relationship between critical thinking logic

Critical Thinking vs. Logical Thinking

What's the difference.

Critical thinking and logical thinking are both important skills that involve analyzing information and making informed decisions. Critical thinking involves evaluating arguments and evidence to determine the validity of a claim or statement. It requires questioning assumptions, considering alternative perspectives, and recognizing biases. Logical thinking, on the other hand, focuses on using reasoning and evidence to draw conclusions and make predictions. It involves following a systematic process of thinking to ensure that conclusions are based on sound reasoning. While critical thinking emphasizes questioning and evaluating information, logical thinking emphasizes the process of reasoning and drawing logical conclusions. Both skills are essential for making well-informed decisions and solving complex problems.

Further Detail

Introduction.

Critical thinking and logical thinking are two important cognitive processes that play a crucial role in problem-solving, decision-making, and overall intellectual development. While they are often used interchangeably, there are distinct differences between the two. In this article, we will explore the attributes of critical thinking and logical thinking, highlighting their unique characteristics and how they contribute to our ability to analyze information and make informed judgments.

Critical thinking is the process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and evaluating information gathered from observation, experience, reflection, reasoning, or communication. It involves questioning assumptions, recognizing biases, and considering multiple perspectives before arriving at a conclusion. Logical thinking, on the other hand, refers to the process of using reasoning consistently to come to a conclusion based on available information. It involves identifying patterns, making connections, and drawing inferences in a systematic and rational manner.

Attributes of Critical Thinking

One of the key attributes of critical thinking is the ability to think independently and objectively. Critical thinkers are able to assess information without being swayed by emotions or personal biases. They are also adept at identifying logical fallacies and inconsistencies in arguments, allowing them to make sound judgments based on evidence and reasoning. Additionally, critical thinkers are open-minded and willing to consider alternative viewpoints, which helps them avoid confirmation bias and make more informed decisions.

• Independent and objective thinking
• Identification of logical fallacies
• Open-mindedness and consideration of alternative viewpoints

Attributes of Logical Thinking

Logical thinking is characterized by a systematic approach to problem-solving and decision-making. Logical thinkers are able to break down complex issues into smaller, more manageable parts, allowing them to analyze each component individually before drawing conclusions. They also excel at recognizing cause-and-effect relationships and understanding how different variables interact with one another. Furthermore, logical thinkers are skilled at using deductive and inductive reasoning to arrive at logical solutions to problems.

• Systematic approach to problem-solving
• Recognition of cause-and-effect relationships
• Proficiency in deductive and inductive reasoning

Relationship Between Critical Thinking and Logical Thinking

While critical thinking and logical thinking are distinct processes, they are closely related and often work in tandem to enhance our cognitive abilities. Critical thinking provides the framework for evaluating information and making informed judgments, while logical thinking helps us organize and analyze that information in a systematic and rational manner. Together, they form a powerful combination that allows us to approach problems and challenges with clarity, precision, and objectivity.

Application in Real-Life Scenarios

Both critical thinking and logical thinking are essential skills that have practical applications in various aspects of our lives. In academic settings, critical thinking is crucial for analyzing complex texts, evaluating arguments, and formulating well-reasoned responses. Logical thinking, on the other hand, is valuable in fields such as mathematics, computer science, and engineering, where problem-solving requires a systematic and logical approach. In professional settings, both skills are highly sought after by employers who value employees who can think critically, solve problems efficiently, and make sound decisions based on evidence.

In conclusion, critical thinking and logical thinking are two distinct but complementary cognitive processes that play a vital role in our ability to analyze information, make informed judgments, and solve complex problems. While critical thinking focuses on evaluating information and considering multiple perspectives, logical thinking emphasizes the systematic and rational analysis of that information. By honing both skills, we can enhance our cognitive abilities, improve our decision-making processes, and approach challenges with clarity and objectivity.

Comparisons may contain inaccurate information about people, places, or facts. Please report any issues.

Success Skills

Critical thinking and logic.

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

Logic’s Relationship to Critical Thinking

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

Questions of Logic in Critical Thinking

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

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

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

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

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

Candela Citations

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

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19 Logic and Critical Thinking

Introduction [1].

This chapter is a primer on basic logical concepts that often appear in various critical thinking textbooks—concepts such as entailment, contraries, contradictories, necessary and sufficient conditions, etc. The chapter will not provide a historical genealogy of these concepts—in some sense critical thinking, argumentation theory, and formal logic all trace their roots back to at least Aristotle over two thousand years ago. As a result, for many of these concepts, determining whether the concept was a logic concept co-opted by critical thinking, or a critical thinking concept co-opted and changed by logic and then co-opted back again, is extremely difficult. Regardless, a brief orientation of the relationship of critical thinking and logic is in order.

Critical thinking, at least as it is most often justified, is a practical, skill-building exercise with the goal of improving our reasoning. This motivation, of understanding and improving our reasoning, has also been the motivation behind the development of logic over the past several thousand years. While we could study and understand each piece of reasoning individually, it is much more efficient to look for reasoning patterns that recur over and over again, to distinguish those patterns that are good from those that are bad, and so to find principles underpinning our reasoning that help us distinguish good reasoning from bad reasoning across the board. This push to generalize and theorize with the patterns of reasoning generated numerous formal logical systems, including the syllogistic and modal logics of Aristotle.

However, logic, especially formal logic, has not been constrained solely by the goal of understanding and improving our reasoning. Like abstract mathematics, the formal structures underpinning logical systems, are rich and complex enough to generate study all their own, with no concern for the original motivation that may have pushed us to study patterns of human reasoning. Regardless, many of logic’s concepts are still useful in organizing any study of reasoning.

In what follows I begin with a fairly substantial discussion of the core concept needed to understand the traditional logical concepts such as entailment or contradictory or necessary condition—the concept of a possibility. Once we have this notion in play, the definitions of the standard logical concepts, which I provide in Section 2, are quite straightforward. In the final section, I discuss the potential for misapplication of various concepts or distinctions.

1. Possibilities

1.1 possibility and reasoning.

The core concept of logic is the concept of a possibility (a case, a scenario, an option, a way things could be). While logicians and philosophers continue to work on illuminating the nature of possibilities, we can, even without a precise definition, still intuitively grasp the notion. You could stop reading right now or you could keep going. England didn’t win, but England could have won, if they had scored their penalty kick. That die, when rolled, will land on one of six possible sides. There are many things that might happen if the bill is passed into law. According to 18th century philosopher Gottfried Leibniz, God surveyed all the possible ways the universe might be and, being omnibenevolent, chose to create the best one (this one!?).

We appeal to possibilities all the time in our reasoning. Indeed, if there were but one way things could be and we knew completely what that way was like, then we would not need to reason at all—we would just know how things were going to unfold. But given that (i) we do not know completely how things are or how the future is going to unfold and (ii) we assume there are multiple possibilities for how the future might unfold, we need to reason about the ways things could be in order to learn how things are and how to best manage whatever the future brings. For example, the detective investigating a suspicious death gets a new piece of evidence—the deceased was killed by a rare poison. As a result, some scenarios are closed off as viable explanations of the death—e.g., the deceased was deliberately killed by someone who did not have access to the poison. Other scenarios, ones that may not have been in the detective’s awareness before the new piece of evidence was acquired, become relevant—e.g., that someone who knew or at least had access to the poison was responsible for the death. As a result, a new line of inquiry opens for the detective: find out who had access to the poison. Similarly, a doctor runs a series of tests to try to eliminate certain possible explanations for a given patient’s symptoms. Given certain results the possible explanations get narrowed down to one (and hopefully a treatment is available); given other results multiple possibilities remain and the doctor has to decide which tests may be required for progress to be made; unexpected results, while eliminating some possibilities may open up new possibilities that the doctor had not originally been considering. Finally, you are trying to decide when and in which order to run a list of errands. You take into account the likely lines at each location at different times of day, and the likely traffic at different times of day. After evaluating the possibilities, you choose the best option for you.

1.2 Types of possibil i ties

1.2 .1 physical & epistemic possibilities.

Given the ubiquity of possibilities in our reasoning, theorists often classify the possibilities. For example, physicists are interested in distinguishing the physical possibilities (the possibilities consistent with the laws of physics) from the physical impossibilities (the possibilities inconsistent with the laws of physics). Other general types of possibilities include epistemic possibilities—scenarios consistent with what we know; moral possibilities—those consistent with a given moral code; legal possibilities—situations consistent with what is permitted by a given legal code. We can even combine these types—epistemic physical possibilities are those that are consistent with the laws of physics as we currently know them. If what we know about the world changes at a fundamental level, what once was epistemically physically possible (measuring time independently of motion or gravity) may become epistemically physically impossible. Like for the detective and the doctor above, new, unexpected evidence may require an adjustment by the scientist in what possibilities are under consideration as viable explanations.

1.2 .2 Equally probable possibilities

Two other sorts of possibilities deserve mention. Probabilistic reasoning depends on possibilities of a very special sort—equally probable possibilities. To determine the probability that a fair coin will come up heads we assume that there are two equally likely possibilities, “heads’’ and “tails” (we usually ignore the extremely unlikely, though still physically possible situation in which the coin lands and stays on its edge). Failing to consider the relevant equally likely possibilities can make our probabilistic reasoning go awry. You will either win the lottery or you will not. There are two possibilities here, but treating them as equally likely is certainly an obvious mistake. Assuming the lottery is fair, the relevant equally likely possibilities are that each individual ticket (or set of numbers) will be the winner. If your ticket is one of many, then the probability you will win the lottery is much lower than the probability of your losing. Less obvious, but equally problematic is the following sort of case:

Three drawers contain the following mixture of coins—one contains two gold coins, one contains two silver coins, and one contains one gold coin and one silver coin. Without looking you pick a drawer, open it, and pick out a coin. When you open your eyes, you see the coin is gold. What is the probability that the other coin in that drawer is gold?

Many will reason as follows. The coin came from either the gold/gold drawer or the gold/silver drawer. Each drawer is equally likely and if it came from the gold/gold drawer the other coin is gold. But if it came from the gold/silver drawer the other coin is silver. Hence, the probability the other coin is gold is ½ or 50%. Unfortunately, the two possible drawers are not the relevant equally likely possibilities (no more than your winning or losing were the relevant equally likely possibilities in the lottery case). The relevant possibilities are opening a drawer and picking out a coin without looking. There are six different equally likely ways that could happen, one for each coin. Once you gain the new evidence that the coin you picked is gold when you open your eyes, you can eliminate three of the six possibilities, i.e. the ones in which you pick a silver coin. Of the three possibilities left two are such that the other coin is gold, i.e., the two possibilities in which you pick one of the two coins from the gold/gold drawer. Only in the gold/silver case is the other coin silver. Hence, the probability of the other coin being gold is 2/3. The moral here is that accurate probabilistic reasoning requires identifying and using the relevant equally likely possibilities from amongst all the sorts of possibilities that may present themselves—not always an easy task.

1.2 .3 Practical possibilities

Another significant type of possibility, especially in our everyday reasoning, is practical possibility—possibilities that are consistent with our means, desires, and will (or perhaps our epistemic practical possibilities—the possibilities that, given what we know or believe, are consistent with our means, desires, and will). When deciding how to get to an important meeting across town you are likely to not even consider the possibility that you flap your arms and fly, or the possibility that you use your personal matter/energy transport device, or even the possibility that you sprint all the way there. The first is physically impossible; the second, while perhaps physically possible, is beyond our current technological means; and the third, while certainly physically possible, is quite likely beyond your will and most certainly contrary to your strong desire to not arrive at the important meeting sweating profusely and gasping for breath. Instead you consider what your actual transportation options are (your own car, Uber, taxi, walk, subway, or some combination), how much time you have, how much money you are willing to spend, and then you try to find the optimal possibility (usually constrained by the desire to not spend too much time actually calculating the optimal possibility). Mundane decisions about which possibility to actualize like this happen all the time: what to eat this week, which movie to go see, what to do after dinner, when to get your hair cut, etc. Though mundane, they are still of interest to critical thinking or argumentation theorists since businesses and advertisers spend billions of dollars and devote millions of work-hours to trying to influence your desires and will in order to persuade you to choose their product.

Of more social significance are your individual choices that impact larger groups—in particular (if you live in a democracy) your voting choices, your decisions about how much effort you put into monitoring the outcome of your voting choices, and what the individuals or policies you voted for end up doing. In an optimal world, your political representatives would enact policies that benefit the most people in the most cost efficient, affordable, and just way. Of course, there may be little agreement about what is the most affordable, or just, or beneficial option, especially if what elected representatives take to be the best option is what will get them re-elected rather than what is actually good for their constituents. Regardless of the complexities and intricacies of public policy debate and decision-making, at the core is an attempt to find and agree upon a practical possibility, from amongst the myriad available, to actualize for our mutual benefit.

Given so many types of overlapping sets of possibilities, many of which differ for different individuals or groups of individuals—your set of practical possibilities does not likely match that of your neighbor even if the two sets overlap significantly; compare your set with someone of quite different socio-economic means and the sets overlap even less—and it is no surprise that numerous problems can arise when reasoning with and about possibilities. Individuals can consider too many possibilities, or more commonly, fail to consider all the relevant possibilities. For example, human beings are quite prone to confirmation bias—taking confirming instances as justifying an already-accepted theory or explanation rather than actively seeking out or testing for disconfirming instances. Detectives, or doctors, or researchers can become so fixated on the explanation they already believe to be correct that they are blind to the alternate explanations that are still consistent with the evidence available. In the case of probabilistic reasoning, we already saw cases of considering the wrong set of possibilities. Reasoners can also illegitimately shift the set of possibilities under consideration or shift the value assigned to various possibilities mid-reasoning. An egregious example can occur in public policy debates over the negative consequences of potential policies. When negative consequence X is a potential consequence of the opposition’s preferred policy it is judged to be likely enough to count as a reason against the policy, but when negative consequence X is a potential consequence of one’s own preferred policy, it is judged not to be likely enough to count as a reason against the policy. Identifying the correct set of possibilities and correct relative values of those possibilities is essential to reasoning correctly in numerous situations and yet identifying and ranking possibilities is often an extremely difficult task.

1.2 .4 Logical possibilities

One way to try to sidestep some of these problems is to determine what reasoning holds no matter what the possibilities in question are—to determine the patterns of reasoning that work in all the possibilities. After all, if a piece of reasoning works no matter what possibility you are considering, then you do not need to worry whether you are considering the right set of possibilities or not. Hence, one goal of formal logic is to be able to identify the structure that defines all the ways things could be, i.e., the logical possibilities.

The rough and ready notion of a “logical possibility” is a possibility that has no contradiction in it. Whilst it is not logically possible for an individual to both exist at a particular time and place and not exist at that time and place, which is contradictory, it is logically possible that the person exist in Montana in one instant, and then exist on one of the moons of Jupiter, say Io, in the next. There is no contradiction in the possibility that you exist in Montana in one instant and on Io in the next. But this possibility, while logically possible, is not physically possible. Given the distance from Montana to Io, we would need to violate the physical restriction on moving matter or energy (currently travelling below the speed of light) faster than the speed of light to get from Montana to Io from one instant to the next, so such travel is physically impossible.

Earlier I said that philosophers are still investigating and debating the nature of possibilities. But, whatever they are, there is one actualized one and lots of unactualized ones. In Leibniz’s argument that this world is the best of all possibilities, God examines all the possibilities and then actualizes the best one. Even if you doubt Leibniz’s argument, of all the myriad ways this universe could be, it is in fact one way, namely, the possibility that is actualized. The detective has numerous possibilities in mind about who is responsible for the deceased’s death; the detective hopes that by finding more evidence the possibilities can be reduced to one, the actual one. When you are deciding what to do tomorrow, you consider numerous possibilities and then engage in actions that make one (hopefully the one you wanted) actual.

But since there are lots of unactualized possibilities and only one actual possibility, how do we distinguish the unactualized possibilities from each other? Quite simply by what is true and false at each possibility. I flip a coin twice. There are four possible outcomes. Heads for the first flip and heads for the second; heads for the first and tails for the second; tails for the first, and heads for the second; and tails for both. Suppose the coin comes up tails on the first and heads on the second—that is the possibility that got actualized. How do we distinguish the three non-actualized possibilities? Well, in the first and second it is true that the coin first came up heads, but in fourth it is false that the coin first came up heads. But possibilities one and two differ in what is true and false of the second coin flip.

1.3 Declarative sentences and propositions

Given that we distinguish possibilities by what is true and false if they are actualized, one proposal for understanding possibilities is just as sets of declarative sentences. For example, the first coin flip possibility would be the set {“the first flip of the coin came up heads”, “the second flip of the coin came up heads”}. While initially appealing, the problem with this proposal is that sentences are not as well behaved as is needed to demarcate possibilities. Why?

Sometimes different sentences describe the same possibility or state of affairs. For example, “George is a bachelor” and “George is an unmarried male of marriageable age” describe the same state of affairs, but are different sentences since they are composed of different words. But since they are different sentences, sets that differ only in regards to which of these two sentences they contain are still different sets, and so different possibilities. Yet, we agreed the sentences were just two different ways of talking about the same possibility.

Alternatively, sometimes the same sentence can be used in different ways to describe different possibilities. For example, the sentence “The movie was a bomb” used in the United States likely describes a state of affairs in which the movie was bad, but the same sentence used in the United Kingdom likely describes a state of affairs in which the movie was good. But if one sentence can be used in different ways to describe different possibilities, then, once again, we cannot identify possibilities merely with sets of sentences.

To avoid the vagaries of sentences, logicians usually resort to propositions—what it is that declarative sentences express. “George is a bachelor” and “George is an unmarried male of marriageable age” express the same proposition about George’s marital status. “England won the World Cup in 1966” expresses a true proposition about the English national soccer team; “2+2 = 5” expresses a false proposition about the sum of 2 and 2. We use declarative sentences to express propositions directly, but other language use often involves them. For instance, when we ask, “did Hungary win the World Cup in 1938?” we wonder whether the proposition that Hungary won the 1938 World Cup is true or false. If we get the correct answer (they did not win—they lost to Italy 4-2 in the finals), then we stop wondering whether it is true or false and start believing it is false (and if the belief if strong enough and acquired in the correct way, we might even know that the proposition is false).

Instead of treating possibilities as sets of sentences, many logicians treat possibilities (or at least model possibilities) as sets of propositions. There are technical details that might require modifying even this proposal, but since the resolution of these details is unlikely to be relevant to the critical thinking project, we can take possibilities to be sets of propositions. The propositions that are members of a particular possibility are said to be true or obtain at that possibility. Propositions that are not members of a particular possibility are false at that possibility or do not obtain at that possibility. Armed with the concepts of (i) a possibility and (ii) propositions being true at or obtaining at possibilities, we can define many of the logical concepts that pervade logic and critical thinking textbooks. So even though some of the logical concepts that are forthcoming are, in some textbooks, defined in terms of sentences, the more common way is to define them in terms of propositions.

2. Logical concepts

2.1 types of propositions.

I begin by discussing some common types of propositions that arise in our reasoning. The most basic is a simple proposition, propositions expressed by such declarative sentences as “George is a bachelor” or “the sky is blue” or “Romeo loves Juliet.” Simple propositions attribute something to some object(s) or thing(s). In the first case, of George, that he is a bachelor, and in the third case, of Romeo, that he loves another object, namely Juliet. N egations of simple propositions, propositions expressed by such declarative sentences as “George is not a bachelor” or “Hungary did not win the 1938 World Cup” say that the simple proposition does not obtain. Of course, we do not speak declaratively solely by affirming either simple propositions or the denial of simple propositions; we combine or modify our simple propositions such as in:

• “George is a bachelor, and so is Todd”;
• “Mary loves Antonio, but he does not love her back”;
• “George went to Sophie’s house or he went to the movies”;
• “If the butler did not do it, then the cook did”;
• “If I take the subway, I will be on time for my meeting”;
• “Every student in this class is eligible”;
• “Someone deliberately killed the deceased”;
• “In order to be on time for your meeting, you must take the subway”;
• “England did not win the game, but they might have if they had scored their penalty kick in the last minute.”

The first two sentences express conjunctions . For a conjunction to be true, both sub-parts of the conjunction have to be true. So for “George is a bachelor and so is Todd” to be true, both “George is a bachelor” and “Todd is a bachelor” must be true. [For ease of exposition I will often omit the phrase “the proposition expressed by” before mentioning sentences as I just did above.]

The sentence “George went to Sophie’s house or he went to the movies” expresses a disjunction. There are two sorts of disjunctions—inclusive and exclusive. For an inclusive disjunction to be true, at least one of the sub-parts must be true. For an exclusive di s junction to be true, exactly one of the sub-parts must be true. If our sentence about George expresses an exclusive disjunction, then for it to be true George needs to be in exactly one of two places—at Sophie’s house or at the movies. This is likely to be the usage of someone trying to tell us where George is at a particular moment. If, on the other hand, the sentence expresses an inclusive disjunction, then it will be true if George went to one of those locations and is still true if George went to both. This is likely to be the usage of someone just trying to lay out where George might have gone over a period of time. While some languages have different words for expressing inclusive and exclusive disjunctions. English relies on context or background knowledge, sometimes with limited success, to try to distinguish which type of disjunction is being expressed. Legal documents, in order to avoid the ambiguity of ‘or’ in English, often spell out exclusive disjunctions as “A or B and not both A and B” while representing inclusive disjunctions as “A and/or B”.

Sentences such as: “ If the butler did not do it, then the cook did” and “ If I take the subway, [ then ] I will be on time for my meeting,” express onditional propositions. Conditionals are frequently used in natural languages such as English, yet there is little agreement on how they are to be analyzed logically. (Some theorists even go so far as to deny that conditional sentences express propositions at all.) Usually the disagreement concerns determining exactly what it takes for conditionals to be true, but there is widespread agreement that declarative conditionals are false if the ‘if’­–part, the antecedent , is true, and the ‘then’–part, the consequent , is false. If it is true that I take the subway, and false that I will be on time for my meeting, then the conditional “If I take the subway, I will be on time for my meeting,” is false. As a consequence, logic has defined a minimal version of the conditional, called the material conditional . Material conditionals are false if the antecedent is true and the consequent is false, but true otherwise—in other words, material conditionals are the most permissive when considering what it takes for a conditional to be true. There has been much debate about whether indicative conditionals such as “If the butler did not do it, then the cook did” just express material conditionals or rather express something stronger. Despite the disagreement, the most common articulation of conditionals in introductory logic texts is in terms of material conditionals, and it is most often this sort of conditional that is coopted into critical thinking texts. One merely needs to keep in mind that the work on understanding conditionals is far from finished.

“ Every student in this class is eligible” expresses a universal proposition—a proposition that attributes something to every member of a specified group. For a universal proposition to be true there can be no instance of a member of the group not having the specified attribute. If “Every student in this class is eligible” is true, then there is no student in the class who is not eligible. Oftentimes the group is not fully identified in the sentence used to express the proposition. For example, saying “All the beer is in the fridge” or “All horses have heads” are unlikely to be taken as expressing that every single beer in the universe is in a particular fridge or that there is no single instance of a headless horse anywhere. Depending on the context of use, likely plausible interpretations of those sentences would be: “All the beer we brought home from the store (and which has not already been drunk) is in the fridge” and “Typical, normal live horses have heads.” But once the group is fully specified, for a universal proposition to be true, every member of the group must have the attributed property or properties.

Instead of saying that everything in a given group has a stated attribute, we often merely want to convey that at least one thing or some things in the group have a particular property as in “Someone deliberately killed the deceased.” Such propositions are existential propositions. They are true when at least one object in a specified group has a specified attribute. For example, “Some student is eligible” is true just so long as at least one student is indeed eligible.

So far, most of our examples of propositions can be true or false given a single possibility. Suppose we restrict ourselves to just the actual possibility—then it is either true that George is a bachelor at the actual possibility or it is not; if, at the actual possibility, there is no student in the class who is not eligible, then the universal “Every student in class is eligible” is true at the actual possibility and otherwise false. But some of our declarative sentences are not just about one possibility; rather, they depend on multiple possibilities. Sentences such as the last two on our list, which express modal propositions are examples. (They are called “modal” because expressions such as “must”, “can”, “might”, “would”, etc., were said to indicate the “mode” of the component proposition.)

Different modal expressions have different truth conditions. Consider, for example, the sentence—“In order to be on time for your meeting you must take the subway.” For it to be true, all the possible ways (probably some set of practical possibilities constrained by the background in which the sentence is uttered) in which you make the meeting on time include your taking the subway. In the case of England losing, but winning if they’d scored their penalty, the first part is a negation that is true just so long as England won is false. So the first part tells us what the actual possibility is like. But the second part tells us what the relevantly similar possibilities except for England scoring their penalty, are like—namely, that England won in at least one of those possibilities. Compare that with the stronger claim that England would have won if they had scored their penalty—that claim will be true just so long as England wins in all the relevantly similar possibilities. (Part of the debate about conditionals is whether even conditionals without explicit modal terms, such as ‘might’ or ‘would’ or ‘must’, etc. are really expressing propositions concerning multiple possibilities, and not just the actual one—again, a debate I will not be able to resolve here.)

This list is not at all meant to be exhaustive of the type of propositions we express via our declarative sentences. Rather, it is meant to give a flavor for the sorts of propositions dealt with in first and second logic courses, the sorts of propositions that logicians attempt to model and define clearly and precisely in their basic systems. Why are logicians interested in these sorts of propositions? Because they show up in many of the reasoning patterns that we use over and over. For example, if I tell you George is either at Sophie’s or at the movies, and you tell me he is not at movies, we both hopefully reason that we should check for George at Sophie’s house. Another example: If the IRS says that all taxpayers satisfying their three specified conditions can claim a particular deduction, and you determine that you satisfy those three conditions, you should reason that you can take that particular deduction. It is by recognizing these types of propositions, and the patterns that result in combining them, that formal logic, which focuses on the patterns, gets its impetus. But regardless of whether one is focusing on the goodness of patterns or more generally on the patterns and content of reasoning, both critical thinking theorists and logicians need to take special care in determining what proposition a given sentence in a particular context expresses, for without understanding the correct proposition we will not be considering and evaluating the correct possibilities.

Even though understanding and classifying what propositions various declarative sentences express is an ongoing project, there is another classification scheme that logicians often appeal to—necessary truths (also called tautologies), necessary falsehoods (also called contradictions), and contingent propositions. The definitions are as follows:

Necessary Truth : A proposition that is true in all possibilities.

Necessary Falsehood : A proposition that is false in all possibilities.

Contingent Proposition : A proposition that is true in some, but not all, possibilities.

Sentences such as: “Either Socrates corrupts the youth of Athens or he does not”, or “If it is raining, then it is raining” express necessary truths. For every possibility there is, either Socrates corrupts the youth of Athens in that possibility or he does not. Some have wondered if there any non-trivial tautologies, since the standard examples, such as the ones I just gave, seem to be pretty trivial, uninformative sentences. Many theoreticians hold that the truths of mathematics are all necessary truths and many of those truths are certainly non-trivial—they often take a lot of work for us to know that they are true. Others point out that even if many necessary truths seem trivial or uninformative, they are still very useful. Plato, for example, uses the Socrates sentence in part of his dialogue concerning whether Socrates should have been found guilty of a particular offense. Plato starts with the obvious truth that either Socrates corrupts the youth or he does not, but proceeds to argue that in either case, Socrates should not be found guilty.

Sentences such as “At a particular moment in time, Socrates is over six-feet tall and Socrates is not over six-feet tall” express necessary falsehoods. For any possibility, and any moment of time in that possibility, Socrates cannot be both over six-feet tall and not over six-feet tall. Necessary falsehoods, or contradictions as they are more commonly called, are useful as sign-posts of something having gone drastically wrong in our reasoning. If we can show that someone’s position contains or leads to a contradiction, then we show that they aren’t even talking about a genuine possibility at all, but rather an impossibility. Good reasoners generally want to avoid being committed to impossibilities, so they try to avoid being committed to contradictions in their reasoning.

Most of the propositions we deal with in our everyday reasoning are contingent ones. “The coin landed heads on the first flip” is true in some possibilities, but false in others. “George will arrive on time” is true in some, but false in others. Even complex propositions, such as “If I take the subway, I will make it to the meeting on time” are likely to be true in some possibilities (smooth running reliable subway system) and false in others (an unreliable or scanty subway system). The challenge for good reasoners, of course, is to try to figure out, on the basis of what we already know, and the acquisition of new evidence, which propositions are in fact true at the actual possibility and which are not true. The detective, the doctor, the scientist, the everyday reasoner, are all reasoning using various possibilities in order to try to determine which propositions are true or false at the actual possibility.

2.2 Relations amongst propositions

Given that reasoning is the moving from given propositions to other propositions, and logicians are trying to understand correct reasoning, many of the important concepts of logic concern not just types of propositions, but the relations amongst propositions, I finish this section with definitions, examples, and discussion of eight such relations.

Necessary condition

One proposition, A , is a necessary condition for another propos i tion , B, if there is no possibility in which B obtains and A does not.

If A is a necessary condition for B, then you cannot have B without A. For example, if it is true that meeting the eligibility requirements is a necessary condition for legitimately holding office, then there is no possibility in which one legitimately holds office and does not meet the eligibility requirements. But if it is false that meeting the eligibility requirements is a necessary condition for legitimately holding office, then there is at least one possibility in which one legitimately holds office and does not meet the eligibility requirements.

Sufficient condition

One proposition, A , is a sufficient condition for another propos i tion B, if there is no possibility in which A obtains and B does not.

If A is a sufficient condition for B, then A guarantees B. For example, if it is true that getting a perfect score on every assessment is sufficient for passing the course, then there is no possibility in which one gets a perfect score on every assessment and one does not pass the course. If it is false, then there is at least one possibility in which one gets a perfect score on every assessment and still does not pass the course.

In many elementary logic or critical thinking textbooks, necessary and sufficient conditions are treated as material conditionals. For example, “George attending class is sufficient for George passing the course” is treated as “If George attends class, then George passes the course.” But necessary and sufficient conditions cannot be material conditionals, since denying a sufficient or necessary condition is not the same as denying a material conditional. For example, saying “George attending class is not sufficient for George passing” is not the same as denying the material conditional “If George attends class, then George passes the course” is true. Denying the material conditional is just saying that it is actually the case that George attends class, but does not pass the course, i.e., that the antecedent is true and the consequent is false. But denying that George’s attending is sufficient for George’s passing is not saying that George attends and does not pass, but rather says that there is a possibility, not necessarily the actual one, in which George attends, but does not pass. In other words, necessary and sufficient conditions are describing what is true of a range of possibilities.

Equivalence

Two propositions are equivalent just so long as there is no poss i bility in which one is true and the other is false .

In other words, for each possibility, the two propositions are either both true or both false. For example, “All Euclidean triangles have three sides” and “All Euclidean triangles have three interior angles” are both true in all possibilities and false in none, so they are logically equivalent to each other. Similarly, “Either Peter failed to make the team or Abigail failed to make the team” is logically equivalent to “Abigail and Peter did not both make the team.” If the first proposition is true, then, on an inclusive disjunction reading, at least one of the two did not make the team, so it is true that they did not both make the team. If on the other hand the first proposition is false, then it is false Pater failed to make the team (and so made it) and it is false Abigail failed to make the team (and so also made it), in which case both made the team and the second proposition is also false. Since the propositions are true in the same possibilities and false in the same possibilities they are logically equivalent.

Equivalence of proposition is not to be confused with the equivalence of sentences. Two sentences are equivalent, such as “George is a bachelor” and “George is a unmarried male of marriageable age” just in case they express the same proposition. Two distinct propositions, on the other hand, are equivalent just in case they are true or false in exactly the same possibilities. Of course, without a clear notion of the identity conditions of propositions, it is often hard to determine whether we have two sentences expressing one proposition, or two sentences expressing two distinct propositions that are equivalent to each other. [Like possibilities, theorists are still debating how to understand propositions. For example, here I have defined possibilities as sets of propositions, but some theorists reverse the order of dependence and define propositions as sets of possibilities, i.e., the possibilities at which they are true. Either way, having defined one concept in the terms of the other, the theorist still owes us an account of the undefined concept—a task theorists continue to pursue.]

Consistency

Two propositions are consistent with each other just in case there is at least one possibility in which both are true.

For example, “Sphere A is completely red” is consistent with “Cube B is completely blue” just so long as there is a possibility in which both are true. But “Sphere A is completely red” is inconsistent with “Sphere A is partly blue” since there is no possibility in which both are true.

Two propositions are contrary to each other if there is no possibi l ity in which both are true.

Contrariness is a kind of inconsistency. As we just saw, “Sphere A is completely red” is inconsistent with “Sphere A is partly blue” because there is no possibility in which both are true, i.e., because they are contrary to each other. But even though both propositions cannot be true together, they both could be false together, such as in possibilities in which “Sphere A is completely green” is true. But there is an even stronger kind of inconsistency, than mere contrariness.

Contradictor y

Two propositions are contradictory to each other if there is no possibility in which both are true or both are false .

“Sphere A is completely red” is contradictory to “Sphere A is not completely red” since if one is true, the other is false and if one is false, the other is true. Similarly, if it is true that “Snow guarantees skiing” then it is false that “There is a possibility in which there is snow and no skiing” and vice versa.

One important reason to keep these two kinds of inconsistency separate is that reasoners sometimes treat inconsistency as if it were just the same as being contradictory—they reason that if two states of affairs are inconsistent, then if one is false, the other one must be true. But such reasoners miss or ignore the possibility that two inconsistent propositions might still both be false, and as we saw in the previous section, ignoring or missing relevant possibilities is prone to generate reasoning errors. Hence, knowing whether two propositions are consistent, or contrary, or contradictory gives us important information about which possibilities are still relevant to whatever inquiry or reasoning we are pursuing using those propositions.

Since logicians are motivated by the goal of distinguishing good reasoning from bad reasoning and at least one part of good reasoning is that what we reason from adequately supports what we reason to, logicians are very interested in relations of adequate support. One very special kind of adequate support is entailment.

Entailmen t

P roposition A entails proposition B just so long as there is no po s sibility in which A is true and B is false.

For example, “Sam’s car weighs over 1000kg” entails “Sam’s car weighs at least 500kg”—any possibility in which Sam’s car is over 1000kg it is clearly at least 500kg. “Sam’s car is a red hatchback” entails “Sam’s car is red” and “Sam possesses a car” and “Sam’s car is a hatchback”. Instead of talking about what a single proposition entails, logicians are often interested in what a group or set of propositions entails. [A set of propositions entails another proposition just so long as there is no possibility in which all members of the set are true and the other proposition is false.] For example, “George went to Sophie’s house or to the movies” and “George did not go to the movies” entail “George went to Sophie’s house.” On the other hand, “If Sally attends class, then she passes the course” and “Sally passes the course” does not entail “Sally attends class,” since there are possibilities in which Sally can study well enough on her own and there is no attendance requirement, such that while it is true that “If Sally attends, then she passes the course” and true that “she passes the course”, it is false that “she attends class”.

Logic, especially formal logic, is primarily interested in entailment and other consequence relations. But at the elementary levels of logic at least the concept of entailment is applied to a concept that is also of interest to critical thinking and argumentation theorists—the concept of an argument. In logic, arguments are often modeled as a set of a set of propositions (the premises) and another proposition (the conclusion). [But see Chapters 8 and 9 of this volume for a more detailed discussion of the concept of an argument.] Logicians define validity , a property of arguments, in terms of whether or not the entailment relation holds between the premises and the other proposition, the conclusion. If the premises entail the conclusion, then the argument is valid, i.e., there is no possibility in which the premises are true and the conclusion false, and otherwise the argument is invalid. [Validity here is not to be confused with the notion of ‘valid’ that is used in everyday speech to signify that something is “good” or “worthy of further consideration”, as in: “She made a valid point, when she said ….”. Nor is it to be confused with the notion of ‘valid’ that is used in survey research to signify the goodness or utility of a measuring instrument or the results of such an instrument—for that concept see Chapter 19 of this volume.]

In the previous section, I said that one of the motivations for studying logic was to try to find properties of good reasoning that would hold in all the possibilities. Entailment (and so validity) is one such property. If the arguments you make are valid, i.e. if your reasons entail your conclusion, then your reasoning, at least in terms of support, is good reasoning. Of course, other aspects of that reasoning might be problematic, but at least you know that your reasons, if true, guarantee your conclusion, no matter what set of possibilities is the relevant set.

But consider: Most of the coins on the table are heads-up and that quarter is a coin on the table, so it is heads up. “Most of the coins on the table are heads-up” and “That quarter is a coin on the table” do not entail that “That quarter is heads up” and yet in many situations we would likely say that the first two propositions give very strong reasons to believe the third. In other words while entailment is a sure sign of inferential goodness in reasoning, the lack of entailment does not necessarily mean there is a lack of inferential goodness. Sometimes we say our reasoning is good enough, even if our reasons do not entail what we infer from them. If, in the possibilities in which our reasons are true, enough of them also have what we infer to be true, then we can say that the inferential link is good because the reasons sufficiently support our conclusion. The general definition of sufficient support is as follows:

Sufficient Support

Propostion A (or a set of propositions) sufficiently supports a proposition B just so long as, in enough of the possibilities in which A (or the set of propositions) is true, B is also true.

What counts as “enough” often varies from context to context. For example, in civil litigation, the conclusion of wrongdoing has to be supported by a preponderance of the evidence, i.e., the possibilities in which the defendant did what they are accused of, should be the case in more than 50% of the possibilities in which the provided evidence is true. But in criminal cases, the conclusion of wrongdoing should be supported beyond a reasonable doubt (which, at least if we take the vast majority of judges’ views on what that means, is above 80%). Statistical significance for supporting various hypotheses in the sciences is often set at 95% or higher. Determining what should count as “enough” in various contexts is often extremely challenging. At the very least, some of what counts as “enough” depends on the importance of the outcome. For example, since criminal sanctions are so much higher than civil sanctions, we demand more assurance that the evidence supports the conclusion of wrongdoing in the criminal case than in the civil case.

Logicians, I said, are primarily interested in consequence relations such as entailment. Different types of logic study these relations in different domains. For example, temporal or tense logics are interested in determining the consequence relations amongst uses of temporal phrases, such as, “in the future”, “in the past” and “now”. Modal logics study the consequence relations amongst propositions containing modal terms such as “must”, “can”, etc. But in addition to distinguishing types of logics by the types of propositions being modeled, logics are also categorized in terms of the type of consequence relation being studied. At the most general level, there are two types of logic—deductive and inductive. Deductive logic is concerned with entailment. Inductive logic is concerned with consequence relations weaker than entailment. Unsurprisingly, since there are many consequence relations weaker than entailment, inductive logic is a much less unified field of study than deductive logic. As we shall see in the next section, there are other uses of the terms ‘deductive’ and ‘inductive’, but these are generally misuses—the key difference between inductive and deductive logic is the type of consequence relation being studied.

I conclude this section with a final point about these eight definitions. They have all been given in terms of possibilities in general, i.e., logical possibilities. But for each definition, we could restrict the possibilities we are talking about and get restricted versions of these definitions. For example, physically necessary truths are those that hold in all the possibilities in which the physical laws hold. Morally necessary truths are those that are true in all the possibilities with the same moral code, etc. A set of propositions would physically entail another proposition if there is no physical possibility in which the propositions in the set are true and the other proposition is false. Two propositions are morally contradictory if there is no moral possibility in which both are true or both are false.

Even though explicit talk of these restricted kinds of logical concepts is rare, the theoretical apparatus is available and useful for trying to get clear on what various reasoners or arguers are in fact claiming. For example, in common discourse, when someone says that A entails B, I suspect they rarely mean that there is no possibility whatsoever in which A obtains and B does not; rather, for some contextually determined (though usually unspecified) group of possibilities there is no possibility in which A obtains and B does not. Similarly, for necessary and sufficient conditions; when someone says that snow is necessary for skiing, they probably do not mean that there is no possibility whatsoever in which there is skiing but no snow (there are in fact numerous possibilities—water skiing, roller skiing, sand skiing, skiing on artificial pellets, etc.), but rather that our typical conception of skiing requires snow. In the sciences, they are rarely concerned with logically necessary and sufficient conditions, but rather with causally necessary and sufficient conditions—conditions that require or guarantee something else in all the possibilities consistent with the causal laws. The moral for critical thinking is that even when one encounters terms such as ‘entails’ or ‘contradictory’ or ‘necessary condition’ they may not be being used in their strictly logical sense, but rather being used over a subset of relevant possibilities.

3. Logic and the activity of reasoning

3.1 logic and reasoning.

I conclude with some final comments about the application, and misapplication, of logical concepts in the study of reasoning. Logical systems are models. In particular, they are models of consequence relations between propositions. Some of the models are quite limited. For example, standard sentential or propositional logic systems ignore the internal structure of simple propositions and focus solely on connectives such as ‘and’, ‘or’ or ‘if,…then’. Others add elements to model ‘must’ and ‘can’ while still ignoring everything else, and so on. The hope is to ultimately get a model, or group of models, that illuminates the standards of good reasoning, at least with regards to inferential support. Like most models, logical models can be very helpful when properly applied within the domain they model. Trying to use the model outside the proper domain, however, can have drastic consequences. For example, claiming that the standard sentential logic system is a good model for explaining instances of good reasoning utilizing modal claims is clearly a mistake. (This is true not just for logical models. For example, using the “model” of the north star as a fixed point is extremely useful for general terrestrial navigation, but using the same model for routing certain sorts of messages, which requires quite precise location determination, gets poor results.) Similarly, since logic focuses on support relations and good reasoning usually involves not just adequate support, but good reasons as well, it is a mistake to think logic is the whole story of good reasoning. Indeed, logic has little to nothing to say about what makes reasons good reasons, but rather focuses on what can legitimately be inferred from whatever good reasons we find.

3.2 Arguments and explanations

Clearly the target domain we are trying to understand and improve—the activities of reasoning, arguing, justifying, persuading, etc., are much more complicated than any of the various logical systems that logicians produce to model certain aspects of those activities. And yet many theorists still try to find distinctions in the models that are really only distinctions in the activities and not really the concern of logic at all. For example, logicians and argumentation theorists have spent a lot of time trying to distinguish arguments from explanations. But suppose I lay out several reasons (including some reasons about what I think will happen in the next six months) why you should believe a particular company will fail in the next six months. Six months go by and the company fails and someone else asks “Why?” and I trot out my reasons again. Nothing has changed about the propositions involved, so, from the perspective of logic, there is one object, one set of propositions, here. Yet, how that object has been used has changed. Initially the reasons are used to argue that the company will fail. After the fact, the reasons are used to explain the company’s failure. We argue for propositions we are not sure of (or to convince others of propositions they are not sure of), but we explain propositions we are sure of, some of which may have been proved to us by argument, in order to understand why they are true. [Note that unlike my example, there are plenty of cases where the reasons one might give to argue for a proposition, which turns out to be true, need not be the reasons given when explaining why the proposition is true. For example, if something unexpected happens in the six months that contributes to the company’s failure that is likely to be a part of the act of explaining even though it was not part of the act of arguing.] The fact that there is a difference between acts of arguing and acts of explaining does not mean that, in the domain of logic, we should find separate kinds of things—arguments on the one hand and explanations on the other.

3.3 Inferring and implying

Going in the other direction, no one doubts that, considered in terms of propositions and support relations the inference from A to B and the implication of B by A are the same thing. But it is a mistake to think that the act of inferring is the same as the act of implying. You assert a group of facts (with the intention that I draw conclusions from those facts). I, being a good reasoner, draw those conclusions. You imply those conclusions and I infer those conclusions. Put another way, if I ask someone what they are inferring, I am asking about reasoning going on in their head, but if I ask someone what they are implying, I am asking about reasoning they hope to be going on in other people’s heads. Put yet another way, reasons do not infer conclusions, but rather imply them. People, when considering those reasons on their own, infer those conclusions, but do not imply them.

3.4 Deductive and inductive

Sometimes concepts are misapplied in both the model and the target domain. For example, some logic textbooks and critical thinking textbooks try to distinguish deductive arguments from inductive arguments, but from the perspective of logic there is nothing about the sets of propositions that compose arguments that make one kind of set deductive and another set inductive. For every group of reasons and a given conclusion we can ask whether the reasons entail the conclusion or not (the domain of deductive logic) or whether those very same reasons offer some support weaker than entailment or not (the domain of inductive logic.) Nor is it clear that we reason deductively or inductively—when we reason, we infer one or more propositions from others. Of that reasoning we did, we might wonder whether it is good or bad. The answer to that question will, in part, depend on what counts as good enough support in the situation in which I am using the reasoning. If the context requires entailment and the reasons do entail the conclusion, then the reasoning is adequate with regards to its support relation. If the reasons do not entail the conclusion, then it will fail to be adequate in such a situation. Similarly for a required support relation weaker than entailment—if reasons support the conclusion at or above the required level, then the support relation is adequate, whereas if it is below the required level the support relation is not adequate. The reasoning is one act of reasoning—whether the actual support relation of that reasoning is adequate or not depends on the situation. But none of this suddenly makes it the case that there are two distinct kinds of reasoning going on (even if there is a felt difference between realizing some reasons entail a conclusion versus realizing some premises only strongly support a conclusion.)

3.5 Linked vs. convergent arguments

One final example. The push for general principles often takes something that may track a real distinction or property in a certain specific set of cases and try to generalize it to all cases. For example, there is a strong intuition that reasons such as: “If you pass the test, then you will pass the course” and “You pass the test” work together to support the conclusion “you will pass the course” whereas reasons such as “You read all the supplemental material” and “You took good notes” and “You went to the tutor consistently” independently support the conclusion that “You are prepared for the test.” This intuition is strong enough, that numerous textbooks, especially those that use argument diagramming as a tool, try to distinguish arguments with linked premise structures from arguments with convergent premise structures. The problem here is two-fold. On the one hand, attempts to actually provide a rule for determining when a set (or subset) of reasons are linked or not have, to date, all failed, at least if we trust the intuitions that generated the drive to generalize the phenomena in the first place. On the other hand, the underlying judgments of whether premises are working together or are independent seem to vary from person to person and context to context enough to suspect that the distinction may not be tracking a real phenomenon that deserves to be represented or captured in our logical models.

4. Last word

Despite these injunctions to take care with the proper application of various concepts that have made their way into various textbooks, the core logical concepts of Section 2, such as sufficient support or co n sistency or necessary condition are useful in any study of reasoning. Even if good reasoners need to be careful and work diligently to determine which propositions are being expressed, and which possibilities are relevant, and what the needed standard of sufficient support is in a given situation, once these tasks are accomplished, we can evaluate our reasoning for inconsistencies and determine whether our reasons entail or at least adequately support our conclusions.

• © G.C. Goddu ↵

Studies in Critical Thinking Copyright © by J. Anthony Blair is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Internet Encyclopedia of Philosophy

Critical thinking.

Critical Thinking is the process of using and assessing reasons to evaluate statements, assumptions, and arguments in ordinary situations. The goal of this process is to help us have good beliefs, where “good” means that our beliefs meet certain goals of thought, such as truth, usefulness, or rationality. Critical thinking is widely regarded as a species of informal logic, although critical thinking makes use of some formal methods. In contrast with formal reasoning processes that are largely restricted to deductive methods—decision theory, logic, statistics—the process of critical thinking allows a wide range of reasoning methods, including formal and informal logic, linguistic analysis, experimental methods of the sciences, historical and textual methods, and philosophical methods, such as Socratic questioning and reasoning by counterexample.

The goals of critical thinking are also more diverse than those of formal reasoning systems. While formal methods focus on deductive validity and truth, critical thinkers may evaluate a statement’s truth, its usefulness, its religious value, its aesthetic value, or its rhetorical value. Because critical thinking arose primarily from the Anglo-American philosophical tradition (also known as “analytic philosophy”), contemporary critical thinking is largely concerned with a statement’s truth. But some thinkers, such as Aristotle (in Rhetoric ), give substantial attention to rhetorical value.

The primary subject matter of critical thinking is the proper use and goals of a range of reasoning methods, how they are applied in a variety of social contexts, and errors in reasoning. This article also discusses the scope and virtues of critical thinking.

Critical thinking should not be confused with Critical Theory. Critical Theory refers to a way of doing philosophy that involves a moral critique of culture. A “critical” theory, in this sense, is a theory that attempts to disprove or discredit a widely held or influential idea or way of thinking in society. Thus, critical race theorists and critical gender theorists offer critiques of traditional views and latent assumptions about race and gender. Critical theorists may use critical thinking methodology, but their subject matter is distinct, and they also may offer critical analyses of critical thinking itself.

Table of Contents

• Argument and Evaluation
• Categorical Logic
• Propositional Logic
• Modal Logic
• Predicate Logic
• Other Formal Systems
• Generalization
• Causal Reasoning
• Formal Fallacies
• Informal Fallacies
• Heuristics and Biases
• The Principle of Charity/Humility
• The Principle of Caution
• The Expansiveness of Critical Thinking
• Productivity and the Limits of Rationality
• Classical Approaches
• The Paul/Elder Model
• Other Approaches
• References and Further Reading

The process of evaluating a statement traditionally begins with making sure we understand it; that is, a statement must express a clear meaning. A statement is generally regarded as clear if it expresses a proposition , which is the meaning the author of that statement intends to express, including definitions, referents of terms, and indexicals, such as subject, context, and time. There is significant controversy over what sort of “entity” propositions are, whether abstract objects or linguistic constructions or something else entirely. Whatever its metaphysical status, it is used here simply to refer to whatever meaning a speaker intends to convey in a statement.

The difficulty with identifying intended propositions is that we typically speak and think in natural languages (English, Swedish, French), and natural languages can be misleading. For instance, two different sentences in the same natural language may express the same proposition, as in these two English sentences:

Jamie is taller than his father. Jamie’s father is shorter than he.

Further, the same sentence in a natural language can express more than one proposition depending on who utters it at a time:

I am shorter than my father right now.

The pronoun “I” is an indexical; it picks out, or “indexes,” whoever utters the sentence and, therefore, expresses a different proposition for each new speaker who utters it. Similarly, “right now” is a temporal indexical; it indexes the time the sentence is uttered. The proposition it is used to express changes each new time the sentence is uttered and, therefore, may have a different truth value at different times (as, say, the speaker grows taller: “I am now five feet tall” may be true today, but false a year from now). Other indexical terms that can affect the meaning of the sentence include other pronouns (he, she, it) and definite articles (that, the).

Further still, different sentences in different natural languages may express the same proposition . For example, all of the following express the proposition “Snow is white”:

Snow is white. (English)

Der Schnee ist weiss. (German)

La neige est blanche. (French)

La neve é bianca. (Italian)

Finally, statements in natural languages are often vague or ambiguous , either of which can obscure the propositions actually intended by their authors. And even in cases where they are not vague or ambiguous, statements’ truth values sometimes vary from context to context. Consider the following example.

The English statement, “It is heavy,” includes the pronoun “it,” which (when used without contextual clues) is ambiguous because it can index any impersonal subject. If, in this case, “it” refers to the computer on which you are reading this right now, its author intends to express the proposition, “The computer on which you are reading this right now is heavy.” Further, the term “heavy” reflects an unspecified standard of heaviness (again, if contextual clues are absent). Assuming we are talking about the computer, it may be heavy relative to other computer models but not to automobiles. Further still, even if we identify or invoke a standard of heaviness by which to evaluate the appropriateness of its use in this context, there may be no weight at which an object is rightly regarded as heavy according to that standard. (For instance, is an object heavy because it weighs 5.3 pounds but not if it weighs 5.2 pounds? Or is it heavy when it is heavier than a mouse but lighter than an anvil?) This means “heavy” is a vague term. In order to construct a precise statement, vague terms (heavy, cold, tall) must often be replaced with terms expressing an objective standard (pounds, temperature, feet).

Part of the challenge of critical thinking is to clearly identify the propositions (meanings) intended by those making statements so we can effectively reason about them. The rules of language help us identify when a term or statement is ambiguous or vague, but they cannot, by themselves, help us resolve ambiguity or vagueness. In many cases, this requires assessing the context in which the statement is made or asking the author what she intends by the terms. If we cannot discern the meaning from the context and we cannot ask the author, we may stipulate a meaning, but this requires charity, to stipulate a plausible meaning, and humility, to admit when we discover that our stipulation is likely mistaken.

2. Argument and Evaluation

Once we are satisfied that a statement is clear, we can begin evaluating it. A statement can be evaluated according to a variety of standards. Commonly, statements are evaluated for truth, usefulness, or rationality. The most common of these goals is truth, so that is the focus of this article.

The truth of a statement is most commonly evaluated in terms of its relation to other statements and direct experiences. If a statement follows from or can be inferred from other statements that we already have good reasons to believe, then we have a reason to believe that statement. For instance, the statement “The ball is blue” can be derived from “The ball is blue and round.” Similarly, if a statement seems true in light of, or is implied by, an experience, then we have a reason to believe that statement. For instance, the experience of seeing a red car is a reason to believe, “The car is red.” (Whether these reasons are good enough for us to believe is a further question about justification , which is beyond the scope of this article, but see “ Epistemic Justification .”) Any statement we derive in these ways is called a conclusion . Though we regularly form conclusions from other statements and experiences—often without thinking about it—there is still a question of whether these conclusions are true: Did we draw those conclusions well? A common way to evaluate the truth of a statement is to identify those statements and experiences that support our conclusions and organize them into structures called arguments . (See also, “ Argument .”)

An argument is one or more statements (called premises ) intended to support the truth of another statement (the conclusion ). Premises comprise the evidence offered in favor of the truth of a conclusion. It is important to entertain any premises that are intended to support a conclusion, even if the attempt is unsuccessful. Unsuccessful attempts at supporting a proposition constitute bad arguments, but they are still arguments. The support intended for the conclusion may be formal or informal. In a formal, or deductive, argument, an arguer intends to construct an argument such that, if the premises are true, the conclusion must be true. This strong relationship between premises and conclusion is called validity . This relationship between the premises and conclusion is called “formal” because it is determined by the form (that is, the structure) of the argument (see §3). In an informal, or inductive , argument, the conclusion may be false even if the premises are true. In other words, whether an inductive argument is good depends on something more than the form of the argument. Therefore, all inductive arguments are invalid, but this does not mean they are bad arguments. Even if an argument is invalid, its premises can increase the probability that its conclusion is true. So, the form of inductive arguments is evaluated in terms of the strength the premises confer on the conclusion, and stronger inductive arguments are preferred to weaker ones (see §4). (See also, “ Deductive and Inductive Arguments .”)

Psychological states, such as sensations, memories, introspections, and intuitions often constitute evidence for statements. Although these states are not themselves statements, they can be expressed as statements. And when they are, they can be used in and evaluated by arguments. For instance, my seeing a red wall is evidence for me that, “There is a red wall,” but the physiological process of seeing is not a statement. Nevertheless, the experience of seeing a red wall can be expressed as the proposition, “I see a red wall” and can be included in an argument such as the following:

• I see a red wall in front of me.
• Therefore, there is a red wall in front of me.

This is an inductive argument, though not a strong one. We do not yet know whether seeing something (under these circumstances) is reliable evidence for the existence of what I am seeing. Perhaps I am “seeing” in a dream, in which case my seeing is not good evidence that there is a wall. For similar reasons, there is also reason to doubt whether I am actually seeing. To be cautious, we might say we seem to see a red wall.

To be good , an argument must meet two conditions: the conclusion must follow from the premises—either validly or with a high degree of likelihood—and the premises must be true. If the premises are true and the conclusion follows validly, the argument is sound . If the premises are true and the premises make the conclusion probable (either objectively or relative to alternative conclusions), the argument is cogent .

Here are two examples:

• Earth is larger than its moon.
• Our sun is larger than Earth.
• Therefore, our sun is larger than Earth’s moon.

In example 1, the premises are true. And since “larger than” is a transitive relation, the structure of the argument guarantees that, if the premises are true, the conclusion must be true. This means the argument is also valid. Since it is both valid and has true premises, this deductive argument is sound.

  Example 2:

• It is sunny in Montana about 205 days per year.
• I will be in Montana in February.
• Hence, it will probably be sunny when I am in Montana.

In example 2, premise 1 is true, and let us assume premise 2 is true. The phrase “almost always” indicates that a majority of days in Montana are sunny, so that, for any day you choose, it will probably be a sunny day. Premise 2 says I am choosing days in February to visit. Together, these premises strongly support (though they do not guarantee) the conclusion that it will be sunny when I am there, and so this inductive argument is cogent.

In some cases, arguments will be missing some important piece, whether a premise or a conclusion. For instance, imagine someone says, “Well, she asked you to go, so you have to go.” The idea that you have to go does not follow logically from the fact that she asked you to go without more information. What is it about her asking you to go that implies you have to go? Arguments missing important information are called enthymemes . A crucial part of critical thinking is identifying missing or assumed information in order to effectively evaluate an argument. In this example, the missing premise might be that, “She is your boss, and you have to do what she asks you to do.” Or it might be that, “She is the woman you are interested in dating, and if you want a real chance at dating her, you must do what she asks.” Before we can evaluate whether her asking implies that you have to go, we need to know this missing bit of information. And without that missing bit of information, we can simply reply, “That conclusion doesn’t follow from that premise.”

The two categories of reasoning associated with soundness and cogency—formal and informal, respectively—are considered, by some, to be the only two types of argument. Others add a third category, called abductive reasoning, according to which one reasons according to the rules of explanation rather than the rules of inference . Those who do not regard abductive reasoning as a third, distinct category typically regard it as a species of informal reasoning. Although abductive reasoning has unique features, here it is treated, for reasons explained in §4d, as a species of informal reasoning, but little hangs on this characterization for the purposes of this article.

3. Formal Reasoning

Although critical thinking is widely regarded as a type of informal reasoning, it nevertheless makes substantial use of formal reasoning strategies. Formal reasoning is deductive , which means an arguer intends to infer or derive a proposition from one or more propositions on the basis of the form or structure exhibited by the premises. Valid argument forms guarantee that particular propositions can be derived from them. Some forms look like they make such guarantees but fail to do so (we identify these as formal fallacies in §5a). If an arguer intends or supposes that a premise or set of premises guarantee a particular conclusion, we may evaluate that argument form as deductive even if the form fails to guarantee the conclusion, and is thus discovered to be invalid.

Before continuing in this section, it is important to note that, while formal reasoning provides a set of strict rules for drawing valid inferences, it cannot help us determine the truth of many of our original premises or our starting assumptions. And in fact, very little critical thinking that occurs in our daily lives (unless you are a philosopher, engineer, computer programmer, or statistician) involves formal reasoning. When we make decisions about whether to board an airplane, whether to move in with our significant others, whether to vote for a particular candidate, whether it is worth it to drive ten miles faster the speed limit even if I am fairly sure I will not get a ticket, whether it is worth it to cheat on a diet, or whether we should take a job overseas, we are reasoning informally. We are reasoning with imperfect information (I do not know much about my flight crew or the airplane’s history), with incomplete information (no one knows what the future is like), and with a number of built-in biases, some conscious (I really like my significant other right now), others unconscious (I have never gotten a ticket before, so I probably will not get one this time). Readers who are more interested in these informal contexts may want to skip to §4.

An argument form is a template that includes variables that can be replaced with sentences. Consider the following form (found within the formal system known as sentential logic ):

• If p, then q.
• Therefore, q.

This form was named modus ponens (Latin, “method of putting”) by medieval philosophers. p and q are variables that can be replaced with any proposition, however simple or complex. And as long as the variables are replaced consistently (that is, each instance of p is replaced with the same sentence and the same for q ), the conclusion (line 3), q , follows from these premises. To be more precise, the inference from the premises to the conclusion is valid . “Validity” describes a particular relationship between the premises and the conclusion, namely: in all cases , the conclusion follows necessarily from the premises, or, to use more technical language, the premises logically guarantee an instance of the conclusion.

Notice we have said nothing yet about truth . As critical thinkers, we are interested, primarily, in evaluating the truth of sentences that express propositions, but all we have discussed so far is a type of relationship between premises and conclusion (validity). This formal relationship is analogous to grammar in natural languages and is known in both fields as syntax . A sentence is grammatically correct if its syntax is appropriate for that language (in English, for example, a grammatically correct simple sentence has a subject and a predicate—“He runs.” “Laura is Chairperson.”—and it is grammatically correct regardless of what subject or predicate is used—“Jupiter sings.”—and regardless of whether the terms are meaningful—“Geflorble rowdies.”). Whether a sentence is meaningful, and therefore, whether it can be true or false, depends on its semantics , which refers to the meaning of individual terms (subjects and predicates) and the meaning that emerges from particular orderings of terms. Some terms are meaningless—geflorble; rowdies—and some orderings are meaningless even though their terms are meaningful—“Quadruplicity drinks procrastination,” and “Colorless green ideas sleep furiously.”.

Despite the ways that syntax and semantics come apart, if sentences are meaningful, then syntactic relationships between premises and conclusions allow reasoners to infer truth values for conclusions. Because of this, a more common definition of validity is this: it is not possible for all the premises to be true and the conclusion false . Formal logical systems in which syntax allows us to infer semantic values are called truth-functional or truth-preserving —proper syntax preserves truth throughout inferences.

The point of this is to note that formal reasoning only tells us what is true if we already know our premises are true. It cannot tell us whether our experiences are reliable or whether scientific experiments tell us what they seem to tell us. Logic can be used to help us determine whether a statement is true, but only if we already know some true things. This is why a broad conception of critical thinking is so important: we need many different tools to evaluate whether our beliefs are any good.

Consider, again, the form modus ponens , and replace p with “It is a cat” and q with “It is a mammal”:

• If it is a cat, then it is a mammal.
• It is a cat.
• Therefore, it is a mammal.

In this case, we seem to “see” (in a metaphorical sense of see ) that the premises guarantee the truth of the conclusion. On reflection, it is also clear that the premises might not be true; for instance, if “it” picks out a rock instead of a cat, premise 1 is still true, but premise 2 is false. It is also possible for the conclusion to be true when the premises are false. For instance, if the “it” picks out a dog instead of a cat, the conclusion “It is a mammal” is true. But in that case, the premises do not guarantee that conclusion; they do not constitute a reason to believe the conclusion is true.

Summing up, an argument is valid if its premises logically guarantee an instance of its conclusion (syntactically), or if it is not possible for its premises to be true and its conclusion false (semantically). Logic is truth-preserving but not truth-detecting; we still need evidence that our premises are true to use logic effectively.

            A Brief Technical Point

Some readers might find it worth noting that the semantic definition of validity has two counterintuitive consequences. First, it implies that any argument with a necessarily true conclusion is valid. Notice that the condition is phrased hypothetically: if the premises are true, then the conclusion cannot be false. This condition is met if the conclusion cannot be false:

• Two added to two equals four.

This is because the hypothetical (or “conditional”) statement would still be true even if the premises were false:

• If it is blue, then it flies.
• It is an airplane.

It is true of this argument that if the premises were true, the conclusion would be since the conclusion is true no matter what.

Second, the semantic formulation also implies that any argument with necessarily false premises is valid. The semantic condition for validity is met if the premises cannot be true:

• Some bachelors are married.
• Earth’s moon is heavier than Jupiter.

In this case, if the premise were true, the conclusion could not be false (this is because anything follows syntactically from a contradiction), and therefore, the argument is valid. There is nothing particularly problematic about these two consequences. But they highlight unexpected implications of our standard formulations of validity, and they show why there is more to good arguments than validity.

Despite these counterintuitive implications, valid reasoning is essential to thinking critically because it is a truth-preserving strategy: if deductive reasoning is applied to true premises, true conclusions will result.

There are a number of types of formal reasoning, but here we review only some of the most common: categorical logic, propositional logic, modal logic, and predicate logic.

a. Categorical Logic

Categorical logic is formal reasoning about categories or collections of subjects, where subjects refers to anything that can be regarded as a member of a class, whether objects, properties, or events or even a single object, property, or event. Categorical logic employs the quantifiers “all,” “some,” and “none” to refer to the members of categories, and categorical propositions are formulated in four ways:

A claims: All As are Bs (where the capitals “A” and “B” represent categories of subjects).

E claims: No As are Bs.

I claims: Some As are Bs.

O claims: Some As are not Bs.

Categorical syllogisms are syllogisms (two-premised formal arguments) that employ categorical propositions. Here are two examples:

• All cats are mammals. (A claim) 1. No bachelors are married. (E claim)
• Some cats are furry. (I claim) 2. All the people in this building are bachelors. (A claim)
• Therefore, some mammals are furry. (I claim) 3. Thus, no people in this building are married. (E claim)

There are interesting limitations on what categorical logic can do. For instance, if one premise says that, “Some As are not Bs,” may we infer that some As are Bs, in what is known as an “existential assumption”? Aristotle seemed to think so ( De Interpretatione ), but this cannot be decided within the rules of the system. Further, and counterintuitively, it would mean that a proposition such as, “Some bachelors are not married,” is false since it implies that some bachelors are married.

Another limitation on categorical logic is that arguments with more than three categories cannot be easily evaluated for validity. The standard method for evaluating the validity of categorical syllogisms is the Venn diagram (named after John Venn, who introduced it in 1881), which expresses categorical propositions in terms of two overlapping circles and categorical arguments in terms of three overlapping circles, each circle representing a category of subjects.

Venn diagram for claim and Venn diagram for argument

A, B, and C represent categories of objects, properties, or events. The symbol “ ∩ ” comes from mathematical set theory to indicate “intersects with.” “A∩B” means all those As that are also Bs and vice versa. 

Though there are ways of constructing Venn diagrams with more than three categories, determining the validity of these arguments using Venn diagrams is very difficult (and often requires computers). These limitations led to the development of more powerful systems of formal reasoning.

b. Propositional Logic

Propositional, or sentential , logic has advantages and disadvantages relative to categorical logic. It is more powerful than categorical logic in that it is not restricted in the number of terms it can evaluate, and therefore, it is not restricted to the syllogistic form. But it is weaker than categorical logic in that it has no operators for quantifying over subjects, such as “all” or “some.” For those, we must appeal to predicate logic (see §3c below).

Basic propositional logic involves formal reasoning about propositions (as opposed to categories), and its most basic unit of evaluation is the atomic proposition . “Atom” means the smallest indivisible unit of something, and simple English statements (subject + predicate) are atomic wholes because if either part is missing, the word or words cease to be a statement, and therefore ceases to be capable of expressing a proposition. Atomic propositions are simple subject-predicate combinations, for instance, “It is a cat” and “I am a mammal.” Variable letters such as p and q in argument forms are replaced with semantically rich constants, indicated by capital letters, such as A and B . Consider modus ponens again (noting that the atomic propositions are underlined in the English argument):

As you can see from premise 1 of the Semantic Replacement, atomic propositions can be combined into more complex propositions using symbols that represent their logical relationships (such as “If…, then…”). These symbols are called “operators” or “connectives.” The five standard operators in basic propositional logic are:

These operations allow us to identify valid relations among propositions: that is, they allow us to formulate a set of rules by which we can validly infer propositions from and validly replace them with others. These rules of inference (such as modus ponens ; modus tollens ; disjunctive syllogism) and rules of replacement (such as double negation; contraposition; DeMorgan’s Law) comprise the syntax of propositional logic, guaranteeing the validity of the arguments employing them.

Two Rules of Inference:

Two Rules of Replacement:

For more, see “ Propositional Logic .”

c. Modal Logic

Standard propositional logic does not capture every type of proposition we wish to express (recall that it does not allow us to evaluate categorical quantifiers such as “all” or “some”). It also does not allow us to evaluate propositions expressed as possibly true or necessarily true, modifications that are called modal operators or modal quantifiers .

Modal logic refers to a family of formal propositional systems, the most prominent of which includes operators for necessity (□) and possibility (◊) (see §3d below for examples of other modal systems). If a proposition, p , is possibly true, ◊ p , it may or may not be true. If p is necessarily true, □ p , it must be true; it cannot be false. If p is necessarily false, either ~◊ p or □~ p , it must be false; it cannot be true.

There is a variety of modal systems, the weakest of which is called K (after Saul Kripke, who exerted important influence on the development of modal logic), and it involves only two additional rules:

Necessitation Rule:   If  A  is a theorem of  K , then so is □ A .

Distribution Axiom:  □( A ⊃ B ) ⊃ (□ A ⊃□ B ).  [If it is necessarily the case that if A, then B , then if it is necessarily the case that A, it is necessarily the case that B .]

Other systems maintain these rules and add others for increasing strength. For instance, the (S4) modal system includes axiom (4):

(4)  □ A ⊃ □□ A   [If it is necessarily the case that A, then it is necessarily necessary that A.]

An influential and intuitive way of thinking about modal concepts is the idea of “possible worlds” (see Plantinga, 1974; Lewis 1986). A world is just the set of all true propositions. The actual world is the set of all actually true propositions—everything that was true, is true, and (depending on what you believe about the future) will be true. A possible world is a way the actual world might have been. Imagine you wore green underwear today. The actual world might have been different in that way: you might have worn blue underwear. In this interpretation of modal quantifiers, there is a possible world in which you wore blue underwear instead of green underwear. And for every possibility like this, and every combination of those possibilities, there is a distinct possible world.

If a proposition is not possible, then there is no possible world in which that proposition is true. The statement, “That object is red all over and blue all over at the same time” is not true in any possible worlds. Therefore, it is not possible (~◊P), or, in other words, necessarily false (□~P). If a proposition is true in all possible worlds, it is necessarily true. For instance, the proposition, “Two plus two equal four,” is true in all possible worlds, so it is necessarily true (□P) or not possibly false (~◊~P).

All modal systems have a number of controversial implications, and there is not space to review them here. Here we need only note that modal logic is a type of formal reasoning that increases the power of propositional logic to capture more of what we attempt to express in natural languages. (For more, see “ Modal Logic: A Contemporary View .”)

d. Predicate Logic

Predicate logic, in particular, first-order predicate logic, is even more powerful than propositional logic. Whereas propositional logic treats propositions as atomic wholes, predicate logic allows reasoners to identify and refer to subjects of propositions, independently of their predicates. For instance, whereas the proposition, “Susan is witty,” would be replaced with a single upper-case letter, say “S,” in propositional logic, predicate logic would assign the subject “Susan” a lower-case letter, s, and the predicate “is witty” an upper-case letter, W, and the translation (or formula ) would be: Ws.

In addition to distinguishing subjects and predicates, first-order predicate logic allows reasoners to quantify over subjects. The quantifiers in predicate logic are “All…,” which is comparable to “All” quantifier in categorical logic and is sometimes symbolized with an upside-down A: ∀ (though it may not be symbolized at all), and “There is at least one…,” which is comparable to “Some” quantifier in categorical logic and is symbolized with a backward E: ∃. E and O claims are formed by employing the negation operator from propositional logic. In this formal system, the proposition, “Someone is witty,” for example, has the form: There is an x , such that x has the property of being witty, which is symbolized: (∃ x)(Wx). Similarly, the proposition, “Everyone is witty,” has the form: For all x, x has the property of being witty, which is symbolized (∀ x )( Wx ) or, without the ∀: ( x )( Wx ).

Predicate derivations are conducted according to the same rules of inference and replacement as propositional logic with the exception of four rules to accommodate adding and eliminating quantifiers.

Second-order predicate logic extends first-order predicate logic to allow critical thinkers to quantify over and draw inferences about subjects and predicates, including relations among subjects and predicates. In both first- and second-order logic, predicates typically take the form of properties (one-place predicates) or relations (two-place predicates), though there is no upper limit on place numbers. Second-order logic allows us to treat both as falling under quantifiers, such as e verything that is (specifically, that has the property of being) a tea cup and everything that is a bachelor is unmarried .

e. Other Formal Systems

It is worth noting here that the formal reasoning systems we have seen thus far (categorical, propositional, and predicate) all presuppose that truth is bivalent , that is, two-valued. The two values critical thinkers are most often concerned with are true and false , but any bivalent system is subject to the rules of inference and replacement of propositional logic. The most common alternative to truth values is the binary code of 1s and 0s used in computer programming. All logics that presuppose bivalence are called classical logics . In the next section, we see that not all formal systems are bivalent; there are non-classical logics . The existence of non-classical systems raises interesting philosophical questions about the nature of truth and the legitimacy of our basic rules of reasoning, but these questions are too far afield for this context. Many philosophers regard bivalent systems as legitimate for all but the most abstract and purely formal contexts. Included below is a brief description of three of the most common non-classical logics.

Tense logic , or temporal logic, is a formal modal system developed by Arthur Prior (1957, 1967, 1968) to accommodate propositional language about time. For example, in addition to standard propositional operators, tense logic includes four operators for indexing times: P “It has at some time been the case that…”; F “It will at some time be the case that…”; H “It has always been the case that…”; and G “It will always be the case that….”

Many-valued logic , or n -valued logic, is a family of formal logical systems that attempts to accommodate intuitions that suggest some propositions have values in addition to true and false. These are often motivated by intuitions that some propositions have neither of the classic truth values; their truth value is indeterminate (not just undeterminable, but neither true nor false), for example, propositions about the future such as, “There will be a sea battle tomorrow.” If the future does not yet exist, there is no fact about the future, and therefore, nothing for a proposition to express.

Fuzzy logic is a type of many-valued logic developed out of Lotfi Zadeh’s (1965) work on mathematical sets. Fuzzy logic attempts to accommodate intuitions that suggest some propositions have truth value in degrees, that is, some degree of truth between true and false. It is motivated by concerns about vagueness in reality, for example whether a certain color is red or some degree of red, or whether some temperature is hot or some degree of hotness.

Formal reasoning plays an important role in critical thinking, but not very often. There are significant limits to how we might use formal tools in our daily lives. If that is true, how do critical thinkers reason well when formal reasoning cannot help? That brings us to informal reasoning.

4. Informal Reasoning

Informal reasoning is inductive , which means that a proposition is inferred (but not derived) from one or more propositions on the basis of the strength provided by the premises (where “strength” means some degree of likelihood less than certainty or some degree of probability less than 1 but greater than 0; a proposition with 0% probability is necessarily false).

Particular premises grant strength to premises to the degree that they reflect certain relationships or structures in the world . For instance, if a particular type of event, p , is known to cause or indicate another type of event, q , then upon encountering an event of type p , we may infer that an event of type q is likely to occur. We may express this relationship among events propositionally as follows:

• Events of type p typically cause or indicate events of type q .
• An event of type p occurred.
• Therefore, an event of type q probably occurred.

If the structure of the world (for instance, natural laws) makes premise 1 true, then, if premise 2 is true, we can reasonably (though not certainly) infer the conclusion.

Unlike formal reasoning, the adequacy of informal reasoning depends on how well the premises reflect relationships or structures in the world. And since we have not experienced every relationship among objects or events or every structure, we cannot infer with certainty that a particular conclusion follows from a true set of premises about these relationships or structures. We can only infer them to some degree of likelihood by determining to the best of our ability either their objective probability or their probability relative to alternative conclusions.

The objective probability of a conclusion refers to how likely, given the way the world is regardless of whether we know it , that conclusion is to be true. The epistemic probability of a conclusion refers to how likely that conclusion is to be true given what we know about the world , or more precisely, given our evidence for its objective likelihood.

Objective probabilities are determined by facts about the world and they are not truths of logic, so we often need evidence for objective probabilities. For instance, imagine you are about to draw a card from a standard playing deck of 52 cards. Given particular assumptions about the world (that this deck contains 52 cards and that one of them is the Ace of Spades), the objective likelihood that you will draw an Ace of Spades is 1/52. These assumptions allow us to calculate the objective probability of drawing an Ace of Spades regardless of whether we have ever drawn a card before. But these are assumptions about the world that are not guaranteed by logic: we have to actually count the cards, to be sure we count accurately and are not dreaming or hallucinating, and that our memory (once we have finished counting) reliably maintains our conclusions. None of these processes logically guarantees true beliefs. So, if our assumptions are correct, we know the objective probability of actually drawing an Ace of Spades in the real world. But since there is no logical guarantee that our assumptions are right, we are left only with the epistemic probability (the probability based on our evidence) of drawing that card. If our assumptions are right, then the objective probability is the same as our epistemic probability: 1/52. But even if we are right, objective and epistemic probabilities can come apart under some circumstances.

Imagine you draw a card without looking at it and lay it face down. What is the objective probability that that card is an Ace of Spades? The structure of the world has now settled the question, though you do not know the outcome. If it is an Ace of Spades, the objective probability is 1 (100%); it is the Ace of Spades. If it is not the Ace of Spades, the objective probability is 0 (0%); it is not the Ace of Spades. But what is the epistemic probability? Since you do not know any more about the world than you did before you drew the card, the epistemic probability is the same as before you drew it: 1/52.

Since much of the way the world is is hidden from us (like the card laid face down), and since it is not obvious that we perceive reality as it actually is (we do not know whether the actual coins we flip are evenly weighted or whether the actual dice we roll are unbiased), our conclusions about probabilities in the actual world are inevitably epistemic probabilities. We can certainly calculate objective probabilities about abstract objects (for instance, hypothetically fair coins and dice—and these calculations can be evaluated formally using probability theory and statistics), but as soon as we apply these calculations to the real world, we must accommodate the fact that our evidence is incomplete.

There are four well-established categories of informal reasoning: generalization, analogy, causal reasoning, and abduction.

a. Generalization

Generalization is a way of reasoning informally from instances of a type to a conclusion about the type. This commonly takes two forms: reasoning from a sample of a population to the whole population , and reasoning from past instances of an object or event to future instances of that object or event . The latter is sometimes called “enumerative induction” because it involves enumerating past instances of a type in order to draw an inference about a future instance. But this distinction is weak; both forms of generalization use past or current data to infer statements about future instances and whole current populations.

A popular instance of inductive generalization is the opinion poll: a sample of a population of people is polled with respect to some statement or belief. For instance, if we poll 57 sophomores enrolled at a particular college about their experiences of living in dorms, these 57 comprise our sample of the population of sophomores at that particular college. We want to be careful how we define our population given who is part of our sample. Not all college students are like sophomores, so it is not prudent to draw inferences about all college students from these sophomores. Similarly, sophomores at other colleges are not necessarily like sophomores at this college (it could be the difference between a liberal arts college and a research university), so it is prudent not to draw inferences about all sophomores from this sample at a particular college.

Let us say that 90% of the 57 sophomores we polled hate the showers in their dorms. From this information, we might generalize in the following way:

• We polled 57 sophomores at Plato’s Academy. (the sample)
• 90% of our sample hates the showers in their dorms. (the polling data)
• Therefore, probably 90% of all sophomores at Plato’s Academy hate the showers in their dorms. (a generalization from our sample to the whole population of sophomores at Plato’s Academy)

Is this good evidence that 90% of all sophomores at that college hate the showers in their dorms?

A generalization is typically regarded as a good argument if its sample is representative of its population. A sample is representative if it is similar in the relevant respects to its population. A perfectly representative sample would include the whole population: the sample would be identical with the population, and thus, perfectly representative. In that case, no generalization is necessary. But we rarely have the time or resources to evaluate whole populations. And so, a sample is generally regarded as representative if it is large relative to its population and unbiased .

In our example, whether our inference is good depends, in part, on how many sophomores there are. Are there 100, 2,000? If there are only 100, then our sample size seems adequate—we have polled over half the population. Is our sample unbiased? That depends on the composition of the sample. Is it comprised only of women or only of men? If this college is not co-ed, that is not a problem. But if the college is co-ed and we have sampled only women, our sample is biased against men. We have information only about female freshmen dorm experiences, and therefore, we cannot generalize about male freshmen dorm experiences.

How large is large enough? This is a difficult question to answer. A poll of 1% of your high school does not seem large enough to be representative. You should probably gather more data. Yet a poll of 1% of your whole country is practically impossible (you are not likely to ever have enough grant money to conduct that poll). But could a poll of less than 1% be acceptable? This question is not easily answered, even by experts in the field. The simple answer is: the more, the better. The more complicated answer is: it depends on how many other factors you can control for, such as bias and hidden variables (see §4c for more on experimental controls).

Similarly, we might ask what counts as an unbiased sample. An overly simple answer is: the sample is taken randomly, that is, by using a procedure that prevents consciously or unconsciously favoring one segment of the population over another (flipping a coin, drawing lottery balls). But reality is not simple. In political polls, it is important not to use a selection procedure that results in a sample with a larger number of members of one political party than another relative to their distribution in the population, even if the resulting sample is random. For example, the two most prominent parties in the U.S. are the Democratic Party and the Republican Party. If 47% of the U.S. is Republican and 53% is Democrat, an unbiased sample would have approximately 47% Republicans and 53% Democrats. But notice that simply choosing at random may not guarantee that result; it could easily occur, just by choosing randomly, that our sample has 70% Democrats and 30% Republicans (suppose our computer chose, albeit randomly, from a highly Democratic neighborhood). Therefore, we want to control for representativeness in some criteria, such as gender, age, and education. And we explicitly want to avoid controlling for the results we are interested in; if we controlled for particular answers to the questions on our poll, we would not learn anything—we would get all and only the answers we controlled for.

Difficulties determining representativeness suggest that reliable generalizations are not easy to construct. If we generalize on the basis of samples that are too small or if we cannot control for bias, we commit the informal fallacy of hasty generalization (see §5b). In order to generalize well, it seems we need a bit of machinery to guarantee representativeness. In fact, it seems we need an experiment, one of the primary tools in causal reasoning (see §4c below).

Argument from Analogy , also called analogical reasoning , is a way of reasoning informally about events or objects based on their similarities. A classic instance of reasoning by analogy occurs in archaeology, when researchers attempt to determine whether a stone object is an artifact (a human-made item) or simply a rock. By comparing the features of an unknown stone with well-known artifacts, archaeologists can infer whether a particular stone is an artifact. Other examples include identifying animals’ tracks by their similarities with pictures in a guidebook and consumer reports on the reliability of products.

To see how arguments from analogy work in detail, imagine two people who, independently of one another, want to buy a new pickup truck. Each chooses a make and model he or she likes, and let us say they decide on the same truck. They then visit a number of consumer reporting websites to read reports on trucks matching the features of the make and model they chose, for instance, the year it was built, the size of the engine (6 cyl. or 8 cyl.), the type of transmission (2WD or 4WD), the fuel mileage, and the cab size (standard, extended, crew). Now, let us say one of our prospective buyers is interested in safety —he or she wants a tough, safe vehicle that will protect against injuries in case of a crash. The other potential buyer is interested in mechanical reliability —he or she does not want to spend a lot of time and money fixing mechanical problems.

With this in mind, here is how our two buyers might reason analogically about whether to purchase the truck (with some fake report data included):

• The truck I have in mind was built in 2012, has a 6-cylinder engine, a 2WD transmission, and a king cab.
• 62 people who bought trucks like this one posted consumer reports and have driven it for more than a year.
• 88% of those 62 people report that the truck feels very safe.
• Therefore, the truck I am looking at will likely be very safe.
• 88% of those 62 people report that the truck has had no mechanical problems.
• Therefore, the truck I am looking at will likely have no mechanical problems.

Are the features of these analogous vehicles (the ones reported on) sufficiently numerous and relevant for helping our prospective truck buyers decide whether to purchase the truck in question (the one on the lot)? Since we have some idea that the type of engine and transmission in a vehicle contribute to its mechanical reliability, Buyer 2 may have some relevant features on which to draw a reliable analogy. Fuel mileage and cab size are not obviously relevant, but engine specifications seem to be. Are these specifications numerous enough? That depends on whether anything else that we are not aware of contributes to overall reliability. Of course, if the trucks having the features we know also have all other relevant features we do not know (if there are any), then Buyer 2 may still be able to draw a reliable inference from analogy. Of course, we do not currently know this.

Alternatively, Buyer 1 seems to have very few relevant features on which to draw a reliable analogy. The features listed are not obviously related to safety. Are there safety options a buyer may choose but that are not included in the list? For example, can a buyer choose side-curtain airbags, or do such airbags come standard in this model? Does cab size contribute to overall safety? Although there are a number of similarities between the trucks, it is not obvious that we have identified features relevant to safety or whether there are enough of them. Further, reports of “feeling safe” are not equivalent to a truck actually being safe. Better evidence would be crash test data or data from actual accidents involving this truck. This information is not likely to be on a consumer reports website.

A further difficulty is that, in many cases, it is difficult to know whether many similarities are necessary if the similarities are relevant. For instance, if having lots of room for passengers is your primary concern, then any other features are relevant only insofar as they affect cab size. The features that affect cab size may be relatively small.

This example shows that arguments from analogy are difficult to formulate well. Arguments from analogy can be good arguments when critical thinkers identify a sufficient number of features of known objects that are also relevant to the feature inferred to be shared by the object in question. If a rock is shaped like a cutting tool, has marks consistent with shaping and sharpening, and has wear marks consistent with being held in a human hand, it is likely that rock is an artifact. But not all cases are as clear.

It is often difficult to determine whether the features we have identified are sufficiently numerous or relevant to our interests. To determine whether an argument from analogy is good, a person may need to identify a causal relationship between those features and the one in which she is interested (as in the case with a vehicle’s mechanical reliability). This usually takes the form of an experiment, which we explore below (§4c).

Difficulties with constructing reliable generalizations and analogies have led critical thinkers to develop sophisticated methods for controlling for the ways these arguments can go wrong. The most common way to avoid the pitfalls of these arguments is to identify the causal structures in the world that account for or underwrite successful generalizations and analogies. Causal arguments are the primary method of controlling for extraneous causal influences and identifying relevant causes. Their development and complexity warrant regarding them as a distinct form of informal reasoning.

c. Causal Reasoning

Causal arguments attempt to draw causal conclusions (that is, statements that express propositions about causes: x causes y ) from premises about relationships among events or objects. Though it is not always possible to construct a causal argument, when available, they have an advantage over other types of inductive arguments in that they can employ mechanisms (experiments) that reduce the risks involved in generalizations and analogies.

The interest in identifying causal relationships often begins with the desire to explain correlations among events (as pollen levels increase, so do allergy symptoms) or with the desire to replicate an event (building muscle, starting a fire) or to eliminate an event (polio, head trauma in football).

Correlations among events may be positive (where each event increases at roughly the same rate) or negative (where one event decreases in proportion to another’s increase). Correlations suggest a causal relationship among the events correlated.

But we must be careful; correlations are merely suggestive—other forces may be at work. Let us say the y-axis in the charts above represents the number of millionaires in the U.S. and the x-axis represents the amount of money U.S. citizens pay for healthcare each year. Without further analysis, a positive correlation between these two may lead someone to conclude that increasing wealth causes people to be more health conscious and to seek medical treatment more often. A negative correlation may lead someone to conclude that wealth makes people healthier and, therefore, that they need to seek medical care less frequently.

Unfortunately, correlations can occur without any causal structures (mere coincidence) or because of a third, as-yet-unidentified event (a cause common to both events, or “common cause”), or the causal relationship may flow in an unexpected direction (what seems like the cause is really the effect). In order to determine precisely which event (if any) is responsible for the correlation, reasoners must eliminate possible influences on the correlation by “controlling” for possible influences on the relationship (variables).

Critical thinking about causes begins by constructing hypotheses about the origins of particular events. A hypothesis is an explanation or event that would account for the event in question. For example, if the question is how to account for increased acne during adolescence, and we are not aware of the existence of hormones, we might formulate a number of hypotheses about why this happens: during adolescence, people’s diets change (parents no longer dictate their meals), so perhaps some types of food cause acne; during adolescence, people become increasingly anxious about how they appear to others, so perhaps anxiety or stress causes acne; and so on.

After we have formulated a hypothesis, we identify a test implication that will help us determine whether our hypothesis is correct. For instance, if some types of food cause acne, we might choose a particular food, say, chocolate, and say: if chocolate causes acne (hypothesis), then decreasing chocolate will decrease acne (test implication). We then conduct an experiment to see whether our test implication occurs.

Reasoning about our experiment would then look like one of the following arguments:

There are a couple of important things to note about these arguments. First, despite appearances, both are inductive arguments. The one on the left commits the formal fallacy of affirming the consequent, so, at best, the premises confer only some degree of probability on the conclusion. The argument on the right looks to be deductive (on the face of it, it has the valid form modus tollens ), but it would be inappropriate to regard it deductively. This is because we are not evaluating a logical connection between H and TI, we are evaluating a causal connection—TI might be true or false regardless of H (we might have chosen an inappropriate test implication or simply gotten lucky), and therefore, we cannot conclude with certainty that H does not causally influence TI. Therefore, “If…, then…” statements in experiments must be read as causal conditionals and not material conditionals (the term for how we used conditionals above).

Second, experiments can go wrong in many ways, so no single experiment will grant a high degree of probability to its causal conclusion. Experiments may be biased by hidden variables (causes we did not consider or detect, such as age, diet, medical history, or lifestyle), auxiliary assumptions (the theoretical assumptions by which evaluating the results may be faulty), or underdetermination (there may be a number of hypotheses consistent with those results; for example, if it is actually sugar that causes acne, then chocolate bars, ice cream, candy, and sodas would yield the same test results). Because of this, experiments either confirm or disconfirm a hypothesis; that is, they give us some reason (but not a particularly strong reason) to believe our hypothesized causes are or are not the causes of our test implications, and therefore, of our observations (see Quine and Ullian, 1978). Because of this, experiments must be conducted many times, and only after we have a number of confirming or disconfirming results can we draw a strong inductive conclusion. (For more, see “ Confirmation and Induction .”)

Experiments may be formal or informal . In formal experiments, critical thinkers exert explicit control over experimental conditions: experimenters choose participants, include or exclude certain variables, and identify or introduce hypothesized events. Test subjects are selected according to control criteria (criteria that may affect the results and, therefore, that we want to mitigate, such as age, diet, and lifestyle) and divided into control groups (groups where the hypothesized cause is absent) and experimental groups (groups where the hypothesized cause is present, either because it is introduced or selected for).

Subjects are then placed in experimental conditions. For instance, in a randomized study, the control group receives a placebo (an inert medium) whereas the experimental group receives the hypothesized cause—the putative cause is introduced, the groups are observed, and the results are recorded and compared. When a hypothesized cause is dangerous (such as smoking) or its effects potentially irreversible (for instance, post-traumatic stress disorder), the experimental design must be restricted to selecting for the hypothesized cause already present in subjects, for example, in retrospective (backward-looking) and prospective (forward-looking) studies. In all types of formal experiments, subjects are observed under exposure to the test or placebo conditions for a specified time, and results are recorded and compared.

In informal experiments, critical thinkers do not have access to sophisticated equipment or facilities and, therefore, cannot exert explicit control over experimental conditions. They are left to make considered judgments about variables. The most common informal experiments are John Stuart Mill’s five methods of inductive reasoning, called Mill’s Methods, which he first formulated in A System of Logic (1843). Here is a very brief summary of Mill’s five methods:

(1) The Method of Agreement

If all conditions containing the event y also contain x , x is probably the cause of y .

For example:

“I’ve eaten from the same box of cereal every day this week, but all the times I got sick after eating cereal were times when I added strawberries. Therefore, the strawberries must be bad.”

(2) The Method of Difference

If all conditions lacking y also lack x , x is probably the cause of y .

“The organization turned all its tax forms in on time for years, that is, until our comptroller, George, left; after that, we were always late. Only after George left were we late. Therefore, George was probably responsible for getting our tax forms in on time.”

(3) The Joint Method of Agreement and Difference

If all conditions containing event y also contain event x , and all events lacking y also lack x , x is probably the cause of y .

“The conditions at the animal shelter have been pretty regular, except we had a string of about four months last year when the dogs barked all night, every night. But at the beginning of those four months we sheltered a redbone coonhound, and the barking stopped right after a family adopted her. All the times the redbone hound wasn’t present, there was no barking. Only the time she was present was there barking. Therefore, she probably incited all the other dogs to bark.”

(4) The Method of Concomitant Variation

If the frequency of event y increases and decreases as event x increases and decreases, respectively, x is probably the cause of y .

“We can predict the amount of alcohol sales by the rate of unemployment. As unemployment rises, so do alcohol sales. As unemployment drops, so do alcohol sales. Last quarter marked the highest unemployment in three years, and our sales last quarter are the highest they had been in those three years. Therefore, unemployment probably causes people to buy alcohol.”

(5) The Method of Residues

If a number of factors x , y , and z , may be responsible for a set of events A , B , and C , and if we discover reasons for thinking that x is the cause of A and y is the cause of B , then we have reason to believe z is the cause of C .

“The people who come through this medical facility are usually starving and have malaria, and a few have polio. We are particularly interested in treating the polio. Take this patient here: she is emaciated, which is caused by starvation; and she has a fever, which is caused by malaria. But notice that her muscles are deteriorating, and her bones are sore. This suggests she also has polio.”

d. Abduction

Not all inductive reasoning is inferential. In some cases, an explanation is needed before we can even begin drawing inferences. Consider Darwin’s idea of natural selection. Natural selection is not an object, like a blood vessel or a cellular wall, and it is not, strictly speaking, a single event. It cannot be detected in individual organisms or observed in a generation of offspring. Natural selection is an explanation of biodiversity that combines the process of heritable variation and environmental pressures to account for biomorphic change over long periods of time. With this explanation in hand, we can begin to draw some inferences. For instance, we can separate members of a single species of fruit flies, allow them to reproduce for several generations, and then observe whether the offspring of the two groups can reproduce. If we discover they cannot reproduce, this is likely due to certain mutations in their body types that prevent them from procreating. And since this is something we would expect if natural selection were true, we have one piece of confirming evidence for natural selection. But how do we know the explanations we come up with are worth our time?

Coined by C. S. Peirce (1839-1914), abduction , also called retroduction, or inference to the best explanation , refers to a way of reasoning informally that provides guidelines for evaluating explanations. Rather than appealing to types of arguments (generalization, analogy, causation), the value of an explanation depends on the theoretical virtues it exemplifies. A theoretical virtue is a quality that renders an explanation more or less fitting as an account of some event. What constitutes fittingness (or “loveliness,” as Peter Lipton (2004) calls it) is controversial, but many of the virtues are intuitively compelling, and abduction is a widely accepted tool of critical thinking.

The most widely recognized theoretical virtue is probably simplicity , historically associated with William of Ockham (1288-1347) and known as Ockham’s Razor . A legend has it that Ockham was asked whether his arguments for God’s existence prove that only one God exists or whether they allow for the possibility that many gods exist. He supposedly responded, “Do not multiply entities beyond necessity.” Though this claim is not found in his writings, Ockham is now famous for advocating that we restrict our beliefs about what is true to only what is absolutely necessary for explaining what we observe.

In contemporary theoretical use, the virtue of simplicity is invoked to encourage caution in how many mechanisms we introduce to explain an event. For example, if natural selection can explain the origin of biological diversity by itself, there is no need to hypothesize both natural selection and a divine designer. But if natural selection cannot explain the origin of, say, the duck-billed platypus, then some other mechanism must be introduced. Of course, not just any mechanism will do. It would not suffice to say the duck-billed platypus is explained by natural selection plus gremlins. Just why this is the case depends on other theoretical virtues; ideally, the virtues work together to help critical thinkers decide among competing hypotheses to test. Here is a brief sketch of some other theoretical virtues or ideals:

Conservatism – a good explanation does not contradict well-established views in a field.

Independent Testability – a good explanation is successful on different occasions under similar circumstances.

Fecundity – a good explanation leads to results that make even more research possible.

Explanatory Depth – a good explanation provides details of how an event occurs.

Explanatory Breadth – a good explanation also explains other, similar events.

Though abduction is structurally distinct from other inductive arguments, it functions similarly in practice: a good explanation provides a probabilistic reason to believe a proposition. This is why it is included here as a species of inductive reasoning. It might be thought that explanations only function to help critical thinkers formulate hypotheses, and do not, strictly speaking, support propositions. But there are intuitive examples of explanations that support propositions independently of however else they may be used. For example, a critical thinker may argue that material objects exist outside our minds is a better explanation of why we perceive what we do (and therefore, a reason to believe it) than that an evil demon is deceiving me , even if there is no inductive or deductive argument sufficient for believing that the latter is false. (For more, see “ Charles Sanders Peirce: Logic .”)

5. Detecting Poor Reasoning

Our attempts at thinking critically often go wrong, whether we are formulating our own arguments or evaluating the arguments of others. Sometimes it is in our interests for our reasoning to go wrong, such as when we would prefer someone to agree with us than to discover the truth value of a proposition. Other times it is not in our interests; we are genuinely interested in the truth, but we have unwittingly made a mistake in inferring one proposition from others. Whether our errors in reasoning are intentional or unintentional, such errors are called fallacies (from the Latin, fallax, which means “deceptive”). Recognizing and avoiding fallacies helps prevent critical thinkers from forming or maintaining defective beliefs.

Fallacies occur in a number of ways. An argument’s form may seem to us valid when it is not, resulting in a formal fallacy . Alternatively, an argument’s premises may seem to support its conclusion strongly but, due to some subtlety of meaning, do not, resulting in an informal fallacy . Additionally, some of our errors may be due to unconscious reasoning processes that may have been helpful in our evolutionary history, but do not function reliably in higher order reasoning. These unconscious reasoning processes are now widely known as heuristics and biases . Each type is briefly explained below.

a. Formal Fallacies

Formal fallacies occur when the form of an argument is presumed or seems to be valid (whether intentionally or unintentionally) when it is not. Formal fallacies are usually invalid variations of valid argument forms. Consider, for example, the valid argument form modus ponens (this is one of the rules of inference mentioned in §3b):

modus ponens (valid argument form)

In modus ponens , we assume or “affirm” both the conditional and the left half of the conditional (called the antecedent ): (p à q) and p. From these, we can infer that q, the second half or consequent , is true. This a valid argument form: if the premises are true, the conclusion cannot be false.

Sometimes, however, we invert the conclusion and the second premise, affirming that the conditional, (p à q), and the right half of the conditional, q (the consequent), are true, and then inferring that the left half, p (the antecedent), is true. Note in the example below how the conclusion and second premise are switched. Switching them in this way creates a problem.

To get an intuitive sense of why “affirming the consequent” is a problem, consider this simple example:

affirming the consequent

• It is a mammal.
• Therefore, it is a cat.(?)

From the fact that something is a mammal, we cannot conclude that it is a cat. It may be a dog or a mouse or a whale. The premises can be true and yet the conclusion can still be false. Therefore, this is not a valid argument form. But since it is an easy mistake to make, it is included in the set of common formal fallacies.

Here is a second example with the rule of inference called modus tollens . Modus tollens involves affirming a conditional, (p à q), and denying that conditional’s consequent: ~q. From these two premises, we can validly infer the denial of the antecedent: ~p. But if we switch the conclusion and the second premise, we get another fallacy, called denying the antecedent .

Technically, all informal reasoning is formally fallacious—all informal arguments are invalid. Nevertheless, since those who offer inductive arguments rarely presume they are valid, we do not regard them as reasoning fallaciously.

b. Informal Fallacies

Informal fallacies occur when the meaning of the terms used in the premises of an argument suggest a conclusion that does not actually follow from them (the conclusion either follows weakly or with no strength at all). Consider an example of the informal fallacy of equivocation , in which a word with two distinct meanings is used in both of its meanings:

• Any law can be repealed by Congress.
• Gravity is a law.
• Therefore, gravity can be repealed by Congress.

In this case, the argument’s premises are true when the word “law” is rightly interpreted, but the conclusion does not follow because the word law has a different referent in premise 1 (political laws) than in premise 2 (a law of nature). This argument equivocates on the meaning of law and is, therefore, fallacious.

Consider, also, the informal fallacy of ad hominem , abusive, when an arguer appeals to a person’s character as a reason to reject her proposition:

“Elizabeth argues that humans do not have souls; they are simply material beings. But Elizabeth is a terrible person and often talks down to children and the elderly. Therefore, she could not be right that humans do not have souls.”

The argument might look like this:

• Elizabeth is a terrible person and often talks down to children and the elderly.
• Therefore, Elizabeth is not right that humans do not have souls.

The conclusion does not follow because whether Elizabeth is a terrible person is irrelevant to the truth of the proposition that humans do not have souls. Elizabeth’s argument for this statement is relevant, but her character is not.

Another way to evaluate this fallacy is to note that, as the argument stands, it is an enthymeme (see §2); it is missing a crucial premise, namely: If anyone is a terrible person, that person makes false statements. But this premise is clearly false. There are many ways in which one can be a terrible person, and not all of them imply that someone makes false statements. (In fact, someone could be terrible precisely because they are viciously honest.) Once we fill in the missing premise, we see the argument is not cogent because at least one premise is false.

Importantly, we face a number of informal fallacies on a daily basis, and without the ability to recognize them, their regularity can make them seem legitimate. Here are three others that only scratch the surface:

Appeal to the People: We are often encouraged to believe or do something just because everyone else does. We are encouraged to believe what our political party believes, what the people in our churches or synagogues or mosques believe, what people in our family believe, and so on. We are encouraged to buy things because they are “bestsellers” (lots of people buy them). But the fact that lots of people believe or do something is not, on its own, a reason to believe or do what they do.

Tu Quoque (You, too!): We are often discouraged from pursuing a conclusion or action if our own beliefs or actions are inconsistent with them. For instance, if someone attempts to argue that everyone should stop smoking, but that person smokes, their argument is often given less weight: “Well, you smoke! Why should everyone else quit?” But the fact that someone believes or does something inconsistent with what they advocate does not, by itself, discredit the argument. Hypocrites may have very strong arguments despite their personal inconsistencies.

Base Rate Neglect: It is easy to look at what happens after we do something or enact a policy and conclude that the act or policy caused those effects. Consider a law reducing speed limits from 75 mph to 55 mph in order to reduce highway accidents. And, in fact, in the three years after the reduction, highway accidents dropped 30%! This seems like a direct effect of the reduction. However, this is not the whole story. Imagine you looked back at the three years prior to the law and discovered that accidents had dropped 30% over that time, too. If that happened, it might not actually be the law that caused the reduction in accidents. The law did not change the trend in accident reduction. If we only look at the evidence after the law, we are neglecting the rate at which the event occurred without the law. The base rate of an event is the rate that the event occurs without the potential cause under consideration. To take another example, imagine you start taking cold medicine, and your cold goes away in a week. Did the cold medicine cause your cold to go away? That depends on how long colds normally last and when you took the medicine. In order to determine whether a potential cause had the effect you suspect, do not neglect to compare its putative effects with the effects observed without that cause.

For more on formal and informal fallacies and over 200 different types with examples, see “ Fallacies .”

c. Heuristics and Biases

In the 1960s, psychologists began to suspect there is more to human reasoning than conscious inference. Daniel Kahneman and Amos Tversky confirmed these suspicions with their discoveries that many of the standard assumptions about how humans reason in practice are unjustified. In fact, humans regularly violate these standard assumptions, the most significant for philosophers and economists being that humans are fairly good at calculating the costs and benefits of their behavior; that is, they naturally reason according to the dictates of Expected Utility Theory. Kahneman and Tversky showed that, in practice, reasoning is affected by many non-rational influences, such as the wording used to frame scenarios (framing bias) and information most vividly available to them (the availability heuristic).

Consider the difference in your belief about the likelihood of getting robbed before and after seeing a news report about a recent robbery, or the difference in your belief about whether you will be bitten by a shark the week before and after Discovery Channel’s “Shark Week.” For most of us, we are likely to regard their likelihood as higher after we have seen these things on television than before. Objectively, they are no more or less likely to happen regardless of our seeing them on television, but we perceive they are more likely because their possibility is more vivid to us. These are examples of the availability heuristic.

Since the 1960s, experimental psychologists and economists have conducted extensive research revealing dozens of these unconscious reasoning processes, including ordering bias , the representativeness heuristic , confirmation bias , attentional bias , and the anchoring effect . The field of behavioral economics, made popular by Dan Ariely (2008; 2010; 2012) and Richard Thaler and Cass Sunstein (2009), emerged from and contributes to heuristics and biases research and applies its insights to social and economic behaviors.

Ideally, recognizing and understanding these unconscious, non-rational reasoning processes will help us mitigate their undermining influence on our reasoning abilities (Gigerenzer, 2003). However, it is unclear whether we can simply choose to overcome them or whether we have to construct mechanisms that mitigate their influence (for instance, using double-blind experiments to prevent confirmation bias).

6. The Scope and Virtues of Good Reasoning

Whether the process of critical thinking is productive for reasoners—that is, whether it actually answers the questions they are interested in answering—often depends on a number of linguistic, psychological, and social factors. We encountered some of the linguistic factors in §1. In closing, let us consider some of the psychological and social factors that affect the success of applying the tools of critical thinking.

Not all psychological and social contexts are conducive for effective critical thinking. When reasoners are depressed or sad or otherwise emotionally overwhelmed, critical thinking can often be unproductive or counterproductive. For instance, if someone’s child has just died, it would be unproductive (not to mention cruel) to press the philosophical question of why a good God would permit innocents to suffer or whether the child might possibly have a soul that could persist beyond death. Other instances need not be so extreme to make the same point: your company’s holiday party (where most people would rather remain cordial and superficial) is probably not the most productive context in which to debate the president’s domestic policy or the morality of abortion.

The process of critical thinking is primarily about detecting truth, and truth may not always be of paramount value. In some cases, comfort or usefulness may take precedence over truth. The case of the loss of a child is a case where comfort seems to take precedence over truth. Similarly, consider the case of determining what the speed limit should be on interstate highways. Imagine we are trying to decide whether it is better to allow drivers to travel at 75 mph or to restrict them to 65. To be sure, there may be no fact of the matter as to which is morally better, and there may not be any difference in the rate of interstate deaths between states that set the limit at 65 and those that set it at 75. But given the nature of the law, a decision about which speed limit to set must be made. If there is no relevant difference between setting the limit at 65 and setting it at 75, critical thinking can only tell us that , not which speed limit to set. This shows that, in some cases, concern with truth gives way to practical or preferential concerns (for example, Should I make this decision on the basis of what will make citizens happy? Should I base it on whether I will receive more campaign contributions from the business community?). All of this suggests that critical thinking is most productive in contexts where participants are already interested in truth.

b. The Principle of Charity/Humility

Critical thinking is also most productive when people in the conversation regard themselves as fallible, subject to error, misinformation, and deception. The desire to be “right” has a powerful influence on our reasoning behavior. It is so strong that our minds bias us in favor of the beliefs we already hold even in the face of disconfirming evidence (a phenomenon known as “confirmation bias”). In his famous article, “The Ethics of Belief” (1878), W. K. Clifford notes that, “We feel much happier and more secure when we think we know precisely what to do, no matter what happens, than when we have lost our way and do not know where to turn. … It is the sense of power attached to a sense of knowing that makes men desirous of believing, and afraid of doubting” (2010: 354).

Nevertheless, when we are open to the possibility that we are wrong, that is, if we are humble about our conclusions and we interpret others charitably, we have a better chance at having rational beliefs in two senses. First, if we are genuinely willing to consider evidence that we are wrong—and we demonstrate that humility—then we are more likely to listen to others when they raise arguments against our beliefs. If we are certain we are right, there would be little reason to consider contrary evidence. But if we are willing to hear it, we may discover that we really are wrong and give up faulty beliefs for more reasonable ones.

Second, if we are willing to be charitable to arguments against our beliefs, then if our beliefs are unreasonable, we have an opportunity to see the ways in which they are unreasonable. On the other hand, if our beliefs are reasonable, then we can explain more effectively just how well they stand against the criticism. This is weakly analogous to competition in certain types of sporting events, such as basketball. If you only play teams that are far inferior to your own, you do not know how good your team really is. But if you can beat a well-respected team on fair terms, any confidence you have is justified.

c. The Principle of Caution

In our excitement over good arguments, it is easy to overextend our conclusions, that is, to infer statements that are not really warranted by our evidence. From an argument for a first, uncaused cause of the universe, it is tempting to infer the existence of a sophisticated deity such as that of the Judeo-Christian tradition. From an argument for the compatibilism of the free will necessary for moral responsibility and determinism, it is tempting to infer that we are actually morally responsible for our behaviors. From an argument for negative natural rights, it is tempting to infer that no violation of a natural right is justifiable. Therefore, it is prudent to continually check our conclusions to be sure they do not include more content than our premises allow us to infer.

Of course, the principle of caution must itself be used with caution. If applied too strictly, it may lead reasoners to suspend all belief, and refrain from interacting with one another and their world. This is not, strictly speaking, problematic; ancient skeptics, such as the Pyrrhonians, advocated suspending all judgments except those about appearances in hopes of experiencing tranquility. However, at least some judgments about the long-term benefits and harms seem indispensable even for tranquility, for instance, whether we should retaliate in self-defense against an attacker or whether we should try to help a loved one who is addicted to drugs or alcohol.

d. The Expansiveness of Critical Thinking

The importance of critical thinking cannot be overstated because its relevance extends into every area of life, from politics, to science, to religion, to ethics. Not only does critical thinking help us draw inferences for ourselves, it helps us identify and evaluate the assumptions behind statements, the moral implications of statements, and the ideologies to which some statements commit us. This can be a disquieting and difficult process because it forces us to wrestle with preconceptions that might not be accurate. Nevertheless, if the process is conducted well, it can open new opportunities for dialogue, sometimes called “critical spaces,” that allow people who might otherwise disagree to find beliefs in common from which to engage in a more productive conversation.

It is this possibility of creating critical spaces that allows philosophical approaches like Critical Theory to effectively challenge the way social, political, and philosophical debates are framed. For example, if a discussion about race or gender or sexuality or gender is framed in terms that, because of the origins those terms or the way they have functioned socially, alienate or disproportionately exclude certain members of the population, then critical space is necessary for being able to evaluate that framing so that a more productive dialogue can occur (see Foresman, Fosl, and Watson, 2010, ch. 10 for more on how critical thinking and Critical Theory can be mutually supportive).

e. Productivity and the Limits of Rationality

Despite the fact that critical thinking extends into every area of life, not every important aspect of our lives is easily or productively subjected to the tools of language and logic. Thinkers who are tempted to subject everything to the cold light of reason may discover they miss some of what is deeply enjoyable about living. The psychologist Abraham Maslow writes, “I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail” (1966: 16). But it is helpful to remember that language and logic are tools, not the projects themselves. Even formal reasoning systems depend on axioms that are not provable within their own systems (consider Euclidean geometry or Peano arithmetic). We must make some decisions about what beliefs to accept and how to live our lives on the basis of considerations outside of critical thinking.

Borrowing an example from William James (1896), consider the statement, “Religion X is true.” James says that, while some people find this statement interesting, and therefore, worth thinking critically about, others may not be able to consider the truth of the statement. For any particular religious tradition, we might not know enough about it to form a belief one way or the other, and even suspending judgment may be difficult, since it is not obvious what we are suspending judgment about.

If I say to you: ‘Be a theosophist or be a Mohammedan,’ it is probably a dead option, because for you neither hypothesis is likely to be alive. But if I say: ‘Be an agnostic or be a Christian,’ it is otherwise: trained as you are, each hypothesis makes some appeal, however small, to your belief (2010: 357).

Ignoring the circularity in his definition of “dead option,” James’s point seems to be that if you know nothing about a view or what statements it entails, no amount of logic or evidence could help you form a reasonable belief about that position.

We might criticize James at this point because his conclusion seems to imply that we have no duty to investigate dead options, that is, to discover if there is anything worth considering in them. If we are concerned with truth, the simple fact that we are not familiar with a proposition does not mean it is not true or potentially significant for us. But James’s argument is subtler than this criticism suggests. Even if you came to learn about a particularly foreign religious tradition, its tenets may be so contrary to your understanding of the world that you could not entertain them as possible beliefs of yours . For instance, you know perfectly well that, if some events had been different, Hitler would not have existed: his parents might have had no children, or his parents’ parents might have had no children. You know roughly what it would mean for Hitler not to have existed and the sort of events that could have made it true that he did not exist. But how much evidence would it take to convince you that, in fact, Hitler did not exist, that is, that your belief that Hitler did exist is false ? Could there be an argument strong enough? Not obviously. Since all the information we have about Hitler unequivocally points to his existence, any arguments against that belief would have to affect a very broad range of statements; they would have to be strong enough to make us skeptical of large parts of reality.

7. Approaches to Improving Reasoning through Critical Thinking

Recall that the goal of critical thinking is not just to study what makes reasons and statements good, but to help us improve our ability to reason, that is, to improve our ability to form, hold, and discard beliefs according to whether they meet the standards of good thinking. Some ways of approaching this latter goal are more effective than others. While the classical approach focuses on technical reasoning skills, the Paul/Elder model encourages us to think in terms of critical concepts, and irrationality approaches use empirical research on instances of poor reasoning to help us improve reasoning where it is least obvious we need it and where we need it most. Which approach or combination of approaches is most effective depends, as noted above, on the context and limits of critical thinking, but also on scientific evidence of their effectiveness. Those who teach critical thinking, of all people, should be engaged with the evidence relevant to determining which approaches are most effective.

a. Classical Approaches

The classic approach to critical thinking follows roughly the structure of this article: critical thinkers attempt to interpret statements or arguments clearly and charitably, and then they apply the tools of formal and informal logic and science, while carefully attempting to avoid fallacious inferences (see Weston, 2008; Walton, 2008; Watson and Arp, 2015). This approach requires spending extensive time learning and practicing technical reasoning strategies. It presupposes that reasoning is primarily a conscious activity, and that enhancing our skills in these areas will improve our ability to reason well in ordinary situations.

There are at least two concerns about this approach. First, it is highly time intensive relative to its payoff. Learning the terminology of systems like propositional and categorical logic and the names of the fallacies, and practicing applying these tools to hypothetical cases requires significant time and energy. And it is not obvious, given the problems with heuristics and biases, whether this practice alone makes us better reasoners in ordinary contexts. Second, many of the ways we reason poorly are not consciously accessible (recall the heuristics and biases discussion in §5c). Our biases, combined with the heuristics we rely on in ordinary situations, can only be detected in experimental settings, and addressing them requires restructuring the ways in which we engage with evidence (see Thaler and Sunstein, 2009).

b. The Paul/Elder Model

Richard Paul and Linda Elder (Paul and Elder, 2006; Paul, 2012) developed an alternative to the classical approach on the assumption that critical thinking is not something that is limited to academic study or to the discipline of philosophy. On their account, critical thinking is a broad set of conceptual skills and habits aimed at a set of standards that are widely regarded as virtues of thinking: clarity, accuracy, depth, fairness, and others. They define it simply as “the art of analyzing and evaluating thinking with a view to improving it” (2006: 4). Their approach, then, is to focus on the elements of thought and intellectual virtues that help us form beliefs that meet these standards.

The Paul/Elder model is made up of three sets of concepts: elements of thought, intellectual standards, and intellectual traits. In this model, we begin by identifying the features present in every act of thought. They use “thought” to mean critical thought aimed at forming beliefs, not just any act of thinking, musing, wishing, hoping, remembering. According to the model, every act of thought involves:

These comprise the subject matter of critical thinking; that is, they are what we are evaluating when we are thinking critically. We then engage with this subject matter by subjecting them to what Paul and Elder call universal intellectual standards. These are evaluative goals we should be aiming at with our thinking:

While in classical approaches, logic is the predominant means of thinking critically, in the Paul/Elder model, it is put on equal footing with eight other standards. Finally, Paul and Elder argue that it is helpful to approach the critical thinking process with a set of intellectual traits or virtues that dispose us to using elements and standards well.

To remind us that these are virtues of thought relevant to critical thinking, they use “intellectual” to distinguish these traits from their moral counterparts (moral integrity, moral courage, and so on).

The aim is that, as we become familiar with these three sets of concepts and apply them in everyday contexts, we become better at analyzing and evaluating statements and arguments in ordinary situations.

Like the classical approach, this approach presupposes that reasoning is primarily a conscious activity, and that enhancing our skills will improve our reasoning. This means that it still lacks the ability to address the empirical evidence that many of our reasoning errors cannot be consciously detected or corrected. It differs from the classical approach in that it gives the technical tools of logic a much less prominent role and places emphasis on a broader, and perhaps more intuitive, set of conceptual tools. Learning and learning to apply these concepts still requires a great deal of time and energy, though perhaps less than learning formal and informal logic. And these concepts are easy to translate into disciplines outside philosophy. Students of history, psychology, and economics can more easily recognize the relevance of asking questions about an author’s point of view and assumptions than perhaps determining whether the author is making a deductive or inductive argument. The question, then, is whether this approach improves our ability to think better than the classical approach.

c. Other Approaches

A third approach that is becoming popular is to focus on the ways we commonly reason poorly and then attempt to correct them. This can be called the Rationality Approach , and it takes seriously the empirical evidence (§5c) that many of our errors in reasoning are not due to a lack of conscious competence with technical skills or misusing those skills, but are due to subconscious dispositions to ignore or dismiss relevant information or to rely on irrelevant information.

One way to pursue this approach is to focus on beliefs that are statistically rare or “weird.” These include beliefs of fringe groups, such as conspiracy theorists, religious extremists, paranormal psychologists, and proponents of New Age metaphysics (see Gilovich, 1992; Vaughn and Schick, 2010; Coady, 2012). If we recognize the sorts of tendencies that lead to these controversial beliefs, we might be able to recognize and avoid similar tendencies in our own reasoning about less extreme beliefs, such as beliefs about financial investing, how statistics are used to justify business decisions, and beliefs about which public policies to vote for.

Another way to pursue this approach is to focus directly on the research on error, those ordinary beliefs that psychologists and behavioral economists have discovered we reason poorly, and to explore ways of changing how we frame decisions about what to believe (see Nisbett and Ross, 1980; Gilovich, 1992; Ariely, 2008; Kahneman, 2011). For example, in one study, psychologists found that judges issue more convictions just before lunch and the end of the day than in the morning or just after lunch (Danzinger, et al., 2010). Given that dockets do not typically organize cases from less significant crimes to more significant crimes, this evidence suggests that something as irrelevant as hunger can bias judicial decisions. Even though hunger has nothing to do with the truth of a belief, knowing that it can affect how we evaluate a belief can help us avoid that effect. This study might suggest something as simple as that we should avoid being hungry when making important decisions. The more we learn ways in which our brains use irrelevant information, the better we can organize our reasoning to avoid these mistakes. For more on how decisions can be improved by restructuring our decisions, see Thaler and Sunstein, 2009.

A fourth approach is to take more seriously the role that language plays in our reasoning. Arguments involve complex patterns of expression, and we have already seen how vagueness and ambiguity can undermine good reasoning (§1). The pragma-dialectics approach (or pragma-dialectical theory) is the view that the quality of an argument is not solely or even primarily a matter of its logical structure, but is more fundamentally a matter of whether it is a form of reasonable discourse (Van Eemeren and Grootendorst, 1992). The proponents of this view contend that, “The study of argumentation should … be construed as a special branch of linguistic pragmatics in which descriptive and normative perspectives on argumentative discourse are methodically integrated” (Van Eemeren and Grootendorst, 1995: 130).

The pragma-dialectics approach is a highly technical approach that uses insights from speech act theory, H. P. Grice’s philosophy of language, and the study of discourse analysis. Its use, therefore, requires a great deal of background in philosophy and linguistics. It has an advantage over other approaches in that it highlights social and practical dimensions of arguments that other approaches largely ignore. For example, argument is often public ( external ), in that it creates an opportunity for opposition, which influences people’s motives and psychological attitudes toward their arguments. Argument is also social in that it is part of a discourse in which two or more people try to arrive at an agreement. Argument is also functional ; it aims at a resolution that can only be accommodated by addressing all the aspects of disagreement or anticipated disagreement, which can include public and social elements. Argument also has a rhetorical role ( dialectical ) in that it is aimed at actually convincing others, which may have different requirements than simply identifying the conditions under which they should be convinced.

These four approaches are not mutually exclusive. All of them presuppose, for example, the importance of inductive reasoning and scientific evidence. Their distinctions turn largely on which aspects of statements and arguments should take precedence in the critical thinking process and on what information will help us have better beliefs.

8. References and Further Reading

• Ariely, Dan. 2008. Predictably Irrational: The Hidden Forces that Shape Our Decisions. New York: Harper Perennial.
• Ariely, Dan. 2010. The Upside of Irrationality. New York: Harper Perennial.
• Ariely, Dan. 2012. The (Honest) Truth about Dishonesty. New York: Harper Perennial.
• Aristotle. 2002. Categories and De Interpretatione, J. L. Akrill, editor. Oxford: University of Oxford Press.
• Clifford, W. K. 2010. “The Ethics of Belief.” In Nils Ch. Rauhut and Robert Bass, eds., Readings on the Ultimate Questions: An Introduction to Philosophy, 3rd ed. Boston: Prentice Hall, 351-356.
• Chomsky, Noam. 1957/2002. Syntactic Structures. Berlin: Mouton de Gruyter.
• Coady, David. What To Believe Now: Applying Epistemology to Contemporary Issues. Malden, MA: Wiley-Blackwell, 2012.
• Danzinger, Shai, Jonathan Levav, and Liora Avnaim-Pesso. 2011. “Extraneous Factors in Judicial Decisions.” Proceedings of the National Academy of Sciences of the United States of America. Vol. 108, No. 17, 6889-6892. doi: 10.1073/pnas.1018033108.
• Foresman, Galen, Peter Fosl, and Jamie Carlin Watson. 2017. The Critical Thinking Toolkit. Malden, MA: Wiley-Blackwell.
• Fogelin, Robert J. and Walter Sinnott-Armstrong. 2009. Understanding Arguments: An Introduction to Informal Logic, 8th ed. Belmont, CA: Wadsworth Cengage Learning.
• Gigerenzer, Gerd. 2003. Calculated Risks: How To Know When Numbers Deceive You. New York: Simon and Schuster.
• Gigerenzer, Gerd, Peter Todd, and the ABC Research Group. 2000. Simple Heuristics that Make Us Smart. Oxford University Press.
• Gilovich, Thomas. 1992. How We Know What Isn’t So. New York: Free Press.
• James, William. “The Will to Believe”, in Nils Ch. Rauhut and Robert Bass, eds., Readings on the Ultimate Questions: An Introduction to Philosophy, 3rd ed. Boston: Prentice Hall, 2010, 356-364.
• Kahneman, Daniel. 2011. Thinking Fast and Slow. New York: Farrar, Strauss and Giroux.
• Lewis, David. 1986. On the Plurality of Worlds. Oxford Blackwell.
• Lipton, Peter. 2004. Inference to the Best Explanation, 2nd ed. London: Routledge.
• Maslow, Abraham. 1966. The Psychology of Science: A Reconnaissance. New York: Harper & Row.
• Mill, John Stuart. 2011. A System of Logic, Ratiocinative and Inductive. New York: Cambridge University Press.
• Nisbett, Richard and Lee Ross. 1980. Human Inference: Strategies and Shortcomings of Social Judgment. Englewood Cliffs, NJ: Prentice Hall.
• Paul, Richard. 2012. Critical Thinking: What Every Person Needs to Survive in a Rapidly Changing World. Tomales, CA: The Foundation for Critical Thinking.
• Paul, Richard and Linda Elder. 2006. The Miniature Guide to Critical Thinking Concepts and Tools, 4th ed. Tomales, CA: The Foundation for Critical Thinking.
• Plantinga, Alvin. 1974. The Nature of Necessity. Oxford Clarendon.
• Prior, Arthur. 1957. Time and Modality. Oxford, UK: Oxford University Press.
• Prior, Arthur. 1967. Past, Present and Future. Oxford, UK: Oxford University Press.
• Prior, Arthur. 1968. Papers on Time and Tense. Oxford, UK: Oxford University Press.
• Quine, W. V. O. and J. S. Ullian. 1978. The Web of Belief, 2nd ed. McGraw-Hill.
• Russell, Bertrand. 1940/1996. An Inquiry into Meaning and Truth, 2nd ed. London: Routledge.
• Thaler, Richard and Cass Sunstein. 2009. Nudge: Improving Decisions about Health, Wealth, and Happiness. New York: Penguin Books.
• van Eemeren, Frans H. and Rob Grootendorst. 1992. Argumentation, Communication, and Fallacies: A Pragma-Dialectical Perspective. London: Routledge.
• van Eemeren, Frans H. and Rob Grootendorst. 1995. “The Pragma-Dialectical Approach to Fallacies.” In Hans V. Hansen and Robert C. Pinto, eds. Fallacies: Classical and Contemporary Readings. Penn State University Press, 130-144.
• Vaughn, Lewis and Theodore Schick. 2010. How To Think About Weird Things: Critical Thinking for a New Age, 6th ed. McGraw-Hill.
• Walton, Douglas. 2008. Informal Logic: A Pragmatic Approach, 2nd ed. New York: Cambridge University Press.
• Watson, Jamie Carlin and Robert Arp. 2015. Critical Thinking: An Introduction to Reasoning Well, 2nd ed. London: Bloomsbury Academic.
• Weston, Anthony. 2008. A Rulebook for Arguments, 4th ed. Indianapolis: Hackett.
• Zadeh, Lofti. 1965. “Fuzzy Sets and Systems.” In J. Fox, ed., System Theory. Brooklyn, NY: Polytechnic Press, 29-39.

Author Information

Jamie Carlin Watson Email: [email protected] University of Arkansas for Medical Sciences U. S. A.

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Logic vs. Critical Thinking: A Comparative Analysis

We all face complex problems daily. Sound reasoning is key to solving them, but many struggle to effectively apply logical and critical thinking skills.

We’re here to help you sharpen your mental toolkit. In this post, we’ll break down the differences between logic and critical thinking, showing you how to use both for better decision-making.

You’ll learn the unique strengths of each approach when to apply them, and how they work together. By the end, you’ll clearly grasp these essential skills and be ready to tackle any challenge confidently.

Let’s get started on this path to smarter problem-solving!

What is Logical Thinking?

Logical thinking is a method of reasoning that uses a structured, rule-based approach to reach conclusions. It’s a valuable skill that helps us make sense of information and solve problems consistently.

This type of thinking is crucial in fields where precision and accuracy are key. It allows us to follow a clear path from premise to conclusion, ensuring our decisions are based on solid grounds.

When we think logically, we create a mental map of ideas and how they connect. We start with known facts or premises and then use established rules of logic to conclude. This process helps us avoid errors in reasoning and arrive at reliable outcomes.

Examples of Logical Thinking in Action

Let’s look at some real-life examples of logical thinking:

• When solving math problems, we use logical steps to reach the correct answer. For instance, when solving an algebraic equation, we follow a series of logical steps to isolate the variable and find its value.
• Coders use logical thinking in computer programming to create algorithms and debug their code. They must consider all possible scenarios and outcomes, using if-then statements and other logical constructs to ensure their programs function correctly under various conditions.
• Scientific experiments also rely on logical processes to test hypotheses and draw conclusions from data. Scientists design experiments with control groups and variables, interpret results using statistical analysis, and draw logical conclusions based on their findings.
• In everyday life, we might use logical thinking when planning a trip. We consider budget, time constraints, and personal preferences to make rational decisions about transportation, accommodation, and activities.

What is Critical Thinking?

Critical thinking carefully examines, analyzes, and evaluates ideas to form well-reasoned judgments. It’s about digging deeper, questioning assumptions, and considering multiple perspectives. It questions assumptions, meaning we don’t take things at face value but examine the underlying beliefs. This involves asking probing questions like “Why is this true?” or “What evidence supports this claim?”

This skill is essential today where we’re bombarded with information from various sources. Critical thinking helps us navigate this complexity and make informed decisions.

Critical thinking also involves metacognition – thinking about our thinking. We reflect on our thought processes, biases, and assumptions, which helps us improve our reasoning skills over time.

Examples of Critical Thinking in Action

Let’s look at some real-life examples of critical thinking:

• Critical thinking plays a crucial role in many real-world scenarios. We analyze arguments, spot logical flaws, and construct persuasive counterarguments when participating in debates. This requires us to think on our feet, evaluate evidence quickly, and articulate our thoughts clearly.
• Tackling ethical dilemmas requires weighing different moral considerations to make tough decisions. For example, a doctor facing a complex medical case might need to balance the potential benefits of a treatment against its risks, considering factors like quality of life, patient wishes, and medical ethics.
• Critical thinking is essential for strategic planning and decision-making in the business world. Managers must analyze market trends, assess risks, and anticipate potential outcomes to make informed choices about their company’s direction.

Key Differences Between Logical and Critical Thinking

Logical thinking emphasizes following set procedures. It’s like following a recipe—you follow the steps to get the expected result.

On the other hand, critical thinking is more about examining and questioning these steps. It asks, “Why are we doing this? Is there a better way?”

The process of logical thinking is like a straight line. You start at point A and move step-by-step to point B. It’s predictable and follows a clear path.

Critical thinking, however, is more like exploring a maze. You might try different routes, backtrack, or even create new paths as you go. It’s flexible and changes based on what you find along the way.

3. Objective

Logical thinking aims to find the right answer. It’s like solving a math problem – there’s usually one correct solution.

Critical thinking, on the other hand, is more interested in the bigger picture. It asks questions like “Is this information reliable?” “Is this fair to everyone involved?” and “What are we not seeing here?” It’s not just about finding an answer but understanding the question and its context.

How Logical and Critical Thinking Can Complement Each Other

When used together.

Combining logical and critical thinking leads to well-rounded decisions. Think of it like building a house. Rational thinking is like following the blueprint – it ensures everything is in the right place and constructed correctly. Critical thinking is like being the architect – it helps you design the best house for your needs, considering factors like the environment, lifestyle, and future changes.

For example, when solving a complex problem, you might use logical thinking to break it into smaller, manageable parts and solve each step. Then, you’d use critical thinking to step back, look at your solution, and ask if it addresses the core issue or if there might be a better approach.

Real-Life Example

Let’s say you’re making a big business decision, like whether to launch a new product. You’d use logical thinking to analyze sales data, production costs, and market research. This gives you a solid foundation of facts and figures.

Then, you’d switch to critical thinking to consider less tangible factors. You might ask How this product might affect our brand image. What could our competitors do in response? Are there any potential ethical concerns? What economic or social trends might impact the product’s success?

Using both types of thinking, you can make a data-driven decision considering the broader context and potential future scenarios.

As we wrap up our logical and critical thinking exploration, we’ve seen how these two approaches shape our problem-solving skills. Each has strengths: logical thinking offers structure and precision, while critical thinking brings depth and flexibility.

In our daily lives, we often need both. Logical thinking helps us navigate structured tasks, while critical thinking allows us to tackle complex issues and ethical dilemmas.

By understanding and applying both methods, we can make more informed decisions and solve problems more effectively. It’s not about choosing one over the other but knowing when and how to use each.

Remember, developing these skills takes practice. As you face challenges, consider which approach might serve you best. Over time, you’ll naturally blend logical and critical thinking for optimal results.

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3.9: Text- Critical Thinking and Logic

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

Logic’s Relationship to Critical Thinking

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

Questions of Logic in Critical Thinking

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

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

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

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

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

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

Introduction to Logic and Critical Thinking

(10 reviews)

Matthew Van Cleave, Lansing Community College

Copyright Year: 2016

Publisher: Matthew J. Van Cleave

Language: English

Formats Available

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Learn more about reviews.

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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