6.3: What is Logic?

Logic, in its most basic sense, is the study of how ideas reasonably fit together. In other words, when you apply logic, you must be concerned with analyzing ideas and arguments by using reason and rational thinking, not emotions or mysticism or belief. As a dedicated field of study, logic belongs primarily to math, philosophy, and computer science; in these fields, one can get professional training in logic. However, all academic disciplines employ logic: to evaluate evidence, to analyze arguments, to explain ideas, and to connect evidence to arguments. One of the most important uses of logic is in composing and evaluating arguments.

The study of logic divides into two main categories: formal and informal. Formal logic is the formal study of logic. In other words, in math or philosophy or computer science, if you were to take a class on logic, you would likely be learning formal logic. The purpose of formal logic is to eliminate any imprecision or lack of objectivity in evaluating arguments. Logicians, scholars who study and apply logic, have devised a number of formal techniques that accomplish this goal for certain classes of arguments. These techniques can include truth tables, Venn diagrams, proofs, syllogisms, and formulae. The different branches of formal logic include, but are not limited to, propositional logic, categorical logic, and first order logic.

Informal logic is logic applied outside of formal study and is most often used in college, business, and life. According to The Stanford Encyclopedia of Philosophy ,

For centuries, the study of logic has inspired the idea that its methods might be harnessed in efforts to understand and improve thinking, reasoning, and argument as they occur in real life contexts: in public discussion and debate; in education and intellectual exchange; in interpersonal relations; and in law, medicine, and other professions. Informal logic is the attempt to build a logic suited to this purpose. It combines the study of argument, evidence, proof and justification with an instrumental outlook which emphasizes its usefulness in the analysis of real life arguing.

When people apply the principles of logic to employ and evaluate arguments in real life situations and studies, they are using informal logic.

Why Is Logic Important?

Logic is one of the most respected elements of scholarly and professional thinking and writing. Consider that logic teaches us how to recognize good and bad arguments—not just arguments about logic, any argument. Nearly every undertaking in life will ultimately require that you evaluate an argument, perhaps several. You are confronted with a question: “Should I buy this car or that car?” “Should I go to this college or that college?” “Did that scientific experiment show what the scientist claims it did?” “Should I vote for the candidate who promises to lower taxes, or for the one who says she might raise them?” Your life is a long parade of choices.

When answering such questions, to make the best choices, you often have only one tool: an argument. You listen to the reasons for and against various options and must choose among them. Thus, the ability to evaluate arguments is an ability useful in everything that you will do—in your work, your personal life, and your deepest reflections. This is the job of logic.

If you are a student, note that nearly every discipline–be it a science, one of the humanities, or a study like business–relies upon arguments. Evaluating arguments is the most fundamental skill common to math, physics, psychology, history, literary studies, and any other intellectual endeavor. Logic alone tells you how to evaluate the arguments of any discipline.

The alternative to developing logic skills is to be always at the mercy of bad reasoning and, as a result, bad choices. Worse, you can be manipulated by deceivers. Speaking in Canandaigua, New York, on August 3, 1857, the escaped slave and abolitionist leader Frederick Douglass observed,

Power concedes nothing without a demand. It never did and it never will. Find out just what any people will quietly submit to and you have found out the exact measure of injustice and wrong which will be imposed upon them, and these will continue till they are resisted with either words or blows, or with both. The limits of tyrants are prescribed by the endurance of those whom they oppress.

Add this to Frederick Douglass’s words: If you find out just how much a person can be deceived, that is just how far she will be deceived. The limits of tyrants are also prescribed by the reasoning abilities of those they aim to oppress. What logic teaches you is how to demand and recognize good reasoning, and, hence, avoid deceit. You are only as free as your powers of reasoning enable.

The remaining part of this logic section will concern two types of logical arguments— inductive and deductive —and the tests of those arguments, including validity, soundness, reliability, and strength, so that you can check your own arguments and evaluate the arguments of others, no matter if those arguments come from the various academic disciplines, politics, the business world, or just discussions with friends and family.

What Is Deductive Argument?

A deductive argument is an argument whose conclusion is supposed to follow from its premises with absolute certainty, thus leaving no possibility that the conclusion doesn’t follow from the premises. If a deductive argument fails to guarantee the truth of the conclusion, then the deductive argument can no longer be called a deductive argument.

The Tests of Deductive Arguments: Validity and Soundness

So far in this chapter, you have learned what arguments are and how to determine their structure, including how to reconstruct arguments in standard form. But what makes an argument good or bad? There are four main ways to test arguments, two of which are for deductive arguments. The first test for deductive arguments is validity, a concept that is central to logical thinking. Validity relates to how well the premises support the conclusion and is the golden standard that every deductive argument should aim for. A valid argument is an argument whose conclusion cannot possibly be false, assuming that the premises are true. Another way to put this is as a conditional statement: A valid argument is an argument in which if the premises are true, the conclusion must be true. Here is an example of a valid argument:

1. Violet is a dog.
2. Therefore, Violet is a mammal. (from 1)

You might wonder whether it is true that Violet is a dog (maybe she’s a lizard or a buffalo—you have no way of knowing from the information given). But, for the purposes of validity, it doesn’t matter whether premise 1 is actually true or false. All that matters for validity is whether the conclusion follows from the premise. You can see that the conclusion—that Violet is a mammal—does seem to follow from the premise—that Violet is a dog. That is, given the truth of the premise, the conclusion has to be true. This argument is clearly valid because if you assume that “Violet is a dog” is true, then, since all dogs are mammals, it follows that “Violet is a mammal” must also be true. Thus, whether an argument is valid has nothing to do with whether the premises of the argument are actually true. Here is an example where the premises are clearly false, yet the argument is valid:

1. Everyone born in France can speak French.
2. Barack Obama was born in France.
3. Therefore, Barack Obama can speak French. (from 1-2)

This is a valid argument. Why? Because when you assume the truth of the premises (everyone born in France can speak French, and Barack Obama was born in France) the conclusion (Barack Obama can speak French) must be true. Notice that this is so even though none of these statements is actually true. Not everyone born in France can speak French (think about people who were born there but then moved somewhere else where they didn’t speak French and never learned it), and Barack Obama was not born in France, but it is also false that Obama can speak French. However, the argument is still valid even though neither the premises nor the conclusion is actually true. That may sound strange, but if you understand the concept of validity, it is not strange at all. Remember: validity describes the relationship between the premises and conclusion, and it means that the premises imply the conclusion, whether or not that conclusion is true.

To better understand the concept of validity, examine this example of an invalid argument:

1. George was President of the United States.
2. Therefore, George was elected President of the United States. (from 1)

This argument is invalid because it is possible for the premise to be true and yet the conclusion false. Here is a counterexample to the argument. Gerald Ford was President of the United States, but he was never elected president because Ford replaced Richard Nixon when Nixon resigned in the wake of the Watergate scandal. Therefore, it does not follow that just because someone is President of the United States that he was elected President of the United States. In other words, it is possible for the premise of the argument to be true and yet the conclusion false. This means that the argument is invalid. If an argument is invalid, it will always be possible to construct a counterexample to show that it is invalid (as demonstrated in the Gerald Ford scenario). A counterexample is simply a description of a scenario in which the premises of the argument are all true while the conclusion of the argument is false.

Exercise 4

Determine whether the following arguments are valid by using an informal test of validity. In other words, ask whether you can imagine a scenario in which the premises are both true and yet the conclusion is false. For each argument do the following: (1) If the argument is valid, explain your reasoning, and (2) if the argument is invalid, provide a counterexample. Remember, this is a test of validity, so you may assume all premises are true (even if you know or suspect they are not in real life) for the purposes of this assignment.

1. Katie is a human being. Therefore, Katie is smarter than a chimpanzee.

2. Bob is a fireman. Therefore, Bob has put out fires.

3. Gerald is a mathematics professor. Therefore, Gerald knows how to teach mathematics.

4. Monica is a French teacher. Therefore, Monica knows how to teach French.

5. Bob is taller than Susan. Susan is taller than Frankie. Therefore, Bob is taller than Frankie.

6. Craig loves Linda. Linda loves Monique. Therefore, Craig loves Monique.

7. Orel Hershizer is a Christian. Therefore, Orel Hershizer communicates with God.

8. All Muslims pray to Allah. Muhammad is a Muslim. Therefore, Muhammad prays to Allah.

9. Some protozoa are predators. No protozoa are animals. Therefore, some predators are not animals.

10. Charlie only barks when he hears a burglar outside. Charlie is barking. Therefore, there must be a burglar outside.

A good deductive argument is not only valid but also sound. A sound argument is a valid argument that has all true premises. That means that the conclusion, or claim, of a sound argument will always be true because if an argument is valid, the premises transmit truth to the conclusion on the assumption of the truth of the premises. If the premises are actually true, as they are in a sound argument, and since all sound arguments are valid, we know that the conclusion of a sound argument is true. The relationship between soundness and validity is easy to specify: all sound arguments are valid arguments, but not all valid arguments are sound arguments .

Professors will expect sound arguments in college writing. Philosophy professors, for the sake of pursuing arguments based on logic alone, may allow students to pursue unsound arguments, but nearly all other professors will want sound arguments. How do you make sure that all the premises of your argument are true? How can we know that Violet is a dog or that littering is harmful to animals and people? Answers to these questions come from evidence, often in the form of research.

Tip

One way to counter another’s argument is to question his premises and test them for soundness. If you find that one or more premise is unsound, you can add that information–and your explanations–to the support of your own argument.

One way to test the accuracy of a premise is to apply the following questions:

• Is there a sufficient amount of data?
• What is the quality of the data?
• Has additional data been missed?
• Is the data relevant?
• Are there additional possible explanations?

Determine whether the starting claim is based upon a sample that is both representative and sufficiently large, and ask yourself whether all relevant factors have been taken into account in the analysis of data that leads to a generalization.

Another way to evaluate a premise is to determine whether its source is credible. Ask yourself,

• Are the authors identified?
• What are their backgrounds?
• Was the claim something you found on an undocumented website?
• Did you find it in a popular publication or a scholarly one?
• How complete, how recent, and how relevant are the studies or statistics discussed in the source?

What Is Inductive Argument?

In contrast to a deductive argument, an inductive argument is an argument whose conclusion is supposed to follow from its premises with a high level of probability, which means that although it is possible that the conclusion doesn’t follow from its premises, it is unlikely that this is the case. Here is an example of an inductive argument:

Tweets is a healthy, normally functioning bird and since most healthy, normally functioning birds fly, Tweets most likely flies.

Notice that the conclusion, “Tweets probably flies,” contains the words “most likely.” This is a clear indicator that the argument is supposed to be inductive, not deductive. Here is the argument in standard form:

1. Tweets is a healthy, normally functioning bird. ( premise)
2. Most healthy, normally functioning birds fly. ( premise)
3. Therefore, Tweets probably flies. ( conclusion)

Given the information provided by the premises, the conclusion does seem to be well supported. That is, the premises provide strong reasons for accepting the conclusion. The inductive argument’s conclusion is a strong one, even though we can imagine a scenario in which the premises are true and yet the conclusion is false.

Remember, inductive arguments cannot guarantee the truth of the conclusion, which means they will look like invalid deductive arguments. Indeed, they are. There will be counterexamples for inductive arguments because an inductive argument never promises absolute truth. We measure inductive arguments by degrees of probability and plausibility, not absolute categories like validity and soundness. Validity and soundness do not allow for a sliding scale of degrees. They are absolute conditions: There is no such thing as being partially valid or somewhat sound.

Do not let this difference between deductive and inductive arguments cause you to privilege deductive and revile inductive because inductive arguments cannot guarantee truth. That is an unfair measure, and it is not practical. The truth is that most arguments we create and evaluate in life are inductive arguments. It might be helpful to think of deductive arguments as those created in perfect lab conditions, where all the ideal parameters can be met. Life is much messier than that, and we rarely get ideal conditions. One main reason is that we rarely ever have all the information we need to form an absolutely true conclusion. When new information is discovered, a scientist or historian or psychologist or business executive or a college student should investigate how it affects previous ideas and arguments, knowing that those previous ideas may need to be adjusted based on new information. For example, suppose that we added the following premise to our earlier argument:

Tweets is 6 feet tall and can run 30 mph. ( premise)

When we add this premise, the conclusion that Tweets can fly would no longer be likely because any bird that is 6 feet tall and can run 30 mph, is not a kind of bird that can fly. That information leads us to believe that Tweets is an ostrich or emu, which are not kinds of birds that can fly.

The Tests of Inductive Arguments: Reliability and Strength

Inductive arguments can never lead to absolute certainty, which is one reason scholars keep studying and trying to add to knowledge. This does not mean, however, that any inductive argument will be a good one. Inductive arguments must still be evaluated and tested, and the two main tests are reliability and strength.

Test of reliability, much like that of validity for deductive arguments, tests an inductive argument’s reason, its internal logic. In other words, just because an inductive argument cannot guarantee a true conclusion doesn’t mean that it should not be logically constructed. One cannot make just any sort of claim, particularly one that does not have a reliable basis. Reliability, unlike validity, can be measured by degree. More reliable arguments are ones that have a more solid basis in reason. Consider this example:

Ninety-seven percent of Banana^TMcomputers work without any glitches. ( premise)

Max has a Banana^TMcomputer. ( premise)

Therefore, Max’s computer works without any glitches. ( conclusion)

This argument has a high degree of reliability. While it may well be true that Max has one of the three percent of computers that have glitches, it is much more likely, given the initial premise that he does not. If the initial premise changes, however, so does the reliability of the argument:

Thirty-three percent of Banana^TMcomputers work without any glitches.

Max has a Banana^TMcomputer.

Therefore, Max’s computer works without any glitches.

Note how the degree of reliability has gone done dramatically. The argument can now be considered unreliable since the conclusion that Max’s computer will work without glitches is improbable given the premises provided. The conclusion still could be true, but it has tipped toward unlikely.

The second test of inductive arguments is strength. Strength, like reliability, can be measured by degree. Strong arguments must have the following conditions: (1) They must be reliable arguments; (2) they draw upon multiple lines of reasoning as support and/or a collection of data. Indeed, the more the data and the more the reasons for a conclusion, the stronger the argument. Consider the following argument:

Susie has walked by Mack the dog every day for ten days. ( premise)

Mack the dog has never bitten Susie. ( premise)

Thus, when Susie walks by Mack the dog today, he will not bite her. ( conclusion)

This argument is reasonable; we can see that the premises may logically lead to the conclusion. However, the argument is not very strong as Susie has only walked by the dog for ten days. Is that enough data to make the conclusion a likely one? What if we had more data, like so—

Susie has walked by Mack the dog every day for five years.

Mack the dog has never bitten Susie.

Thus, when Susie walks by Mack the dog today, he will not bite her.

This argument, with more data to consider (five years of information instead of just ten days), is much stronger. An argument also gets stronger when reasons are added:

Susie has walked by Mack the dog every day for five years.

Mack the dog has never bitten Susie.

Mack’s owners trained him to be friendly to people. ( additional premise)

Mack the dog’s breed is not known for aggression. ( additional premise)

Thus, when Susie walks by Mack the dog today, he will not bite her.

This argument is even stronger. Not only does it have more data, but it also has additional reasons for Mack’s gentle nature.

Remember these tests when writing your own essays. You are most likely going to be using inductive arguments, and you should make them as reliable and strong as you can because you can bet your professors will be evaluating your arguments by those criteria as well.

What Are Logical Fallacies, and Why Should You Avoid Them?

Fallacies are errors or tricks of reasoning. A fallacy is an error of reasoning if it occurs accidentally; it is a trick of reasoning if a speaker or writer uses it to deceive or manipulate his audience. Fallacies can be either formal or informal.

Whether a fallacy is an error or a trick, whether it is formal or informal, its use undercuts the validity and soundness of any argument. At the same time, fallacious reasoning can damage the credibility of the speaker or writer and improperly manipulate the emotions of the audience or reader. This is a consideration you must keep in mind as a writer who is trying to maintain credibility ( ethos ) with the reader. Moreover, being able to recognize logical fallacies in the speech and writing of others can greatly benefit you as both a college student and a participant in civic life. Not only does this awareness increase your ability to think and read critically—and thus not be manipulated or fooled—but it also provides you with a strong basis for counter arguments.

Even more important, using faulty reasoning is unethical and irresponsible. Using logical fallacies can be incredibly tempting. The unfortunate fact is they work. Every day—particularly in politics and advertising—we can see how using faults and tricks of logic effectively persuade people to support certain individuals, groups, and ideas and, conversely, turn them away from others. Furthermore, logical fallacies are easy to use. Instead of doing the often difficult work of carefully supporting an argument with facts, logic, and researched evidence, the lazy debater turns routinely to the easy path of tricky reasoning. Human beings too often favor what is easy and effective, even if morally questionable, over what is ethical, particularly if difficult. However, your college professors’ task is not to teach you how to join the Dark Side. Their job is to teach you how to write, speak, and argue effectively and ethically . To do so, you must recognize and avoid the logical fallacies.

What Are Formal Fallacies?

Most formal fallacies are errors of logic: The conclusion does not really “follow from” (is not supported by) the premises. Either the premises are untrue, or the argument is invalid. Below is an example of an invalid deductive argument:

Premise: All black bears are omnivores.

Premise: All raccoons are omnivores.

Conclusion: All raccoons are black bears.

Bears are a subset of omnivores. Raccoons also are a subset of omnivores. But these two subsets do not overlap, and that fact makes the conclusion illogical. The argument is invalid—that is, the relationship between the two premises does not support the conclusion.

“Raccoons are black bears” is instantaneously recognizable as fallacious and may seem too silly to be worth bothering about. However, that and other forms of poor logic play out on a daily basis, and they have real world consequences. Below is an example of a common fallacious argument:

Premise: All Arabs are Muslims.

Premise: All Iranians are Muslims.

Conclusion: All Iranians are Arabs.

This argument fails on two levels. First, the premises are untrue because, although many Arabs and Iranians are Muslim, not all are. Second, the two ethnic groups (Iranians and Arabs) are sets that do not overlap; nevertheless, the two groups are confounded because they (largely) share one quality in common (being Muslim). One only has to look at comments on the web to realize that the confusion is widespread and that it influences attitudes and opinions about US foreign policy. The logical problems make this both an invalid and an unsound argument.

What Are Informal Fallacies?

Informal fallacies take many forms and are widespread in everyday discourse. Very often they involve bringing irrelevant information into an argument, or they are based on assumptions that, when examined, prove to be incorrect. Formal fallacies are created when the relationship between premises and conclusion does not hold up or when premises are unsound; informal fallacies are more dependent on misuse of language and of evidence.

It is easy to find lists of informal fallacies, but that does not mean that it is always easy to spot them.

How Can You Check for Logical Fallacies?

One way to go about evaluating an argument for fallacies is to return to the concept of the three fundamental appeals: ethos, logos, and pathos. As a quick reminder,

• Ethos is an appeal to ethics, authority, and/or credibility.
• Logos is an appeal to logic.
• Pathos is an appeal to emotion.

Once you have refreshed your memory of the basics, you may begin to understand how ethos, logos, and pathos can be used appropriately to strengthen your argument or inappropriately to manipulate an audience through the use of fallacies. Classifying fallacies as fallacies of ethos, logos, or pathos will help you to understand their nature and to recognize them. Please keep in mind, however, that some fallacies may fit into multiple categories. For more details and examples on errors in the rhetorical appeals, see Chapter 2, “Rhetorical Analysis.”

Fallacies of ethos relate to credibility. These fallacies may unfairly build up the credibility of the author (or his allies) or unfairly attack the credibility of the author’s opponent (or her allies). Some fallacies give an unfair advantage to the claims of the speaker or writer or an unfair disadvantage to his opponent’s claims. These are fallacies of logos. Fallacies of pathos rely excessively upon emotional appeals, attaching positive associations to the author’s argument and negative ones to his opponent’s position.

Key Takeaways: Logic

• Logic —shows how ideas fit together by using reason.
• Formal Logic —a formal and rigorous study of logic, such as in math and philosophy.
• Informal Logic —the application of logic to arguments of all types: in scholarship, in business, and in life. Informal logic is what this part of the chapter covers.
• Deductive Argument —guarantees a true conclusion based on the premises. The tests for deductive arguments are validity and soundness.
• Validity —a way to evaluate a deductive argument; a valid argument is one which, if the premises are true, the conclusion must be true.
• Soundness —the second way to evaluate a deductive argument; a sound argument is one where the argument is valid AND the premises have been shown to be true (via support).
• Inductive Argument —cannot guarantee a true conclusion but can only assert what is most likely to be true based on the premises and the support. The tests for inductive arguments are reliability and strength.
• Reliability —a test of reason for inductive arguments. Inductive arguments must still be reasonable, must still have a reliable basis in logic.
• Strength —another test for inductive arguments. Inductive arguments are stronger when they have more reasons and more data to support them.
• Logical Fallacy —a flaw or trick of logic to be avoided at all costs. Fallacies can be formal or informal. See the Repository of Logical Fallacies below for individual examples.