Proponents of empirical subject-specificity tend to argue that transfer is more likely to occur if there is critical thinking instruction in a variety of domains, with explicit attention to dispositions and abilities that cut across domains. But evidence for this claim is scanty. There is a need for well-designed empirical studies that investigate the conditions that make transfer more likely.

It is common ground in debates about the generality or subject-specificity of critical thinking dispositions and abilities that critical thinking about any topic requires background knowledge about the topic. For example, the most sophisticated understanding of the principles of hypothetico-deductive reasoning is of no help unless accompanied by some knowledge of what might be plausible explanations of some phenomenon under investigation.

Critics have objected to bias in the theory, pedagogy and practice of critical thinking. Commentators (e.g., Alston 1995; Ennis 1998) have noted that anyone who takes a position has a bias in the neutral sense of being inclined in one direction rather than others. The critics, however, are objecting to bias in the pejorative sense of an unjustified favoring of certain ways of knowing over others, frequently alleging that the unjustly favoured ways are those of a dominant sex or culture (Bailin 1995). These ways favour:

• reinforcement of egocentric and sociocentric biases over dialectical engagement with opposing world-views (Paul 1981, 1984; Warren 1998)
• distancing from the object of inquiry over closeness to it (Martin 1992; Thayer-Bacon 1992)
• indifference to the situation of others over care for them (Martin 1992)
• orientation to thought over orientation to action (Martin 1992)
• being reasonable over caring to understand people’s ideas (Thayer-Bacon 1993)
• being neutral and objective over being embodied and situated (Thayer-Bacon 1995a)
• doubting over believing (Thayer-Bacon 1995b)
• reason over emotion, imagination and intuition (Thayer-Bacon 2000)
• solitary thinking over collaborative thinking (Thayer-Bacon 2000)
• written and spoken assignments over other forms of expression (Alston 2001)
• attention to written and spoken communications over attention to human problems (Alston 2001)
• winning debates in the public sphere over making and understanding meaning (Alston 2001)

A common thread in this smorgasbord of accusations is dissatisfaction with focusing on the logical analysis and evaluation of reasoning and arguments. While these authors acknowledge that such analysis and evaluation is part of critical thinking and should be part of its conceptualization and pedagogy, they insist that it is only a part. Paul (1981), for example, bemoans the tendency of atomistic teaching of methods of analyzing and evaluating arguments to turn students into more able sophists, adept at finding fault with positions and arguments with which they disagree but even more entrenched in the egocentric and sociocentric biases with which they began. Martin (1992) and Thayer-Bacon (1992) cite with approval the self-reported intimacy with their subject-matter of leading researchers in biology and medicine, an intimacy that conflicts with the distancing allegedly recommended in standard conceptions and pedagogy of critical thinking. Thayer-Bacon (2000) contrasts the embodied and socially embedded learning of her elementary school students in a Montessori school, who used their imagination, intuition and emotions as well as their reason, with conceptions of critical thinking as

thinking that is used to critique arguments, offer justifications, and make judgments about what are the good reasons, or the right answers. (Thayer-Bacon 2000: 127–128)

Alston (2001) reports that her students in a women’s studies class were able to see the flaws in the Cinderella myth that pervades much romantic fiction but in their own romantic relationships still acted as if all failures were the woman’s fault and still accepted the notions of love at first sight and living happily ever after. Students, she writes, should

be able to connect their intellectual critique to a more affective, somatic, and ethical account of making risky choices that have sexist, racist, classist, familial, sexual, or other consequences for themselves and those both near and far… critical thinking that reads arguments, texts, or practices merely on the surface without connections to feeling/desiring/doing or action lacks an ethical depth that should infuse the difference between mere cognitive activity and something we want to call critical thinking. (Alston 2001: 34)

Some critics portray such biases as unfair to women. Thayer-Bacon (1992), for example, has charged modern critical thinking theory with being sexist, on the ground that it separates the self from the object and causes one to lose touch with one’s inner voice, and thus stigmatizes women, who (she asserts) link self to object and listen to their inner voice. Her charge does not imply that women as a group are on average less able than men to analyze and evaluate arguments. Facione (1990c) found no difference by sex in performance on his California Critical Thinking Skills Test. Kuhn (1991: 280–281) found no difference by sex in either the disposition or the competence to engage in argumentative thinking.

The critics propose a variety of remedies for the biases that they allege. In general, they do not propose to eliminate or downplay critical thinking as an educational goal. Rather, they propose to conceptualize critical thinking differently and to change its pedagogy accordingly. Their pedagogical proposals arise logically from their objections. They can be summarized as follows:

• Focus on argument networks with dialectical exchanges reflecting contesting points of view rather than on atomic arguments, so as to develop “strong sense” critical thinking that transcends egocentric and sociocentric biases (Paul 1981, 1984).
• Foster closeness to the subject-matter and feeling connected to others in order to inform a humane democracy (Martin 1992).
• Develop “constructive thinking” as a social activity in a community of physically embodied and socially embedded inquirers with personal voices who value not only reason but also imagination, intuition and emotion (Thayer-Bacon 2000).
• In developing critical thinking in school subjects, treat as important neither skills nor dispositions but opening worlds of meaning (Alston 2001).
• Attend to the development of critical thinking dispositions as well as skills, and adopt the “critical pedagogy” practised and advocated by Freire (1968 [1970]) and hooks (1994) (Dalgleish, Girard, & Davies 2017).

A common thread in these proposals is treatment of critical thinking as a social, interactive, personally engaged activity like that of a quilting bee or a barn-raising (Thayer-Bacon 2000) rather than as an individual, solitary, distanced activity symbolized by Rodin’s The Thinker . One can get a vivid description of education with the former type of goal from the writings of bell hooks (1994, 2010). Critical thinking for her is open-minded dialectical exchange across opposing standpoints and from multiple perspectives, a conception similar to Paul’s “strong sense” critical thinking (Paul 1981). She abandons the structure of domination in the traditional classroom. In an introductory course on black women writers, for example, she assigns students to write an autobiographical paragraph about an early racial memory, then to read it aloud as the others listen, thus affirming the uniqueness and value of each voice and creating a communal awareness of the diversity of the group’s experiences (hooks 1994: 84). Her “engaged pedagogy” is thus similar to the “freedom under guidance” implemented in John Dewey’s Laboratory School of Chicago in the late 1890s and early 1900s. It incorporates the dialogue, anchored instruction, and mentoring that Abrami (2015) found to be most effective in improving critical thinking skills and dispositions.

What is the relationship of critical thinking to problem solving, decision-making, higher-order thinking, creative thinking, and other recognized types of thinking? One’s answer to this question obviously depends on how one defines the terms used in the question. If critical thinking is conceived broadly to cover any careful thinking about any topic for any purpose, then problem solving and decision making will be kinds of critical thinking, if they are done carefully. Historically, ‘critical thinking’ and ‘problem solving’ were two names for the same thing. If critical thinking is conceived more narrowly as consisting solely of appraisal of intellectual products, then it will be disjoint with problem solving and decision making, which are constructive.

Bloom’s taxonomy of educational objectives used the phrase “intellectual abilities and skills” for what had been labeled “critical thinking” by some, “reflective thinking” by Dewey and others, and “problem solving” by still others (Bloom et al. 1956: 38). Thus, the so-called “higher-order thinking skills” at the taxonomy’s top levels of analysis, synthesis and evaluation are just critical thinking skills, although they do not come with general criteria for their assessment (Ennis 1981b). The revised version of Bloom’s taxonomy (Anderson et al. 2001) likewise treats critical thinking as cutting across those types of cognitive process that involve more than remembering (Anderson et al. 2001: 269–270). For details, see the Supplement on History .

As to creative thinking, it overlaps with critical thinking (Bailin 1987, 1988). Thinking about the explanation of some phenomenon or event, as in Ferryboat , requires creative imagination in constructing plausible explanatory hypotheses. Likewise, thinking about a policy question, as in Candidate , requires creativity in coming up with options. Conversely, creativity in any field needs to be balanced by critical appraisal of the draft painting or novel or mathematical theory.

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4 Sentential Logic

This chapter introduces a logical language called SL. It is a version of sentential logic , because the basic units of the language will represent entire sentences.

I. Sentence letters

In SL, capital letters are used to represent basic sentences. Considered only as a symbol of SL, the letter A could mean any sentence. So when translating from English into SL, it is important to provide a symbolization key . The key provides an English language sentence for each sentence letter used in the symbolization.

For example, consider this argument:

There is an apple on the desk.

If there is an apple on the desk, then Jenny made it to class.

. ˙ . Jenny made it to class.

This is obviously a valid argument in English. In symbolizing it, we want to preserve the structure of the argument that makes it valid. What happens if we replace each sentence with a letter? Our symbolization key would look like this:

A: There is an apple on the desk.

B: If there is an apple on the desk, then Jenny made it to class.

C: Jenny made it to class.

We would then symbolize the argument in this way:

There is no necessary connection between some sentence A , which could be any sentence, and some other sentences B and C , which could be any sentences. The structure of the argument has been completely lost in this translation.

The important thing about the argument is that the second premise is not merely any sentence, logically divorced from the other sentences in the argument. The second premise contains the first premise and the conclusion as parts . Our symbolization key for the argument only needs to include meanings for A and C , and we can build the second premise from those pieces. So we symbolize the argument this way:

This preserves the structure of the argument that makes it valid, but it still makes use of the English expression ‘If . . . then . . . .’ Although we ultimately want to replace all of the English expressions with logical notation, this is a good start.

The sentences that can be symbolized with sentence letters are called atomic sentences , because they are the basic building blocks out of which more complex sentences can be built. Whatever logical structure a sentence might have is lost when it is translated as an atomic sentence. From the point of view of SL, the sentence is just a letter. It can be used to build more complex sentences, but it cannot be taken apart.

Keep in mind that each of these is a different sentence letter. When there are subscripts in the symbolization key, it is important to keep track of them.

II. Connectives 

Logical connectives are used to build complex sentences from atomic components. There are five logical connectives in SL. This table summarizes them, and they are explained below.

Consider how we might symbolize these sentences:

• Mary is in Barcelona.
• Mary is not in Barcelona.
• Mary is somewhere besides Barcelona.

In order to symbolize sentence 1, we will need one sentence letter. We can provide a symbolization key:

B: Mary is in Barcelona.

Note that here we are giving B a different interpretation than we did in the previous section. The symbolization key only specifies what B means in a specific context . It is vital that we continue to use this meaning of B so long as we are talking about Mary and Barcelona. Later, when we are symbolizing different sentences, we can write a new symbolization key and use B to mean something else.

Now, sentence 1 is simply B .

Since sentence 2 is obviously related to the sentence 1, we do not want to introduce a different sentence letter. To put it partly in English, the sentence means ‘Not B .’ In order to symbolize this, we need a symbol for logical negation. We will use ‘¬.’ Now we can translate ‘Not B ’ to ¬ B .

Sentence 3 is about whether or not Mary is in Barcelona, but it does not contain the word ‘not.’ Nevertheless, it is obviously logically equivalent to sentence 2.

They both mean: It is not the case that Mary is in Barcelona. As such, we can translate both sentence 2 and sentence 3 as ¬ B .

Consider these further examples:

• The widget can be replaced if it breaks.
• The widget is irreplaceable.
• The widget is not irreplaceable.

If we let R mean ‘The widget is replaceable’, then sentence 4 can be translated as R .

What about sentence 5? Saying the widget is irreplaceable means that it is not the case that the widget is replaceable. So even though sentence 5 is not negative in English, we symbolize it using negation as ¬ R .

Sentence 6 can be paraphrased as ‘It is not the case that the widget is irreplaceable.’ Using negation twice, we translate this as ¬¬ R . The two negations in a row each work as negations, so the sentence means ‘It is not the case that . . . it is not the case that . . . R .’ If you think about the sentence in English, it is logically equivalent to sentence 4. So when we define logical equivalence in SL, we will make sure that R and ¬¬ R are logically equivalent.

More examples:

• Elliott is happy.
• Elliott is unhappy.

If we let H mean ‘Elliot is happy’, then we can symbolize sentence 7 as H .

However, it would be a mistake to symbolize sentence 8 as ¬ H . If Elliott is unhappy, then he is not happy— but sentence 8 does not mean the same thing as ‘It is not the case that Elliott is happy.’ It could be that he is not happy but that he is not unhappy either. Perhaps he is somewhere between the two. In order to allow for the possibility that he is indifferent, we would need a new sentence letter to symbolize sentence 8.

For any sentence A : If A is true, then ¬ A is false. If ¬ A is true, then A is false. Using ‘T’ for true and ‘F’ for false, we can summarize this in a characteristic truth table for negation:

We will discuss truth tables at greater length in the next chapter.

Conjunction 

Consider these sentences:

9.  Adam is athletic.

10. Barbara is athletic.

11. Adam is athletic, and Barbara is also athletic.

We will need separate sentence letters for 9 and 10, so we define this symbolization key:

A: Adam is athletic.

B: Barbara is athletic.

Sentence 9 can be symbolized as A .

Sentence 10 can be symbolized as B .

Sentence 11 can be paraphrased as ‘ A and B .’ In order to fully symbolize this sentence, we need another symbol. We will use ‘ & .’ We translate ‘ A and B ’ as A & B . The logical connective ‘ & ’ is called CONJUNCTION, and A and B are each called CONJUNCTS.

Notice that we make no attempt to symbolize ‘also’ in sentence 11. Words like ‘both’ and ‘also’ function to draw our attention to the fact that two things are being conjoined. They are not doing any further logical work, so we do not need to represent them in SL.

Some more examples:

12. Barbara is athletic and energetic.

13. Barbara and Adam are both athletic.

14. Although Barbara is energetic, she is not athletic.

15. Barbara is athletic, but Adam is more athletic than she is.

Sentence 12 is obviously a conjunction. The sentence says two things about Barbara, so in English it is permissible to refer to Barbara only once. It might be tempting to try this when translating the argument: Since B means ‘Barbara is athletic’, one might paraphrase the sentences as ‘ B and energetic.’ This would be a mistake. Once we translate part of a sentence as B , any further structure is lost. B is an atomic sentence; it is nothing more than true or false. Conversely, ‘energetic’ is not a sentence; on its own it is neither true nor false. We should instead paraphrase the sentence as ‘ B and Barbara is energetic.’ Now we need to add a sentence letter to the symbolization key. Let E mean ‘Barbara is energetic.’ Now the sentence can be translated as B & E .

Sentence 13 says one thing about two different subjects. It says of both Barbara and Adam that they are athletic, and in English we use the word ‘athletic’ only once. In translating to SL, it is important to realize that the sentence can be paraphrased as, ‘Barbara is athletic, and Adam is athletic.’ This translates as B & A .

Sentence 14 is a bit more complicated. The word ‘although’ sets up a contrast between the first part of the sentence and the second part. Nevertheless, the sentence says both that Barbara is energetic and that she is not athletic. In order to make each of the conjuncts an atomic sentence, we need to replace ‘she’ with ‘Barbara.’

So we can paraphrase sentence 14 as, ‘ Both Barbara is energetic, and Barbara is not athletic.’ The second conjunct contains a negation, so we paraphrase further: ‘ Both Barbara is energetic and it is not the case that Barbara is athletic.’ This translates as E & ¬ B .

Sentence 15 contains a similar contrastive structure. It is irrelevant for the purpose of translating to SL, so we can paraphrase the sentence as ‘ Both Barbara is athletic, and Adam is more athletic than Barbara.’ (Notice that we once again replace the pronoun ‘she’ with her name.) How should we translate the second conjunct? We already have the sentence letter A which is about Adam’s being athletic and B which is about Barbara’s being athletic, but neither is about one of them being more athletic than the other. We need a new sentence letter. Let R mean ‘Adam is more athletic than Barbara.’ Now the sentence translates as B & R .

It is important to keep in mind that the sentence letters A , B , and R are atomic sentences. Considered as symbols of SL, they have no meaning beyond being true or false. We have used them to symbolize different English language sentences that are all about people being athletic, but this similarity is completely lost when we translate to SL. No formal language can capture all the structure of the English language, but as long as this structure is not important to the argument there is nothing lost by leaving it out.

For any sentences A and B , A & B is true if and only if both A and B are true. We can summarize this in the characteristic truth table for conjunction:

Conjunction is symmetrical because we can swap the conjuncts without changing the truth-value of the sentence. Regardless of what A and B are, A & B is logically equivalent to B & A .

Disjunction 

16. Either Denison will play golf with me, or he will watch movies.

17. Either Denison or Ellery will play golf with me.

For these sentences we can use this symbolization key:

D: Denison will play golf with me.

E: Ellery will play golf with me.

M: Denison will watch movies.

Sentence 16 is ‘Either D or M .’ To fully symbolize this, we introduce a new symbol. The sentence becomes D ∨ M . The ‘∨’ connective is called DISJUNCTION, and D and M are called DISJUNCTS.

Sentence 17 is only slightly more complicated. There are two subjects, but the English sentence only gives the verb once. In translating, we can paraphrase it as. ‘Either Denison will play golf with me, or Ellery will play golf with me.’ Now it obviously translates as D ∨ E .

Sometimes in English, the word ‘or’ excludes the possibility that both disjuncts are true. This is called an EXCLUSIVE OR. An exclusive or is clearly intended when it says, on a restaurant menu, ‘Entrees come with either soup or salad.’ You may have soup; you may have salad; but, if you want both soup and salad, then you have to pay extra.

At other times, the word ‘or’ allows for the possibility that both disjuncts might be true. This is probably the case with sentence 17, above. I might play with Denison, with Ellery, or with both Denison and Ellery. Sentence 17 merely says that I will play with at least one of them. This is called an INCLUSIVE OR.

The symbol ‘∨’ represents an inclusive  or .  So D  E is true if D is true, if E is true, or if both D and E are true. It is false only if both D and E are false. We can summarize this with the characteristic truth table for disjunction:

Like conjunction, disjunction is symmetrical. A ∨ B is logically equivalent to B ∨ A .

These sentences are somewhat more complicated:

18.    Either you will not have soup, or you will not have salad.

19.    You will have neither soup nor salad.

20.    You get either soup or salad, but not both.

We let S 1 mean that you get soup and S 2 mean that you get salad.

Sentence 18 can be paraphrased in this way: ‘Either it is not the case that you get soup, or it is not the case that you get salad.’ Translating this requires both disjunction and negation. It becomes ¬ S 1 ∨ ¬ S 2 .

Sentence 19 also requires negation. It can be paraphrased as, ‘ It is not the case that either that you get soup or that you get salad.’ We need some way of indicating that the negation does not just negate the right or left disjunct, but rather negates the entire disjunction. In order to do this, we put parentheses around the disjunction: ‘It is not the case that ( S 1 ∨ S 2 ).’ This becomes simply ¬( S 1 ∨ S 2). Notice that the parentheses are doing important work here. The sentence ¬ S 1 ∨ S 2 would mean ‘Either you will not have soup, or you will have salad.’

Sentence 20 is an exclusive or . We can break the sentence into two parts. The first part says that you get one or the other. We translate this as ( S 1 ∨ S 2 ). The second part says that you do not get both. We can paraphrase this as, ‘It is not the case both that you get soup and that you get salad.’ Using both negation and conjunction, we translate this as ¬( S 1 & S 2). Now we just need to put the two parts together. As we saw above, ‘but’ can usually be translated as a conjunction. Sentence 20 can thus be translated as ( S 1 ∨ S 2) & ¬( S 1 & S 2).

Although ‘∨’ is an inclusive or , we can symbolize an exclusive or in SL. We just need more than one connective to do it.

Conditional 

For the following sentences, let R mean ‘You will cut the red wire’ and B mean ‘The bomb will explode.’

21.    If you cut the red wire, then the bomb will explode.

22.    The bomb will explode only if you cut the red wire.

Sentence 21 can be translated partially as ‘If R , then B .’ We will use the symbol ‘→’ to represent logical entailment. The sentence becomes R → B . The connective is called a CONDITIONAL. The sentence on the left-hand side of the conditional ( R in this example) is called the ANTECEDENT. The sentence on the right-hand side ( B ) is called the CONSEQUENT.

Sentence 22 is also a conditional. Since the word ‘if’ appears in the second half of the sentence, it might be tempting to symbolize this in the same way as sentence 21. That would be a mistake.

The conditional R → B says that if R were true, then B would also be true. It does not say that your cutting the red wire is the only way that the bomb could explode. Someone else might cut the wire, or the bomb might be on a timer. The sentence R → B does not say anything about what to expect if R is false. Sentence 22 is different. It says that the only conditions under which the bomb will explode involve your having cut the red wire; i.e., if the bomb explodes, then you must have cut the wire. As such, sentence 22 should be symbolized as B → R .

It is important to remember that the connective ‘ → ’ says only that, if the antecedent is true, then the consequent is true. It says nothing about the causal connection between the two events. Translating sentence 22 as B → R does not mean that the bomb exploding would somehow have caused your cutting the wire. Both sentence 21 and 22 suggest that, if you cut the red wire, your cutting the red wire would be the cause of the bomb exploding. They differ on the logical connection. If sentence 22 were true, then an explosion would tell us— those of us safely away from the bomb— that you had cut the red wire. Without an explosion, sentence 22 tells us nothing.

The conditional is asymmetrical . You cannot swap the antecedent and consequent without changing the meaning of the sentence, because A→B and B→A are not logically equivalent.

Not all sentences of the form ‘If . . . then . . . ’ are conditionals. Consider this sentence:

23.    If anyone wants to see me, then I will be on the porch.

If I say this, it means that I will be on the porch, regardless of whether anyone wants to see me or not— but if someone did want to see me, then they should look for me there. If we let P mean ‘I will be on the porch,’ then sentence 23 can be translated simply as P .

Biconditional 

24.    The figure on the board is a triangle only if it has exactly three sides.

25.    The figure on the board is a triangle if it has exactly three sides.

26.    The figure on the board is a triangle if and only if it has exactly three sides.

Let T mean ‘The figure is a triangle’ and S mean ‘The figure has three sides.’

Sentence 24, for reasons discussed above, can be translated as T → S .

Sentence 25 is importantly different. It can be paraphrased as, ‘If the figure has three sides, then it is a triangle.’ So it can be translated as S → T .

Sentence 26 says that T is true if and only if S is true; we can infer S from T , and we can infer T from S . This is called a biconditional, because it entails the two conditionals S → T and T → S . We will use ‘↔’ to represent the biconditional; sentence 26 can be translated as S ↔ T .

We could abide without a new symbol for the biconditional. Since sentence 26 means ‘ T → S and S → T ,’ we could translate it as ( T → S ) & ( S → T ). We would need parentheses to indicate that ( T → S ) and ( S → T ) are separate conjuncts; the expression T → S & S → T would be ambiguous.

Because we could always write ( A → B ) & ( B → A ) instead of A ↔ B , we do not strictly speaking need to introduce a new symbol for the biconditional. Nevertheless, logical languages usually have such a symbol. SL will have one, which makes it easier to translate phrases like ‘if and only if.

A ↔ B is true if and only if A and B have the same truth value. This is the characteristic truth table for the biconditional:

III. Other symbolization 

We have now introduced all of the connectives of SL. We can use them together to translate many kinds of sentences. Consider these examples of sentences that use the English-language connective ‘unless’:

27.    Unless you wear a jacket, you will catch cold.

28.    You will catch cold unless you wear a jacket.

Let J mean ‘You will wear a jacket’ and let D mean ‘You will catch a cold.’

We can paraphrase sentence 27 as ‘Unless J , D .’ This means that if you do not wear a jacket, then you will catch cold; with this in mind, we might translate it as ¬ J → D . It also means that if you do not catch a cold, then you must have worn a jacket; with this in mind, we might translate it as ¬ D → J .

Which of these is the correct translation of sentence 27? Both translations are correct, because the two translations are logically equivalent in SL.

Sentence 28, in English, is logically equivalent to sentence 27. It can be translated as either ¬ J → D or ¬ D → J .

When symbolizing sentences like sentence 27 and sentence 28, it is easy to get turned around. Since the conditional is not symmetric, it would be wrong to translate either sentence as J →¬D . Fortunately, there are other logically equivalent expressions. Both sentences mean that you will wear a jacket or— if you do not wear a jacket— then you will catch a cold. So we can translate them as J ∨ D . (You might worry that the ‘or’ here should be an exclusive or . However, the sentences do not exclude the possibility that you might both wear a jacket and catch a cold; jackets do not protect you from all the possible ways that you might catch a cold.)

IV. Sentences of SL 

The sentence ‘Apples are red, or berries are blue’ is a sentence of English, and the sentence ‘( A ∨ B )’ is a sentence of SL. Although we can identify sentences of English when we encounter them, we do not have a formal definition of ‘sentence of English’. In SL, it is possible to formally define what counts as a sentence. This is one respect in which a formal language like SL is more precise than a natural language like English.

It is important to distinguish between the logical language SL, which we are developing, and the language that we use to talk about SL. When we talk about a language, the language that we are talking about is called the object language. The language that we use to talk about the OBJECT LANGUAGE is called the METALANGUAGE.

The object language in this chapter is SL. The metalanguage is English— not conversational English, but English supplemented with some logical and mathematical vocabulary. The sentence ‘( A ∨ B )’ is a sentence in the object language, because it uses only symbols of SL. The word ‘sentence’ is not itself part of SL, however, so the sentence ‘This expression is a sentence of SL’ is not a sentence of SL. It is a sentence in the metalanguage, a sentence that we use to talk about SL.

In this section, we will give a formal definition for ‘sentence of SL.’ The definition itself will be given in mathematical English, the metalanguage.

Expressions 

There are three kinds of symbols in SL:

We define an EXPRESSION of SL as any string of symbols of SL. Take any of the symbols of SL and write them down, in any order, and you have an expression.

Well-formed formulae 

Since any sequence of symbols is an expression, many expressions of SL will be gobbledegook. A meaningful expression is called a well-formed formula . It is common to use the acronym wff ; the plural is wffs.

Obviously, individual sentence letters like A and G 13 will be wffs. We can form further wffs out of these by using the various connectives. Using negation, we can get ¬ A and ¬ G 13 . Using conjunction, we can get A & G 13 , G 13 & A , A & A , and G 13 & G 13 . We could also apply negation repeatedly to get wffs ¬¬ A or apply negation along with conjunction to get wffs like ¬( A & G 13 ) and ( G 13 & G 13 ). The possible combinations are endless, even starting with just these two sentence letters, and there are infinitely many sentence letters. So there is no point in trying to list all the wffs.

Instead, we will describe the process by which wffs can be constructed. Consider negation: Given any wff A of SL, A is a wff of SL. It is important here that A is not the sentence letter A . Rather, it is a variable that stands in for any wff at all. Notice that this variable A is not a symbol of SL, so A is not an expression of SL. Instead, it is an expression of the metalanguage that allows us to talk about infinitely many expressions of SL: all of the expressions that start with the negation symbol. Because A is part of the metalanguage, it is called a metavariable .We can say similar things for each of the other connectives. For instance, if A and B are wffs of SL, then ( A & B ) is a wff of SL. Providing clauses like this for all of the connectives, we arrive at the following formal definition for a well-formed formula of SL:

1.    Every atomic sentence is a wff.

2.    If A is a wff, then ¬ A is a wff of SL.

3.    If A and B are wffs, then ( A & B ) is a wff.

4.    If A and B are wffs, then ( A ∨ B ) is a wff.

5.    If A and B are wffs, then ( A → B ) is a wff.

6.    If A and B are wffs, then ( A ↔ B ) is a wff.

7.    All and only wffs of SL can be generated by applications of these rules.

Notice that we cannot immediately apply this definition to see whether an arbitrary expression is a wff.  Suppose we want to know whether or not    D is a wff of SL. Looking at the second clause of the definition, we know that¬¬¬ D is a wff if ¬¬ D is a wff. So now we need to ask whether or not ¬¬ D is a wff. Again looking at the second clause of the definition,     D is a wff if D is. Again, D is a wff if D is a wff. Now D is a sentence letter, an atomic sentence of SL, so we know that D is a wff by the first clause of the definition. So for a compound formula like  D , we must apply the definition repeatedly. Eventually we arrive at the atomic sentences from which the wff is built up.

Definitions like this are called recursive . Recursive definitions begin with some specifiable base elements and define ways to indefinitely compound the base elements. Just as the recursive definition allows complex sentences to be built up from simple parts, you can use it to decompose sentences into their simpler parts. To determine whether or not something meets the definition, you may have to refer back to the definition many times.

The connective that you look to first in decomposing a sentence is called the MAIN LOGICAL OPERATOR of that sentence. For example: The main logical operator of ¬( E ∨ ( F → G )) is negation, ¬. The main logical operator of (¬ E ∨ ( F → G )) is disjunction, ∨.

Recall that a sentence is a meaningful expression that can be true or false. Since the meaningful expressions of SL are the wffs and since every wff of SL is either true or false, the definition for a sentence of SL is the same as the definition for a wff. Not every formal language will have this nice feature. In the language QL, which is developed later in the book, there are wffs which are not sentences.

The recursive structure of sentences in SL will be important when we consider the circumstances under which a particular sentence would be true or false. The sentence D is true if and only if the sentence D is false, and so on through the structure of the sentence until we arrive at the atomic components: ¬¬¬D is true if and only if the atomic sentence D is false. We will return to this point in the next chapter.

Notational conventions

A wff like ( Q & R ) must be surrounded by parentheses, because we might apply the definition again to use this as part of a more complicated sentence. If we negate ( Q & R ), we get ( Q & R ). If we just had Q & R without the parentheses and put a negation in front of it, we would have Q & R . It is most natural to read this as meaning the same thing as ( Q & R ), something very different than ( Q & R ). The sentence ( Q & R ) means that it is not the case that both Q and R are true; Q might be false or R might be false, but the sentence does not tell us which. The sentence (¬ Q & R ) means specifically that Q is false and that R is true. As such, parentheses are crucial to the meaning of the sentence.

So, strictly speaking, Q & R without parentheses is not a sentence of SL. When using SL, however, we will often be able to relax the precise definition so as to make things easier for ourselves. We will do this in several ways.

First, we understand that Q & R means the same thing as ( Q & R ). As a matter of convention, we can leave off parentheses that occur around the entire sentence .

Second, it can sometimes be confusing to look at long sentences with many, nested pairs of parentheses. We adopt the convention of using square brackets ‘[’ and ‘]’ in place of parenthesis. There is no logical difference between ( P ∨ Q ) and [ P ∨ Q ], for example. The unwieldy sentence ((( H → I ) ∨ ( I → H )) & ( J ∨ K )) could be written in this way: [( H → I ) ∨ ( I → H )] & ( J ∨ K ).

Third, we will sometimes want to translate the conjunction of three or more sentences. For the sentence ‘Alice, Bob, and Candice all went to the party’, suppose we let A mean ‘Alice went’, B mean ‘Bob went’, and C mean ‘Candice went.’ The definition only allows us to form a conjunction out of two sentences, so we can translate it as ( A & B ) & C or as A & ( B & C ). There is no reason to distinguish between these, since the two translations are logically equivalent. There is no logical difference between the first, in which ( A & B ) is conjoined with C , and the second, in which A is conjoined with ( B & C ). So we might as well just write A & B & C . As a matter of convention, we can leave out parentheses when we conjoin three or more sentences.

Fourth, a similar situation arises with multiple disjunctions. ‘Either Alice, Bob, or Candice went to the party’ can be translated as ( A ∨ B ) ∨ C or as A ∨( B ∨ C ). Since these two translations are logically equivalent, we may write A ∨ B ∨ C .

These latter two conventions only apply to multiple conjunctions or multiple disjunctions. If a series of connectives includes both disjunctions and conjunctions, then the parentheses are essential; as with ( A & B ) C and A & ( B C ). The parentheses are also required if there is a series of conditionals or biconditionals; as with ( A → B ) → C and A ↔ ( B ↔ C ).

We have adopted these four rules as notational conventions , not as changes to the definition of a sentence. Strictly speaking, A B C is still not a sentence. Instead, it is a kind of shorthand. We write it for the sake of convenience, but we really mean the sentence ( A ∨ ( B ∨ C )).

If we had given a different definition for a wff, then these could count as wffs. We might have written rule 3 in this way: “If A , B , . . . Z are wffs, then ( A & B & . . . & Z ), is a wff.” This would make it easier to translate some English sentences, but would have the cost of making our formal language more complicated. We would have to keep the complex definition in mind when we develop truth tables and a proof system. We want a logical language that is expressively simple and allows us to translate easily from English, but we also want a formally simple language. Adopting notational conventions is a compromise between these two desires.

V. Practice Exercises 

*  Part A Using the symbolization key given, translate each English-language sentence into SL.

M: Those creatures are men in suits.

C: Those creatures are chimpanzees.

G: Those creatures are gorillas.

• Those creatures are not men in suits.
• Those creatures are men in suits, or they are not.
• Those creatures are either gorillas or chimpanzees.
• Those creatures are neither gorillas nor chimpanzees.
• If those creatures are chimpanzees, then they are neither gorillas nor men in suits.
• Unless those creatures are men in suits, they are either chimpanzees or they are gorillas.

Part B Using the symbolization key given, translate each English-language sentence into SL.

A: Mister Ace was murdered.

B: The butler did it.

C: The cook did it.

D: The Duchess is lying.

E: Mister Edge was murdered.

F: The murder weapon was a frying pan.

• Either Mister Ace or Mister Edge was murdered.
• If Mister Ace was murdered, then the cook did it.
• If Mister Edge was murdered, then the cook did not do it.
• Either the butler did it, or the Duchess is lying.
• The cook did it only if the Duchess is lying.
• If the murder weapon was a frying pan, then the culprit must have been the cook.
• If the murder weapon was not a frying pan, then the culprit was either the cook or the butler.
• Mister Ace was murdered if and only if Mister Edge was not murdered.
• The Duchess is lying, unless it was Mister Edge who was murdered.
• If Mister Ace was murdered, he was done in with a frying pan.
• Since the cook did it, the butler did not.
• Of course the Duchess is lying!

*  Part C Using the symbolization key given, translate each English-language sentence into SL.

E 1 : Ava is an electrician.

E 2 : Harrison is an electrician.

F 1 : Ava is a firefighter.

F 2 : Harrison is a firefighter.

S 1 : Ava is satisfied with her career.

S 2 : Harrison is satisfied with his career.

• Ava and Harrison are both electricians.
• If Ava is a firefighter, then she is satisfied with her career.
• Ava is a firefighter, unless she is an electrician.
• Harrison is an unsatisfied electrician.
• Neither Ava nor Harrison is an electrician.
• Both Ava and Harrison are electricians, but neither of them find it satisfying.
• Harrison is satisfied only if he is a firefighter.
• If Ava is not an electrician, then neither is Harrison, but if she is, then he is too.
• Ava is satisfied with her career if and only if Harrison is not satisfied with his.
• If Harrison is both an electrician and a firefighter, then he must be satisfied with his work.
• It cannot be that Harrison is both an electrician and a firefighter.
• Harrison and Ava are both firefighters if and only if neither of them is an electrician.

*  Part D Give a symbolization key and symbolize the following sentences in SL.

• Alice and Bob are both spies.
• If either Alice or Bob is a spy, then the code has been broken.
• If neither Alice nor Bob is a spy, then the code remains unbroken.
• The German embassy will be in an uproar, unless someone has broken the code.
• Either the code has been broken or it has not, but the German embassy will be in an uproar regardless.
• Either Alice or Bob is a spy, but not both.

Part E Give a symbolization key and symbolize the following sentences in SL.

• If Gregor plays first base, then the team will lose.
• The team will lose unless there is a miracle.
• The team will either lose or it won’t, but Gregor will play first base regardless.
• Gregor’s mom will bake cookies if and only if Gregor plays first base.
• If there is a miracle, then Gregor’s mom will not bake cookies.

Part F For each argument, write a symbolization key and translate the argument as well as possible into SL.

• If Dorothy plays the piano in the morning, then Roger wakes up cranky. Dorothy plays piano in the morning unless she is distracted. So if Roger does not wake up cranky, then Dorothy must be distracted.
• It will either rain or snow on Tuesday. If it rains, Neville will be sad. If it snows, Neville will be cold. Therefore, Neville will either be sad or cold on Tuesday.
• If Zoog remembered to do his chores, then things are clean but not neat. If he forgot, then things are neat but not clean. Therefore, things are either neat or clean— but not both.

*  Part G For each of the following: (a) Is it a wff of SL? (b) Is it a sentence of SL, allowing for notational conventions?

• J 374 ∨ ¬ J 374
• ( G & ¬ G )
• ( A → ( A & ¬ F )) ∨ ( D ↔ E )
• [( Z ↔ S ) → W ] & [ J ∨ X ]
• ( F ↔ ¬ D → J ) ∨ ( C & D )
• 1. Are there any wffs of SL that contain no sentence letters? Why or why not?
• 2. In the chapter, we symbolized an exclusive or using ∨, & , and ¬. How could you translate an exclusive or using only two connectives? Is there any way to translate an exclusive or using only one connective?

Critical Thinking Copyright © 2019 by Brian Kim is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Unlocking The Power: How Symbols Fuel Critical Thinking

Critical thinking is a crucial skill that enables individuals to analyze information, evaluate arguments, and make informed decisions. In today’s complex and information-driven world, the ability to think critically is more important than ever. However, critical thinking is not a natural ability; it is a skill that needs to be developed and nurtured.

Symbols play a significant role in critical thinking as they have the power to convey meaning beyond their literal representation. They are visual representations that can evoke emotions, provoke thought, and stimulate critical thinking. In this article, we will explore the concept of symbols and their role in critical thinking.

Table of Contents

Understanding Symbols

Symbols are visual representations that carry meaning and significance. They can be words, images, gestures, or objects that represent ideas, concepts, or emotions. Symbols are an integral part of communication, allowing us to convey complex ideas and emotions in a concise and powerful way.

Examples of commonly recognized symbols include the peace sign, the red cross, and the dollar sign. These symbols have become universally understood and can convey a wealth of meaning with just a simple image or gesture.

Symbols have the unique ability to convey meaning beyond their literal representation. They can evoke emotions, trigger memories, and stimulate critical thinking. For example, a red traffic light is a symbol that signifies the need to stop. It not only conveys a literal message but also prompts drivers to think critically about their actions and make a decision based on the symbol’s meaning.

Symbols and Critical Thinking

Symbols have the power to stimulate critical thinking by engaging our minds and emotions. They can evoke curiosity, challenge our assumptions, and encourage us to question the status quo. Symbols can be used in various contexts, such as advertising, politics, and art, to influence critical thinking.

In advertising, symbols are often used to create emotional connections with consumers and persuade them to make purchasing decisions. Advertisers use symbols to tap into our desires, fears, and aspirations, prompting us to think critically about the products or services being promoted.

In politics, symbols are used to rally support, convey ideologies, and influence public opinion. Political parties and movements often adopt symbols that represent their values and goals, aiming to stimulate critical thinking and engage voters on an emotional level.

In art, symbols are used to convey deeper meanings and provoke thought. Artists use symbols to express complex ideas, challenge societal norms, and encourage viewers to think critically about the world around them.

The Power of Symbols in Education

Symbols are also widely used in education to enhance critical thinking skills. They can be used to teach complex concepts and abstract ideas in a visual and engaging way. Symbols help students make connections, understand relationships, and think critically about the subject matter.

For example, in mathematics, symbols such as +, -, ×, and ÷ represent operations and help students understand mathematical concepts. In science, symbols are used to represent elements, compounds, and chemical reactions, enabling students to think critically about the properties and interactions of substances.

Symbols are incorporated into educational materials and activities to make learning more interactive and meaningful. They can be used in diagrams, charts, and graphs to visually represent information and facilitate critical thinking.

Unlocking the Power of Symbols

To unlock the power of symbols and enhance critical thinking, it is essential to develop strategies for utilizing symbols effectively. Here are some tips for recognizing and interpreting symbols:

Observe : Pay attention to symbols in your environment, such as signs, logos, and advertisements. Take note of how they make you feel and what thoughts they provoke.

Analyze : Consider the context and meaning behind symbols. Reflect on their intended message and the emotions they evoke. Ask yourself why certain symbols were chosen and what impact they have on your critical thinking.

Research : Explore the symbolism used in different fields, such as literature, art, and culture. Learn about the historical and cultural significance of symbols to gain a deeper understanding of their meaning.

Practice : Engage in activities that involve interpreting symbols, such as analyzing artworks, decoding advertisements, or discussing political symbols. This will help sharpen your critical thinking skills and enhance your ability to recognize and interpret symbols effectively.

By studying real-life examples of individuals or organizations that have successfully used symbols to fuel critical thinking, we can gain insights into the power of symbols and their potential impact on society.

Critical thinking is a vital skill in today’s complex world, and symbols play a significant role in stimulating critical thinking. Symbols have the power to convey meaning beyond their literal representation, evoke emotions, and provoke thought. By recognizing and interpreting symbols effectively, we can enhance our critical thinking skills and make more informed decisions. The power of symbols in education, advertising, politics, and art demonstrates their potential impact on society. So, let us embrace the power of symbols and harness their ability to fuel critical thinking.

Symbols play a significant role in communication and are essential for conveying meaning beyond their literal representation. Understanding symbols is crucial for developing critical thinking skills and interpreting messages effectively. In this section, we will explore the definition of symbols, their significance in communication, and provide examples of commonly recognized symbols.

Definition of symbols and their significance in communication

Symbols can be defined as visual representations that stand for or represent something else. They are used to communicate ideas, concepts, and emotions. Symbols have been used throughout history to convey messages and are an integral part of human communication.

Symbols are significant in communication because they have the power to transcend language barriers. They can be universally recognized and understood, making them a powerful tool for conveying complex ideas and emotions. For example, the peace symbol, a simple combination of a circle and three lines, is recognized globally as a symbol of peace and unity.

Examples of commonly recognized symbols

There are numerous symbols that are widely recognized and understood across different cultures and societies. Some examples include:

• The heart symbol: universally recognized as a symbol of love and affection.
• The dollar sign ($): represents currency and wealth.
• The cross: a symbol of Christianity and faith.
• The stop sign: universally recognized as a symbol of halting or stopping.
• The thumbs-up gesture: commonly used to indicate approval or agreement.

These symbols have become ingrained in our collective consciousness and are instantly recognizable, regardless of language or cultural background.

Explanation of how symbols convey meaning beyond their literal representation

Symbols have the power to convey meaning beyond their literal representation. They can evoke emotions, provoke thought, and convey abstract concepts. For example, a red rose is not just a flower; it is often associated with love and romance. Similarly, a dove is not just a bird; it is a symbol of peace.

Symbols can also be used to convey complex ideas and concepts that may be difficult to express through words alone. For instance, traffic signs use symbols to communicate instructions quickly and effectively. The symbol of a person walking with a crosswalk indicates that pedestrians have the right of way.

In summary, symbols are visual representations that go beyond their literal meaning. They have the power to convey emotions, provoke thought, and communicate complex ideas.

Understanding symbols is essential for developing critical thinking skills and interpreting messages effectively. By recognizing and interpreting symbols, individuals can gain a deeper understanding of the world around them and engage in more meaningful and thoughtful communication.

Symbols play a crucial role in stimulating critical thinking. They have the power to evoke emotions, provoke thought, and influence our perception of the world around us. In this section, we will explore how symbols are used in various contexts such as advertising, politics, and art to shape our critical thinking abilities.

Symbols as Stimulants

Symbols have the ability to stimulate critical thinking by engaging our minds on a deeper level. When we encounter a symbol, it triggers associations and connections in our brains, leading us to think critically about its meaning and significance. For example, a red stop sign is a symbol that immediately communicates the need to halt or pause. This simple symbol prompts us to think critically about the importance of following traffic rules and considering the safety of ourselves and others.

Symbols and Emotional Provocation

Symbols have the power to evoke emotions, which in turn can fuel critical thinking. For instance, a dove is often used as a symbol of peace. When we see a dove, it triggers emotions associated with peace, such as calmness and harmony. This emotional response can prompt us to think critically about the importance of peace in our lives and the world at large.

In addition, symbols can provoke thought by challenging our existing beliefs and perspectives. They can serve as a catalyst for questioning and reevaluating our assumptions. For example, a political cartoon that uses symbols to depict a controversial issue can prompt us to critically analyze different viewpoints and consider alternative perspectives.

Symbols in Advertising, Politics, and Art

Symbols are extensively used in advertising, politics, and art to influence critical thinking. Advertisers often employ symbols to create associations between their products and desirable qualities or emotions. For instance, a luxury brand may use a symbol like a crown to convey a sense of elegance and exclusivity. This symbol prompts consumers to think critically about the brand’s image and whether it aligns with their own values and aspirations.

In politics, symbols are used to convey messages and shape public opinion. Political parties often adopt symbols that represent their ideologies or values. These symbols can evoke emotions and provoke critical thinking about the party’s stance on various issues. For example, a political party that uses a symbol of a handshake may be seen as promoting unity and cooperation.

Artists also utilize symbols to convey deeper meanings and provoke critical thinking. Symbolism in art allows artists to communicate complex ideas and emotions that may be difficult to express through words alone. By engaging with symbols in art, viewers are encouraged to think critically about the artist’s intended message and interpret the artwork in their own unique way.

In conclusion, symbols are powerful tools that stimulate critical thinking. They have the ability to evoke emotions, provoke thought, and influence our perception of the world. Whether used in advertising, politics, or art, symbols play a significant role in shaping our critical thinking abilities. By recognizing and interpreting symbols effectively, we can unlock their potential to enhance our understanding of complex concepts and encourage deeper reflection on the issues that matter most to us.

Symbols play a crucial role in education, as they have the power to enhance critical thinking skills and facilitate the understanding of complex concepts and abstract ideas. By incorporating symbols into educational materials and activities, educators can create a more engaging and effective learning experience for students. In this section, we will explore how symbols are used in education and their impact on critical thinking.

Explanation of how symbols are used in education to enhance critical thinking skills

Symbols are used in education to enhance critical thinking skills by providing visual representations of concepts and ideas. They serve as a bridge between abstract concepts and concrete understanding, making it easier for students to grasp complex information. For example, mathematical symbols such as +, -, ×, and ÷ are used to represent operations and help students solve equations. By using symbols, students can visualize the process and develop a deeper understanding of mathematical concepts.

Moreover, symbols can be used to represent different perspectives and viewpoints, encouraging students to think critically and analyze various interpretations. For instance, in literature classes, symbols are often used to represent themes, characters, or ideas. Students are encouraged to interpret these symbols and analyze their significance, fostering critical thinking skills and encouraging deeper engagement with the text.

Discussion on how symbols can be used to teach complex concepts and abstract ideas

Symbols are particularly effective in teaching complex concepts and abstract ideas. They simplify complex information and make it more accessible to students. For example, in science classes, symbols are used to represent elements, compounds, and chemical reactions. These symbols condense complex chemical formulas into concise representations, making it easier for students to understand and remember.

Symbols can also be used to teach abstract ideas such as ethics, values, and social issues. By using symbols, educators can create visual representations that evoke emotions and provoke thought. This helps students connect with the subject matter on a deeper level and encourages critical thinking and reflection.

Examples of how symbols are incorporated into educational materials and activities

Symbols are incorporated into educational materials and activities in various ways. In textbooks, symbols are often used to highlight key points or concepts, making them stand out and facilitating comprehension. Diagrams, charts, and graphs are also common examples of symbols used to represent data and information visually.

In addition, educational activities often involve the use of symbols to engage students and promote critical thinking. For instance, in history classes, students may analyze political cartoons or propaganda posters to understand the symbols used and their impact on public opinion. This exercise encourages students to think critically about the messages conveyed through symbols and their influence on society.

Symbols have a significant impact on education and critical thinking. By incorporating symbols into educational materials and activities, educators can enhance students’ understanding of complex concepts, stimulate critical thinking skills, and foster deeper engagement with the subject matter. Symbols serve as powerful tools for simplifying information, representing different perspectives, and evoking emotions. As we continue to explore the potential of symbols in education, we unlock new opportunities for effective teaching and learning.

Symbols have a profound impact on our lives, often influencing our thoughts, emotions, and actions. They play a crucial role in communication, advertising, politics, art, and even education. Understanding and interpreting symbols can enhance our critical thinking skills and enable us to navigate the complexities of the world around us. In this section, we will explore strategies for unlocking the power of symbols and harnessing them to fuel critical thinking.

Strategies for Utilizing Symbols to Enhance Critical Thinking

Develop Symbol Awareness : The first step in unlocking the power of symbols is to develop an awareness of their presence in our daily lives. Pay attention to the symbols you encounter in various contexts, such as advertisements, logos, signs, and artworks. By consciously observing and analyzing symbols, you can begin to understand their intended meanings and the messages they convey.

Interpret Symbols in Context : Symbols derive their meaning from the context in which they are used. Consider the surrounding elements, cultural references, and historical significance when interpreting symbols. This contextual understanding will help you grasp the deeper layers of meaning and critically analyze the intended message.

Explore Multiple Perspectives : Symbols can be interpreted in different ways, depending on an individual’s background, experiences, and beliefs. Embrace diverse perspectives and engage in discussions with others to gain a broader understanding of symbols. This practice encourages critical thinking by challenging your own assumptions and expanding your worldview.

Analyze Symbolic Language : Symbols often communicate complex ideas and abstract concepts more effectively than words alone. Analyze the symbolic language used in literature, art, and media to uncover hidden meanings and provoke deeper thought. Look for recurring symbols and motifs that contribute to the overall message or theme.

Apply Symbolic Thinking : Symbolic thinking involves using symbols as tools for problem-solving and decision-making. Apply this approach by using symbols to represent ideas, relationships, or processes. This technique can help simplify complex information and facilitate critical thinking by allowing you to visualize and manipulate abstract concepts.

Tips for Recognizing and Interpreting Symbols Effectively

Research Symbolic Meanings : Symbols often carry cultural, historical, or universal meanings. Conduct research to understand the symbolism associated with specific objects, colors, animals, or gestures. This knowledge will enable you to interpret symbols accurately and avoid misinterpretations.

Consider Emotional Responses : Symbols have the power to evoke emotions and trigger personal associations. Pay attention to your emotional responses when encountering symbols, as they can provide valuable insights into their intended impact. Reflect on why certain symbols resonate with you and how they influence your thoughts and behaviors.

Question Symbolic Intentions : Symbols can be used manipulatively to influence opinions or shape narratives. Question the intentions behind the use of symbols in various contexts, such as advertising or politics. Analyze whether the symbols are being employed to inform, persuade, or manipulate, and critically evaluate their impact on your own thinking.

Real-Life Examples of Symbolic Thinking

Apple’s Logo : The bitten apple logo of Apple Inc. is a powerful symbol that represents innovation, creativity, and simplicity. It has become synonymous with the brand and conveys a message of cutting-edge technology and user-friendly design.

Peace Symbol : The peace symbol, a combination of the semaphore signals for “N” and “D” (nuclear disarmament), has become an iconic symbol of peace and anti-war movements. It represents the desire for a world free from nuclear weapons and conflict.

Traffic Signs : Traffic signs, such as stop signs and pedestrian crossing symbols, use simple yet universally recognized symbols to convey important messages quickly and effectively. These symbols enhance critical thinking by prompting immediate responses and ensuring road safety.

Unlocking the power of symbols can revolutionize our critical thinking abilities and enable us to navigate the complexities of the world with greater clarity. By developing symbol awareness, interpreting symbols in context, exploring multiple perspectives, and applying symbolic thinking, we can harness the power of symbols to enhance our critical thinking skills. Additionally, by researching symbolic meanings, considering emotional responses, and questioning symbolic intentions, we can become more adept at recognizing and interpreting symbols effectively. Real-life examples, such as Apple’s logo, the peace symbol, and traffic signs, demonstrate the profound impact symbols have on our society. Embrace the power of symbols and unlock your full potential for critical thinking.

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Symbols Made Simple

A quick and friendly introduction to symbolic logic by stephen szanto ..

Most non-professional philosophers are deterred from attending lectures and reading books by academics who use symbolic logic. Some even claim it is an elitist attempt to make presentations deliberately inaccessible to the uninitiated. In any case, I believe it is worth studying and needn’t be a scary as it at first looks. I hope this ‘child’s guide’ to modern philosophical formalism will provide a bridge between these two groups of philosophers.

The two most intimidating symbols are ‘∃’, standing for ‘one’ or ‘some’ or ‘somebody’, and ‘∀’, standing for ‘all’ or ‘every’ or ‘everybody’. They were designed by the Italian mathematician and logician Giuseppe Peano (1858-1912) and they are usually combined with another letter or letters which stand for the statement of our choice. Don’t panic, here is an example:

(∀x) (∃y) (y causes x)

This means “for all x there exists a y such that y causes x,” or, to put it in more everyday terms, “everything has a cause.” Which is a true statement, if we forget about quantum mechanics which has no bearing on our logic in an everyday sense. ∃ and ∀ are called ‘quantifiers’; ∃ the existential quantifier and ∀ the universal quantifier. Let’s now change the order of the quantifiers and we get:

(∃y) (∀x) (y causes x)

This means “there is one y such that for all x, y causes x,” or, more simply put, “there is one cause for everything.” This is considered false by atheistic humanists and materialists, but is passionately declared true by monotheistic devotees, who believe in one God as the maker and cause of the universe. However, if we restrict the statement to nature alone without bringing in metaphysical concepts of theology, we can say that the second proposition is false. For this reason, the illusion that we can reverse the quantifiers in a true statement and end up with another true statement is called the Quantifier Shift Fallacy. Even before we used philosophical symbols this type of fallacy was known to philosophers and theologians. The great Aristotle himself was accused of having committed it at the beginning of his Nicomachean Ethics as was the equally great Thomas Aquinas whose argument from causation is also held to be fallacious in this way.

“Hah”, I can hear you exclaiming, “So if all this could be done without those confusing symbols, what’s their use then?” “Well,” the formalists would say in defence of their abbreviated symbolic expressions, “isn’t it simpler, neater and more elegant to express our arguments in this concise way than scribbling pages and pages of convoluted arguments?”

The philosopher Immanuel Kant hated his contemporary psychiatrists (whose introspective experiments he regarded as philosophically untenable) so much that he wanted philosophers to attend court cases instead of the experts in the field. One can only imagine how much he would have loved to translate large volumes of psychiatric wisdom on the paranoid personality type who thinks that everybody hates him – but whose paranoia actually stems from his unconscious hatred of everybody else – into the following two elegant formulations:

(∀x) (∃y) (x hates y)

meaning “everybody hates me” and

(∃x) (∀y) (x hates y)

meaning “I hate everybody”, which is the cause of the person’s psychiatric problem. Unfortunately in Kant’s time such elegant symbolic formulations had not been invented. I hope you realised that this is another example of the Quantifier Shift Fallacy.

“Yes, yes,” you moan sadly, “but I turned to philosophy to find out what it is all about . Can symbolic logic help me with this?” Well, Frege, Russell and Wittgenstein, in his earlier ‘phase’, as well as the recently deceased Quine and several other outstanding contemporary philosophers think that the answer is ‘yes’. By introducing this strict new logical language we can learn a great deal about the world as it really is. To make this more tangible, let me quote a widely-known example. For a long time mankind was aware of the morning star and the evening star and regarded them as two different bright heavenly bodies. They were mysterious and many stories were attached to them. Then astronomers discovered that they were just one star, the planet Venus. Similarly, with the help of formal logic, some modern philosophers hope to purge our language of obscurity and confusion and show the world in its stark reality.

On the other side more and more equally brilliant thinkers claim that this enterprise has failed and is ‘dead’. Whatever the case, the enterprise is an outstanding achievement in the history of philosophy, and it is well worth the effort to form at least an intuitive insight of its essence. This brief introduction was meant to give you some confidence that it can be understood ‘without tears’ and perhaps also to whet your appetite.

© Dr Stephen Szanto 2005

Stephen Szanto is a medical doctor and also has a PhD in philosophy from London University.

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