What does deductively valid mean

Construct A Deductively Valid Argument/Deductive Logic

Discussion 1

Construct a Deductively Valid Argument

The topic of this week is deductive reasoning. Accordingly, in this discussion your task is to create a deductively valid argument for your position (the same position that you defended in the Week One discussion). (Topic: Is it important to teach arts and humanities to children?) This is the position.

Prepare: To prepare to respond to this prompt, make sure to read carefully over the required portions of Chapter 3 and Chapter 4. See attached file)

View the deLaplante (2013) video What Is a Valid Argument? (see attached file)

Based on the sources, create a deductively valid argument for the position you defended in the Week One discussion. (Topic: Is it important to teach arts and humanities to children?) This is the position.

To make your argument deductively valid, you will need to make sure that there is no possible way that your premises could be true and your conclusion false. Your premises must lead logically to the truth of your conclusion. Make sure that your argument is sound that is in addition to being valid, make sure that the premises are true as far as you can tell. If your argument is invalid or if it has a false premise, revise it until you get an argument that you can stand behind.

Write: Identify the components and structure of your argument by presenting your deductively valid argument in standard form, and explain how your conclusion follows from your premises.

Discussion 2

Week 2, Prompt option #2: Fill in the Missing Premises

We have learned this week about deductive reasoning, including what it takes for an argument to be valid. (See attached file examples only) This discussion allows us to get more practice with the concept through making arguments valid. You will see a list of arguments here. These arguments are not presented in standard form, and each is missing a premise that would be necessary to make it valid. Your tasks will be to put the argument into standard form and add the missing premise that would validly link the premises to the conclusion.

Prepare: To prepare to respond to this prompt, reread the section from Chapter 2 of our book titled “Extracting Arguments in Standard form,” all required portions of Chapters 3 and 4, as well as the guidance and required media for this week. (see attached files for chapters) and attached video file

Reflect: Look at the list of argument options below. Choose an argument that has not yet been chosen by any of your classmates. ( I have taken out the ones that have already been taken.)Think through the reasoning and determine what premise is (or premises are) missing that would be needed to make the argument valid. You might also consider challenging yourself by choosing from the more difficult examples in the list (at the bottom).

Choose from the following list of argument options.
2. Football is dumb because it is a waste of time.
3. If he loved you he would have shown up on time with flowers. He must not love you.
4. All mammals bear live young, so dragons are not mammals.
6. He broke the record for rushing yards in a game on that last play. Therefore he holds the record.
7. He won the election. So he will be the next governor.
8. He won’t go to the wedding since he doesn’t like mushy stuff and weddings are mushy.
9. I can’t go to the movies with you – I have a test tomorrow and I have to study.
11. You shouldn’t go out with that guy. He rides a motorcycle and goes to bars.
12. Capital punishment is wrong because it is killing and it doesn’t save anyone’s life.
13. You shouldn’t use drugs because they are addictive and can ruin people’s lives.
14. To fix your care you will need money. However, to have money you have to have money. It appears that you need to get a job.
15. To go to the movie you have to have a ticket. To buy a ticket you must pay money. Thus, to go the movie you must pay money.
16. If you don’t do your chores then you can’t have any dessert. You really like dessert, so you will certainly do your chores.
17. You will get an A if you study hard and always come to class. You came to class every time and studied. You are bound to get an A.
18. Julie is allergic to gluten. So she won’t be having any bread.
19. Only women can have babies, so women are more important to the survival of the species.
20. If I wear that cologne then women will love me. I bought that cologne, so women are going to love me.
21. I can’t go to the party because there will be alcohol there, and I am a Mormon.
22. You shouldn’t force me to wear a seat belt because that would violate my rights.
23. In order to buy a car you will need money. But to have money you need to get a job. But to go to a job you will need to be able to get to work. So you will not be able to buy a car.
24. Capital punishment kills a human being. It is wrong to kill a human being except in self-defense. So capital punishment is wrong.
25. You shouldn’t tell someone to do something unless you would be willing to do it yourself. You’ve never gone to war. So you shouldn’t vote for others to go to war.
26. If you talk to Mike about politics then he will yell at you. If he yells at you then you will be hurt and it will damage your friendship. Therefore, you shouldn’t talk to Mike about politics.
27. Either the maid or the butler did it. For the butler to have done it he would have had to have been at the mansion yesterday. The butler was away all day yesterday. So, the maid did it.
28. If the maid was guilty then she would have had to been at the scene during the crime. However, she was seen a mile away only minutes before the crime, and she has no car. She must be innocent.
29. It is always wrong to kill a human being unless it is necessary to save somebody’s life. Abortion kills a human being. So abortion is wrong unless the mother’s life is in danger due to the pregnancy.
30. Government intervention is justified if it is necessary to protect the welfare of the people and does not violate anyone’s constitutional rights. Therefore, government intervention is justified in this specific case because it is necessary to protect the welfare of the people.

.1 Basic Concepts in Deductive Reasoning

As noted in Chapter 2, at the broadest level there are two types of arguments: deductive and inductive. The difference between these types is largely a matter of the strength of the connection between premises and conclusion. Inductive arguments are defined and discussed in Chapter 5; this chapter focuses on deductive arguments. In this section we will learn about three central concepts: validity, soundness, and deduction.

Validity

Deductive arguments aim to achieve validity, which is an extremely strong connection between the premises and the conclusion. In logic, the word valid is only applied to arguments; therefore, when the concept of validity is discussed in this text, it is solely in reference to arguments, and not to claims, points, or positions. Those expressions may have other uses in other fields, but in logic, validity is a strict notion that has to do with the strength of the connection between an argument’s premises and conclusion.

To reiterate, an argument is a collection of sentences, one of which (the conclusion) is supposed to follow from the others (the premises). A valid argument is one in which the truth of the premises absolutely guarantees the truth of the conclusion; in other words, it is an argument in which it is impossible for the premises to be true while the conclusion is false. Notice that the definition of valid does not say anything about whether the premises are actually true, just whether the conclusion could be false if the premises were true. As an example, here is a silly but valid argument:

Everything made of cheese is tasty.

The moon is made of cheese.

Therefore, the moon is tasty.

No one, we hope, actually thinks that the moon is made of cheese. You may or may not agree that everything made of cheese is tasty. But you can see that if everything made of cheese were tasty, and if the moon were made of cheese, then the moon would have to be tasty. The truth of that conclusion simply logically follows from the truth of the premises.

Here is another way to better understand the strictness of the concept of validity: You have probably seen some far-fetched movies or read some bizarre books at some point. Books and movies have magic, weird science fiction, hallucinations, and dream sequences—almost anything can happen. Imagine that you were writing a weird, bizarre novel, a novel as far removed from reality as possible. You certainly could write a novel in which the moon was made of cheese. You could write a novel in which everything made of cheese was tasty. But you could not write a novel in which both of these premises were true, but in which the moon turned out not to be tasty. If the moon were made of cheese but was not tasty, then there would be at least one thing that was made of cheese and was not tasty, making the first premise false.

Therefore, if we assume, even hypothetically, that the premises are true (even in strange hypothetical scenarios), it logically follows that the conclusion must be as well. Therefore, the argument is valid. So when thinking about whether an argument is valid, think about whether it would be possible to have a movie in which all the premises were true but the conclusion was false. If it is not possible, then the argument is valid.

Here is another, more realistic, example:

All whales are mammals.

All mammals breathe air.

Therefore, all whales breathe air.

Is it possible for the premises to be true and the conclusion false? Well, imagine that the conclusion is false. In that case there must be at least one whale that does not breathe air. Let us call that whale Fred. Is Fred a mammal? If he is, then there is at least one mammal that does not breathe air, so the second premise would be false. If he isn’t, then there is at least one whale that is not a mammal, so the first premise would be false. Again, we see that it is impossible for the conclusion to be false and still have all the premises be true. Therefore, the argument is valid.

Here is an example of an invalid argument:

All whales are mammals.

No whales live on land.

Therefore, no mammals live on land.

In this case we can tell that the truth of the conclusion is not guaranteed by the premises because the premises are actually true and the conclusion is actually false. Because a valid argument means that it is impossible for the premises to be true and the conclusion false, we can be sure that an argument in which the premises are actually true and the conclusion is actually false must be invalid. Here is a trickier example of the same principle:

All whales are mammals.

Some mammals live in the water.

Therefore, some whales live in the water.

A wet, tree-lined road.

Consider the following argument: “If it is raining, then the streets are wet. The streets are wet. Therefore, it is raining.” Is this a valid argument? Could there be another reason why the road is wet?

This one is trickier because both premises are true, and the conclusion is true as well, so many people may be tempted to call it valid. However, what is important is not whether the premises and conclusion are actually true but whether the premises guarantee that the conclusion is true. Think about making a movie: Could you make a movie that made this argument’s premises true and the conclusion false?

Suppose you make a movie that is set in a future in which whales move back onto land. It would be weird, but not any weirder than other ideas movies have presented. If seals still lived in the water in this movie, then both premises would be true, but the conclusion would be false, because all the whales would live on land.

Because we can create a scenario in which the premises are true and the conclusion is false, it follows that the argument is invalid. So even though the conclusion isn’t actually false, it’s enough that it is possible for it to be false in some situation that would make the premises true. This mere possibility means the argument is invalid.

Soundness

Once you understand what valid means in logic, it is very easy to understand the concept of soundness. A sound argument is just a valid argument in which all the premises are true. In defining validity, we saw two examples of valid arguments; one of them was sound and the other was not. Since both examples were valid, the one with true premises was the one that was sound.

We also saw two examples of invalid arguments. Both of those are unsound simply because they are invalid. Sound arguments have to be valid and have all true premises. Notice that since only arguments can be valid, only arguments can be sound. In logic, the concept of soundness is not applied to principles, observations, or anything else. The word sound in logic is only applied to arguments.

Here is an example of a sound argument, similar to one you may recall seeing in Chapter 2:

All men are mortal.

Bill Gates is a man.

Therefore, Bill Gates is mortal.

There is no question about the argument’s validity. Therefore, as long as these premises are true, it follows that the conclusion must be true as well. Since the premises are, in fact, true, we can reason the conclusion is too.

It is important to note that having a true conclusion is not part of the definition of soundness. If we were required to know that the conclusion was true before deciding whether the argument is sound, then we could never use a sound argument to discover the truth of the conclusion; we would already have to know that the conclusion was true before we could judge it to be sound. The magic of how deductive reasoning works is that we can judge whether the reasoning is valid independent of whether we know that the premises or conclusion are actually true. If we also notice that the premises are all true, then we may infer, by the power of pure reasoning, the truth of the conclusion.

Therefore, knowledge of the truth of the premises and the ability to reason validly enable us to arrive at some new information: that the conclusion is true as well. This is the main way that logic can add to our bank of knowledge.

Although soundness is central in considering whether to accept an argument’s conclusion, we will not spend much time worrying about it in this book. This is because logic really deals with the connections between sentences rather than the truth of the sentences themselves. If someone presents you with an argument about biology, a logician can help you see whether the argument is valid—but you will need a biologist to tell you whether the premises are true. The truth of the premises themselves, therefore, is not usually a matter of logic. Because the premises can come from any field, there would be no way for logic alone to determine whether such premises are true or false. The role of logic—specifically, deductive reasoning—is to determine whether the reasoning used is valid.

Deduction

You have likely heard the term deduction used in other contexts: As Chapter 2 noted, the detective Sherlock Holmes (and others) uses deduction to refer to any process by which we infer a conclusion from pieces of evidence. In rhetoric classes and other places, you may hear deduction used to refer to the process of reasoning from general principles to a specific conclusion. These are all acceptable uses of the term in their respective contexts, but they do not reflect how the concept is defined in logic.

In logic, deduction is a technical term. Whatever other meanings the word may have in other contexts, in logic, it has only one meaning: A deductive argument is one that is presented as being valid. In other words, a deductive argument is one that is trying to be valid. If an argument is presented as though it is supposed to be valid, then we may infer it is deductive. If an argument is deductive, then the argument can be evaluated in part on whether it is, in fact, valid. A deductive argument that is not found to be valid has failed in its purpose of demonstrating its conclusion to be true.

In Chapters 5 and 6, we will look at arguments that are not trying to be valid. Those are inductive arguments. As noted in Chapter 2, inductive arguments simply attempt to establish their conclusion as probable—not as absolutely guaranteed. Thus, it is not important to assess whether inductive arguments are valid, since validity is not the goal. However, if a deductive argument is not valid, then it has failed in its goal; therefore, for deductive reasoning, validity is a primary concern.

Consider someone arguing as follows:

All donuts have added sugar.

All donuts are bad for you.

Therefore, everything with added sugar is bad for you.

Two men engage in a discussion.

Interpreting the intention of the person making an argument is a key step in determining whether the argument is deductive.

Even though the argument is invalid—exactly why this is so will be clearer in the next section—it seems clear that the person thinks it is valid. She is not merely suggesting that maybe things with added sugar might be bad for you. Rather, she is presenting the reasoning as though the premises guarantee the truth of the conclusion. Therefore, it appears to be an attempt at deductive reasoning, even though this one happens to be invalid.

Because our definition of validity depends on understanding the author’s intention, this means that deciding whether something is a deductive argument requires a bit of interpretation—we have to figure out what the person giving the argument is trying to do. As noted briefly in Chapter 2, we ought to seek to provide the most favorable possible interpretation of the author’s intended reasoning. Once we know that an argument is deductive, the next question in evaluating it is whether it is valid. If it is deductive but not valid, we really do not need to consider anything further; the argument fails to demonstrate the truth of its conclusion in the intended sense.

3.2 Evaluating Deductive Arguments

In addition to his well-known literary works, Lewis Carroll wrote several mathematical works, including three books on logic: Symbolic Logic Parts 1 and 2, and The Game of Logic, which was intended to introduce logic to children.

If validity is so critical in evaluating deductive arguments, how do we go about determining whether an argument is valid or invalid? In deductive reasoning, the key is to look at the pattern of an argument , which is called its logical form. As an example, see if you can tell whether the following argument is valid:

All quidnuncs are shunpikers.

All shunpikers are flibbertigibbets.

Therefore, all quidnuncs are flibbertigibbets.

You could likely tell that the argument is valid even though you do not know the meanings of the words. This is an important point. We can often tell whether an argument is valid even if we are not in a position to know whether any of its propositions are true or false. This is because deductive validity typically depends on certain patterns of argument. In fact, even nonsense arguments can be valid. Lewis Carroll (a pen name for C. L. Dodgson) was not only the author of Alice’s Adventures in Wonderland, but also a clever logician famous for both his use of nonsense words and his tricky logic puzzles.

We will look at some of Carroll’s puzzles in this chapter’s sections on categorical logic, but for now, let us look at an argument using nonsense words from his poem “Jabberwocky.” See if you can tell whether the following argument is valid:

All bandersnatches are slithy toves.

All slithy toves are uffish.

Therefore, all bandersnatches are uffish.

If you could tell the argument about quidnuncs was valid, you were probably able to tell that this argument is valid as well. Both arguments have the same pattern, or logical form.

Representing Logical Form

Logical form is generally represented by using variables or other symbols to highlight the pattern. In this case the logical form can be represented by substituting capital letters for certain parts of the propositions. Our argument then has the form:

All S are M.

All M are P.

Therefore, all S are P.

Any argument that follows this pattern, or form, is valid. Try it for yourself. Think of any three plural nouns; they do not have to be related to each other. For example, you could use submarines, candy bars, and mountains. When you have thought of three, substitute them for the letters in the pattern given. You can put them in any order you like, but the same word has to replace the same letter. So you will put one noun in for S in the first and third lines, one noun for both instances of M, and your last noun for both cases of P. If we use the suggested nouns, we would get:

All submarines are candy bars.

All candy bars are mountains.

Therefore, all submarines are mountains.

This argument may be close to nonsense, but it is logically valid. It would not be possible to make up a story in which the premises were true but the conclusion was false. For example, if one wizard turns all submarines into candy bars, and then a second wizard turns all candy bars into mountains, the story would not make any sense (nor would it be logical) if, in the end, all submarines were not mountains. Any story that makes the premises true would have to also make the conclusion true, so that the argument is valid.

As mentioned, the form of an argument is what you get when you remove the specific meaning of each of the nonlogical words in the argument and talk about them in terms of variables. Sometimes, however, one has to change the wording of a claim to make it fit the required form. For example, consider the premise “All men like dogs.” In this case the first category would be “men,” but the second category is not represented by a plural noun but by a predicate phrase, “like dogs.” In such cases we turn the expression “like dogs” into the noun phrase “people who like dogs.” In that case the form of the sentence is still “All A are B,” in which B is “people who like dogs.” As another example, the argument:

All whales are mammals.

Some mammals live in the water.

Therefore, at least some whales live in the water.

can be rewritten with plural nouns as:

All whales are mammals.

Some mammals are things that live in the water.

Therefore, at least some whales are things that live in the water.

and has the form:

All A are B.

Some B are C.

Therefore, at least some A are C.

The variables can represent anything (anything that fits grammatically, that is). When we substitute specific expressions (of the appropriate grammatical category) for each of the variables, we get an instance of that form. So another instance of this form could be made by replacing A with Apples, B with Bananas, and C with Cantaloupes. This would give us

All Apples are Bananas.

Some Bananas are Cantaloupes.

Therefore, at least some Apples are Cantaloupes.

It does not matter at this stage whether the sentences are true or false or whether the reasoning is valid or invalid. All we are concerned with is the form or pattern of the argument.

We will see many different patterns as we study deductive logic. Different kinds of deductive arguments require different kinds of forms. The form we just used is based on categories; the letters represented groups of things, like dogs, whales, mammals, submarines, or candy bars. That is why in these cases we use plural nouns. Other patterns will require substituting entire sentences for letters. We will study forms of this type in Chapter 4. The patterns you need to know will be introduced as we study each kind of argument, so keep your eyes open for them.

Using the Counterexample Method

By definition, an argument form is valid if and only if all of its instances are valid. Therefore, if we can show that a logical form has even one invalid instance, then we may infer that the argument form is invalid. Such an instance is called a counterexample to the argument form’s validity; thus, the counterexample method for showing that an argument form is invalid involves creating an argument with the exact same form but in which the premises are true and the conclusion is false. (We will examine other methods in this chapter and in later chapters.) In other words, finding a counterexample demonstrates the invalidity of the argument’s form.

Consider the invalid argument example from the prior section:

All donuts have added sugar.

All donuts are bad for you.

Therefore, everything with added sugar is bad for you.

By replacing predicate phrases with noun phrases, this argument has the form:

All A are B.

All A are C.

Therefore, all B are C.

This is the same form as that of the following, clearly invalid argument:

All birds are animals.

All birds have feathers.

Therefore, all animals have feathers.

Because we can see that the premises of this argument are true and the conclusion is false, we know that the argument is invalid. Since we have identified an invalid instance of the form, we know that the form is invalid. The invalid instance is a counterexample to the form. Because we have a counterexample, we have good reason to think that the argument about donuts is not valid.

One of our recent examples has the form:

All A are B.

Some B are C.

Therefore, at least some A are C.

Here is a counterexample that challenges this argument form’s validity:

All dogs are mammals.

Some mammals are cats.

Therefore, at least some dogs are cats.

By substituting dogs for A, mammals for B, and cats for C, we have found an example of the argument’s form that is clearly invalid because it moves from true premises to a false conclusion. Therefore, the argument form is invalid.

Here is another example of an argument:

All monkeys are primates.

No monkeys are reptiles.

Therefore, no primates are reptiles.

The conclusion is true in this example, so many may mistakenly think that the reasoning is valid. However, to better investigate the validity of the reasoning, it is best to focus on its form. The form of this argument is:

All A are B.

No A are C.

Therefore, no B are C.

To demonstrate that this form is invalid, it will suffice to demonstrate that there is an argument of this exact form that has all true premises and a false conclusion. Here is such a counterexample:

All men are human.

No men are women.

Therefore, no humans are women.

Clearly, there is something wrong with this argument. Though this is a different argument, the fact that it is clearly invalid, even though it has the exact same form as our original argument, means that the original argument’s form is also invalid.

3.3 Types of Deductive Arguments

Once you learn to look for arguments, you will see them everywhere. Deductive arguments play very important roles in daily reasoning. This section will discuss some of the most important types of deductive arguments.

Mathematical Arguments

Arguments about or involving mathematics generally use deductive reasoning. In fact, one way to think about deductive reasoning is that it is reasoning that tries to establish its conclusion with mathematical certainty. Let us consider some examples.

A mathematical proof is a valid deductive argument that attempts to prove the conclusion. Because mathematical proofs are deductively valid, mathematicians establish mathematical truth with complete certainty (as long as they agree on the premises).

Suppose you are splitting the check for lunch with a friend. In calculating your portion, you reason as follows:

I had the chicken sandwich plate for $8.49.

I had a root beer for $1.29.

I had nothing else.

$8.49 + $1.29 = $9.78.

Therefore, my portion of the bill, excluding tip and tax, is $9.78.

Notice that if the premises are all true, then the conclusion must be true also. Of course, you might be mistaken about the prices, or you might have forgotten that you had a piece of pie for dessert. You might even have made a mistake in how you added up the prices. But these are all premises. So long as your premises are correct and the argument is valid, then the conclusion is certain to be true.

But wait, you might say—aren’t we often mistaken about things like this? After all, it is common for people to make mistakes when figuring out a bill. Your friend might even disagree with one of your premises: For example, he might think the chicken sandwich plate was really $8.99. How can we say that the conclusion is established with mathematical certainty if we are willing to admit that we might be mistaken?

These are excellent questions, but they pertain to our certainty of the truth of the premises. The important feature of valid arguments is that the reasoning is so strong that the conclusion is just as certain to be true as the premises. It would be a very strange friend indeed who agreed with all of your premises and yet insisted that your portion of the bill was something other than $9.78. Still, no matter how good our reasoning, there is almost always some possibility that we are mistaken about our premises.

Arguments From Definitions

Another common type of deductive argument is argument from definition. This type of argument typically has two premises. One premise gives the definition of a word; the second premise says that something meets the definition. Here is an example:

Bachelor means “unmarried male.”

John is an unmarried male.

Therefore, John is a bachelor.

Notice that as with arguments involving math, we may disagree with the premises, but it is very hard to agree with the premises and disagree with the conclusion. When the argument is set out in standard form, it is typically relatively easy to see that the argument is valid.

On the other hand, it can be a little tricky to tell whether the argument is sound. Have we really gotten the definition right? We have to be very careful, as definitions often sound right even though they are a little bit off. For example, the stated definition of bachelor is not quite right. At the very least, the definition should apply only to human males, and probably only adult ones. We do not normally call children or animals “bachelors.”

When crafting or evaluating a deductive argument via definition, special attention should be paid to the clarity of the definition.

An interesting feature of definitions is that they can be understood as going both ways. In other words, if bachelor means “unmarried male,” then we can reason either from the man being an unmarried male to his being a bachelor, as in the previous example, or from the man being a bachelor to his being an unmarried male, as in the following example.

Bachelor means “unmarried male.”

John is a bachelor.

Therefore, John is an unmarried male.

Arguments from definition can be very powerful, but they can also be misused. This typically happens when a word has two meanings or when the definition is not fully accurate. We will learn more about this when we study fallacies in Chapter 7, but here is an example to consider:

Murder is the taking of an innocent life.

Abortion takes an innocent life.

Therefore, abortion is murder.

This is an argument from definition, and it is valid—the premises guarantee the truth of the conclusion. However, are the premises true? Both premises could be disputed, but the first premise is probably not right as a definition. If the word murder really just meant “taking an innocent life,” then it would be impossible to commit murder by killing someone who was not innocent. Furthermore, there is nothing in this definition about the victim being a human or the act being intentional. It is very tricky to get definitions right, and we should be very careful about reaching conclusions based on oversimplified definitions. We will come back to this example from a different angle in the next section when we study syllogisms.

Categorical Arguments

Historically, some of the first arguments to receive a detailed treatment were categorical arguments, having been thoroughly explained by Aristotle himself (Smith, 2014). Categorical arguments are arguments whose premises and conclusions are statements about categories of things. Let us revisit an example from earlier in this chapter:

All whales are mammals.

All mammals breathe air.

Therefore, all whales breathe air.

In each of the statements of this argument, the membership of two categories is compared. The categories here are whales, mammals, and air breathers. As discussed in the previous section on evaluating deductive arguments, the validity of these arguments depends on the repetition of the category terms in certain patterns; it has nothing to do with the specific categories being compared. You can test this by changing the category terms whales, mammals, and air breathers with any other category terms you like. Because this argument’s form is valid, any other argument with the same form will be valid. The branch of deductive reasoning that deals with categorical arguments is known as categorical logic. We will discuss it in the next two sections.

Propositional Arguments

Propositional arguments are a type of reasoning that relates sentences to each other rather than relating categories to each other. Consider this example:

Either Jill is in her room, or she’s gone out to eat.

Jill is not in her room.

Therefore, she’s gone out to eat.

Notice that in this example the pattern is made by the sentences “Jill is in her room” and “she’s gone out to eat.” As with categorical arguments, the validity of propositional arguments can be determined by examining the form, independent of the specific sentences used. The branch of deductive reasoning that deals with propositional arguments is known as propositional logic, which we will discuss in Chapter 4.

3.4 Categorical Logic: Introducing Categorical Statements

The field of deductive logic is a rich and productive one; one could spend an entire lifetime studying it. (See A Closer Look: More Complicated Types of Deductive Reasoning.) Because the focus of this book is critical thinking and informal logic (rather than formal logic), we will only look closely at categorical and propositional logic, which focus on the basics of argument. If you enjoy this introductory exposure, you might consider looking for more books and courses in logic.

A Closer Look: More Complicated Types of Deductive Reasoning

As noted, deductive logic deals with a precise kind of reasoning in which logical validity is based on logical form. Within logical forms, we can use letters as variables to replace English words. Logicians also frequently replace other words that occur within arguments—such as all, some, or, and not—to create a kind of symbolic language. Formal logic represented in this type of symbolic language is called symbolic logic.

Because of this use of symbols, courses in symbolic logic end up looking like math classes. An introductory course in symbolic logic will typically begin with propositional logic and then move to something called predicate logic. Predicate logic combines everything from categorical and propositional logic but allows much more flexibility in the use of some and all. This flexibility allows it to represent much more complex and powerful statements.

Predicate logic forms the basis for even more advanced types of logic. Modal logic, for example, can be used to represent many deductive arguments about possibility and necessity that cannot be symbolized using predicate logic alone. Predicate logic can even help provide a foundation for mathematics. In particular, when predicate logic is combined with a mathematical field called set theory, it is possible to prove the fundamental truths of arithmetic. From there it is possible to demonstrate truths from many important fields of mathematics, including calculus, without which we could not do physics, engineering, or many other fascinating and useful fields. Even the computers that now form such an essential part of our lives are founded, ultimately, on deductive logic.

Categorical arguments have been studied extensively for more than 2,000 years, going back to Aristotle. Categorical logic is the logic of argument made up of categorical statements. It is a logic that is concerned with reasoning about certain relationships between categories of things. To learn more about how categorical logic works, it will be useful to begin by analyzing the nature of categorical statements, which make up the premises and conclusions of categorical arguments. A categorical statement talks about two categories or groups. Just to keep things simple, let us start by talking about dogs, cats, and animals.

One thing we can say about these groups is that all dogs are animals. Of course, all cats are animals, too. So we have the following true categorical statements:

All dogs are animals.

All cats are animals.

In categorical statements, the first group name is called the subject term; it is what the sentence is about. The second group name is called the predicate term. In the categorical sentences just mentioned, dogs and cats are both in the subject position, and animals is in the predicate position. Group terms can go in either position, but of course, the sentence might be false. For example, in the sentence “All animals are dogs” the term dogs is in the predicate position.

You may recall that we can represent the logical form of these types of sentences by replacing the category terms with single letters. Using this method, we can represent the form of these categorical statements in the following way:

All D are A.

All C are A.

Another true statement we can make about these groups is “No dogs are cats.” Which term is in subject position, and which is in predicate position? If you said that dogs is the subject and cats is the predicate, you’re right! The logical form of “No dogs are cats” can be given as “No D are C.”

We now have two sentences in which the category dogs is the subject: “All dogs are animals” and “No dogs are cats.” Both of these statements tell us something about every dog. The first, which starts with all, tells us that each dog is an animal. The second, which begins with no, tells us that each dog is not a cat. We say that both of these types of sentences are universal because they tell us something about every member of the subject class.

Not all categorical statements are universal. Here are two statements about dogs that are not universal:

Some dogs are brown.

Some dogs are not tall.

Statements that talk about some of the things in a category are called particular statements. The distinction between a statement being universal or particular is a distinction of quantity.

Another distinction is that we can say that the things mentioned are in or not in the predicate category. If we say the things are in that category, our statement is affirmative. If we say the things are not in that category, our statement is negative. The distinction between a statement being affirmative or negative is a distinction of quality. For example, when we say “Some dogs are brown,” the thing mentioned (dogs) is in the predicate category (brown things), making this an affirmative statement. When we say “Some dogs are not tall,” the thing mentioned (dogs) is not in the predicate category (tall things), and so this is a negative statement.

Taking both of these distinctions into account, there are four types of categorical statements: universal affirmative, universal negative, particular affirmative, and particular negative. Table 3.1 shows the form of each statement along with its quantity and quality.

Table 3.1: Types of categorical statements

Quantity

Quality

All S is P

Universal

Affirmative

No S is P

Universal

Negative

Some S is P

Particular

Affirmative

Some S is not P

Particular

Negative

To abbreviate these categories of statement even further, logicians over the millennia have used letters to represent each type of statement. The abbreviations are as follows:

A: Universal affirmative (All S is P)

E: Universal negative (No S is P)

I: Particular positive (Some S is P)

O: Particular negative (Some S is not P)

Accordingly, the statements are known as A propositions, E propositions, I propositions, and O propositions. Remember that the single capital letters in the statements themselves are just placeholders for category terms; we can fill them in with any category terms we like. Figure 3.1 shows a traditional way to arrange the four types of statements by quantity and quality.

Now we need to get just a bit clearer on what the four statements mean. Granted, the meaning of categorical statements seems clear: To say, for example, that “no dogs are reptiles” simply means that there are no things that are both dogs and reptiles. However, there are certain cases in which the way that logicians understand categorical statements may differ somewhat from how they are commonly understood in everyday language. In particular, there are two specific issues that can cause confusion.

Clarifying Particular Statements

The first issue is with particular statements (I and O propositions). When we use the word some in everyday life, we typically mean more than one. For example, if someone says that she has some apples, we generally think that this means that she has more than one. However, in logic, we take the word some simply to mean at least one. Therefore, when we say that some S is P, we mean only that at least one S is P. For example, we can say “Some dogs live in the White House” even if only one does.

Clarifying Universal Statements

The second issue involves universal statements (A and E propositions). It is often called the “issue of existential presupposition”—the issue concerns whether a universal statement implies a particular statement. For example, does the fact that all dogs are animals imply that some dogs are animals? The question really becomes an issue only when we talk about things that do not really exist. For example, consider the claim that all the survivors of the Civil War live in New York. Given that there are no survivors of the Civil War anymore, is the statement true or not?

The Greek philosopher Aristotle, the inventor of categorical logic, would have said the statement is false. He thought that “All S is P” could only be true if there was at least one S (Parsons, 2014). Modern logicians, however, hold that that “All S is P” is true even when no S exists. The reasons for the modern view are somewhat beyond the scope of this text—see A Closer Look: Existential Import for a bit more of an explanation—but an example will help support the claim that universal statements are true when no member of the subject class exists.

Suppose we are driving somewhere and stop for snacks. We decide to split a bag of M&M’s. For some reason, one person in our group really wants the brown M&M’s, so you promise that he can have all of them. However, when we open the bag, it turns out that there are no brown candies in it. Since this friend did not get any brown M&M’s, did you break your promise? It seems clear that you did not. He did get all of the brown M&M’s that were in the bag; there just weren’t any. In order for you to have broken your promise, there would have to be a brown M&M that you did not let your friend have. Therefore, it is true that your friend got all the brown M&M’s, even though he did not get any.

This is the way that modern logicians think about universal propositions when there are no members of the subject class. Any universal statement with an empty subject class is true, regardless of whether the statement is positive or negative. It is true that all the brown M&M’s were given to your friend and also true that no brown M&M’s were given to your friend.

A Closer Look: Existential Import

It is important to remember that particular statements in logic (I and O propositions) refer to things that actually exist. The statement “Some dogs are mammals” is essentially saying, “There is at least one dog that exists in the universe, and that dog is a mammal.” The way that logicians refer to this attribute of I and O statements is that they have “existential import.” This means that for them to be true, there must be something that actually exists that has the property mentioned in the statement.

The 19th-century mathematician George Boole, however, presented a problem. Boole agreed with Aristotle that the existential statements I and O had to refer to existing things to be true. Also, for Aristotle, all A statements that are true necessarily imply the truth of their corresponding I statements. The same goes with E and O statements.

George Boole, for whom Boolean logic is named, challenged Aristotle’s assertion that the truth of A statements implies the truth of corresponding I statements. Boole suggested that some valid forms of syllogisms had to be excluded.

Boole pointed out that some true A and E statements refer to things that do not actually exist. Consider the statement “All vampires are creatures that drink blood.” This is a true statement. That means that the corresponding I statement, “Some vampires are creatures that drink blood,” would also be true, according to Aristotle. However, Boole noted that there are no existing things that are vampires. If vampires do not exist, then the I statement, “Some vampires are creatures that drink blood,” is not true: The truth of this statement rests on the idea that there is an actually existing thing called a vampire, which, at this point, there is no evidence of.

Boole reasoned that Aristotle’s ideas did not work in cases where A and E statements refer to nonexisting classes of objects. For example, the E statement “No vampires are time machines” is a true statement. However, both classes in this statement refer to things that do not actually exist. Therefore, the statement “Some vampires are not time machines” is not true, because this statement could only be true if vampires and time machines actually existed.

Boole reasoned that Aristotle’s claim that true A and E statements led necessarily to true I and O statements was not universally true. Hence, Boole claimed that there needed to be a revision of the forms of categorical syllogisms that are considered valid. Because one cannot generally claim that an existential statement (I or O) is true based on the truth of the corresponding universal (A or E), there were some valid forms of syllogisms that had to be excluded under the Boolean (modern) perspective. These syllogisms were precisely those that reasoned from universal premises to a particular conclusion.

Of course, we all recognize that in everyday life we can logically infer that if all dogs are mammals, then it must be true that some dogs are mammals. That is, we know that there is at least one existing dog that is a mammal. However, because our logical rules of evaluation need to apply to all instances of syllogisms, and because there are other instances where universals do not lead of necessity to the truth of particulars, the rules of evaluation had to be reformed after Boole presented his analysis. It is important to avoid committing the existential fallacy, or assuming that a class has members and then drawing an inference about an actually existing member of the class.

Accounting for Conversational Implication

These technical issues likely sound odd: We usually assume that some implies that there is more than one and that all implies that something exists. This is known as conversational implication (as opposed to logical implication). It is quite common in everyday life to make a conversational implication and take a statement to suggest that another statement is true as well, even though it does not logically imply that the other must be true. In logic, we focus on the literal meaning.

One of the common reasons that a statement is taken to conversationally imply another is that we are generally expected to make the most fully informative statement that we can in response to a question. For example, if someone asks what time it is and you say, “Sometime after 3,” your statement seems to imply that you do not know the exact time. If you knew it was 3:15 exactly, then you probably should have given this more specific information in response to the question.

For example, we all know that all dogs are animals. Suppose, however, someone says, “Some dogs are animals.” That is an odd thing to say: We generally would not say that some dogs are animals unless we thought that some of them are not animals. However, that would be making a conversational implication, and we want to make logical implications. For the purposes of logic, we want to know whether the statement “some dogs are animals” is true or false. If we say it is false, then we seem to have stated it is not true that some dogs are animals; this, however, would seem to mean that there are no dogs that are animals. That cannot be right. Therefore, logicians take the statement “Some dogs are animals” simply to mean that there is at least one dog that is an animal, which is true. The statement “Some dogs are not animals” is not part of the meaning of the statement “Some dogs are animals.” In the language of logic, the statement that some S are not P is not part of the meaning of the statement that some S are P.

Of course, it would be odd to make the less informative statement that some dogs are animals, since we know that all dogs are animals. Because we tend to assume someone is making the most informative statement possible, the statement “Some dogs are animals” may conversationally imply that they are not all animals, even though that is not part of the literal meaning of the statement.

In short, a particular statement is true when there is at least one thing that makes it true, even if the universal statement would also be true. In fact, sometimes we emphasize that we are not talking about the whole category by using the words at least, as in, “At least some planets orbit stars.” Therefore, it appears to be nothing more than conversational implication, not literal meaning, that leads our statement “Some dogs are animals” to suggest that some also are not. When looking at categorical statements, be sure that you are thinking about the actual meaning of the sentence rather than what might be conversationally implied.

3.5 Categorical Logic: Venn Diagrams as Pictures of Meaning

Given that it is sometimes tricky to parse out the meaning and implications of categorical statements, a logician named John Venn devised a method that uses diagrams to clarify the literal meanings and logical implications of categorical claims. These diagrams are appropriately called Venn diagrams (Stapel, n.d.). Venn diagrams not only give a visual picture of the meanings of categorical statements, they also provide a method by which we can test the validity of many categorical arguments.

Drawing Venn Diagrams

Here is how the diagramming works: Imagine we get a bunch of people together and all go to a big field. We mark out a big circle with rope on the field and ask everyone with brown eyes to stand in the circle. Would you stand inside the circle or outside it? Where would you stand if we made another circle and asked everyone with brown hair to stand inside? If your eyes or hair are sort of brownish, just pick whether you think you should be inside or outside the circles. No standing on the rope allowed! Remember your answers to those two questions.

Here is an image of the brown-eye circle, labeled “E” for “eyes”; touch inside or outside the circle indicating where you would stand.

A circle labeled “E.”

Here is a picture of the brown-hair circle, labeled “H” for “hair”; touch inside or outside the circle indicating where you would stand.

A circle labeled “H.”

Notice that each circle divides the people into two groups: Those inside the circle have the feature we are interested in, and those outside the circle do not.

Where would you stand if we put both circles on the ground at the same time?

Two circles. The circle on the left is labeled “E,” and the circle on the right is labeled “H.”

As long as you do not have both brown eyes and brown hair, you should be able to figure out where to stand. But where would you stand if you have brown eyes and brown hair? There is not any spot that is in both circles, so you would have to choose. In order to give brown-eyed, brown-haired people a place to stand, we have to overlap the circles.

Two circles, labeled “E” and “H,” that overlap a small portion of each other.

Now there is a spot where people who have both brown hair and brown eyes can stand: where the two circles overlap. We noted earlier that each circle divides our bunch of people into two groups, those inside and those outside. With two circles, we now have four groups. Figure 3.2 shows what each of those groups are and where people from each group would stand.

Figure 3.2: Sample Venn diagram

Two overlapping circles showing where the following groups fall: People with brown eyes and not brown hair are in the portion of the left hand circle that does not overlap the right hand circle. People with brown hair but not brown eyes are in the portion of the right hand circle that does not overlap the left hand circle. People with both brown eyes and brown hair are in the portion of both circles that overlap. People with neither brown hair nor brown eyes fall outside of both circles.

With this background, we can now draw a picture for each categorical statement. When we know a region is empty, we will darken it to show there is nobody there. If we know for sure that someone is in a region, we will put an x in it to represent a person standing there. Figure 3.3 shows the pictures for each of the four kinds of statements.

Figure 3.3: Venn diagrams of categorical statements

Each of the four categorical statements can be represented visually with a Venn diagram.

Four Venn diagrams. The top left diagram shows two circles intersecting. The circle on the left is shaded gray where it does not intersect the other circle. The top right diagram shows two circles intersecting, and the area of intersection is shaded gray. The bottom left diagram shows an X where the two circles intersect. The bottom right diagram shows two circles intersecting, with the area of the left circle labeled with an X.

In drawing these pictures, we adopt the convention that the subject term is on the left and the predicate term is on the right. There is nothing special about this way of doing it, but diagrams are easier to understand if we draw them the same way as much as possible. The important thing to remember is that a Venn diagram is just a picture of the meaning of a statement. We will use this fact in our discussion of inferences and arguments.

Drawing Immediate Inferences

As mentioned, Venn diagrams help us determine what inferences are valid. The most basic of such inferences, and a good place to begin, is something called immediate inference. Immediate inferences are arguments from one categorical statement as premise to another as conclusion. In other words, we immediately infer one statement from another. Despite the fact that these inferences have only one premise, many of them are logically valid. This section will use Venn diagrams to help discern which immediate inferences are valid.

The basic method is to draw a diagram of the premises of the argument and determine if the diagram thereby shows the conclusion is true. If it does, then the argument is valid. In other words, if drawing a diagram of just the premises automatically creates a diagram of the conclusion, then the argument is valid. The diagram shows that any way of making the premises true would also make the conclusion true; it is impossible for the conclusion to be false when the premises are true. We will see how to use this method with each of the immediate inferences and later extend the method to more complicated arguments.

Conversion

Conversion is just a matter of switching the positions of the subject and predicate terms. The resulting statement is called the converse of the original statement. Table 3.2 shows the converse of each type of statement.

Table 3.2: Conversion

Statement

Converse

All S is P.

All P is S.

No S is P.

No P is S.

Some S is P.

Some P is S.

Some S is not P.

Some P is not S.

Forming the converse of a statement is easy; just switch the subject and predicate terms with each other. The question now is whether the immediate inference from a categorical statement to its converse is valid or not. It turns out that the argument from a statement to its converse is valid for some statement types, but not for others. In order to see which, we have to check that the converse is true whenever the original statement is true.

An easy way to do this is to draw a picture of the two statements and compare them. Let us start by looking at the universal negative statement, or E proposition, and its converse. If we form an argument from this statement to its converse, we get the following:

No S is P.

Therefore, no P is S.

Figure 3.4 shows the Venn diagrams for these statements.

As you can see, the same region is shaded in both pictures—the region that is inside both circles. It does not matter which order the circles are in, the picture is the same. This means that the two statements have the same meaning; we call such statements equivalent.

The Venn diagrams for these statements demonstrate that all of the information in the conclusion is present in the premise. We can therefore infer that the inference is valid. A shorter way to say it is that conversion is valid for universal negatives.

We see the same thing when we look at the particular affirmative statement, or I proposition.

In the case of particular affirmatives as well, we can see that all of the information in the conclusion is contained within the premises. Therefore, the immediate inference is valid. In fact, because the diagram for “Some S is P” is the same as the diagram for its converse, “Some P is S” (see Figure 3.5), it follows that these two statements are equivalent as well.

Figure 3.4: Universal negative statement and its converse

In this representation of “No S is P. Therefore, no P is S,” the areas shaded are the same, meaning the statements are equivalent.

Two identical Venn diagrams. Both diagrams have the left circle labeled “S” and the right circle labeled “P.” They both have the overlapping section shaded. The diagram on the left is labeled “No S is P,” and the diagram on the right is labeled “No P is S.”

Figure 3.5: Particular affirmative statement and its converse

As with the E proposition, all of the information contained in the conclusion of the I proposition is also contained within the premises, making the inference valid.

Two identical Venn diagrams. The circle on the left is labeled “S,” the circle on the right is labeled “P,” and the overlapping section is labeled “x.” The diagram on the left is labeled “Some S is P,” and the diagram on the right is labeled, “Some P is S.”

However, there will be a big difference when we draw pictures of the universal affirmative (A proposition), the particular negative (O proposition), and their converses (see Figure 3.6 and Figure 3.7).

In these two cases we get different pictures, so the statements do not mean the same thing. In the original statements, the marked region is inside the S circle but not in the P circle. In the converse statements, the marked region is inside the P circle but not in the S circle. Because there is information in the conclusions of these arguments that is not present in the premises, we may infer that conversion is invalid in these two cases.

Figure 3.6: Universal affirmative statement and its converse

Unlike Figures 3.4 and 3.5 where the diagrams were identical, we get two different diagrams for A propositions. This tells us that there is information contained in the conclusion that was not included in the premises, making the inference invalid.

Two Venn diagrams. The left circle of both diagrams is labeled “S,” and the right circle of both diagrams is labeled “P.” The diagram on the left is labeled “All S is P,” and shows the left circle shaded while the overlapping area and circle labeled “P” are unshaded. The diagram on the right is labeled “All P is S,” and shows the right circle shaded while the overlapping area and circle labeled “S” are unshaded.

Figure 3.7: Particular negative statement and its converse

As with A propositions, O propositions present information in the conclusion that was not present in the premises, rendering the inference invalid.

Two Venn diagrams. The left circle of both diagrams is labeled “S,” and the right circle of both diagrams is labeled “P.” The diagram on the left is labeled “Some S is not P,” and shows a red x in the S circle. The diagram on the right is labeled “Some P is not S,” and shows a red x in the P circle.

Let us consider another type of immediate inference.

Contraposition

Before we can address contraposition, it is necessary to introduce the idea of a complement class. Remember that for any category, we can divide things into those that are in the category and those that are out of the category. When we imagined rope circles on a field, we asked all the brown-haired people to step inside one of the circles. That gave us two groups: the brown-haired people inside the circle, and the non-brown-haired people outside the circle. These two groups are complements of each other. The complement of a group is everything that is not in the group. When we have a term that gives us a category, we can just add non- before the term to get a term for the complement group. The complement of term S is non-S, the complement of term animal is nonanimal, and so on. Let us see what complementing a term does to our Venn diagrams.

Recall the diagram for brown-eyed people. You were inside the circle if you have brown eyes, and outside the circle if you do not. (Remember, we did not let people stand on the rope; you had to be either in or out.) So now consider the diagram for non-brown-eyed people.

If you were inside the brown-eyed circle, you would be outside the non-brown-eyed circle. Similarly, if you were outside the brown-eyed circle, you would be inside the non-brown-eyed circle. The same would be true for complementing the brown-haired circle. Complementing just switches the inside and outside of the circle.

A Venn diagram. The circle on the left is labeled “non-brown eyes,” and the circle on the right is labeled “non-brown hair.” The region in the middle reads Non-brown eyes and non-brown hair.

Do you remember the four regions from Figure 3.2? See if you can find the regions that would have the same people in the complemented picture. Where would someone with blue eyes and brown hair stand in each picture? Where would someone stand if he had red hair and green eyes? How about someone with brown hair and brown eyes?

In Figure 3.8, the regions are colored to indicate which ones would have the same people in them. Use the diagram to help check your answers from the previous paragraph. Notice that the regions in both circles and outside both circles trade places and that the region in the left circle only trades places with the region in the right circle.

Figure 3.8: Complement class

Two Venn diagrams. The circles in the diagram on the left are labeled “S” (left) and “P” (right). The S circle is purple, the overlapping section is off-white, the P circle is green, and the area outside the circles is red. The circles in the diagram on the right are labeled “Non-S” (left) and “Non-P” (right). The Non-S circle is green, the overlapping section is red, the Non-P circle is purple, and the area outside the circles is off-white.

Now that we know what a complement is, we are ready to look at the immediate inference of contraposition. Contraposition combines conversion and complementing; to get the contrapositive of a statement, we first get the converse and then find the complement of both terms.

Let us start by considering the universal affirmative statement, “All S is P.” First we form its converse, “All P is S,” and then we complement both class terms to get the contrapositive, “All non-P is non-S.” That may sound like a mouthful, but you should see that there is a simple, straightforward process for getting the contrapositive of any statement. Table 3.3 shows the process for each of the four types of categorical statements.

Table 3.3: Contraposition

Original

Converse

Contrapositive

All S is P.

All P is S.

All non-P is non-S.

No S is P.

No P is S.

No non-P is non-S.

Some S is P.

Some P is S.

Some non-P is non-S.

Some S is not P.

Some P is not S.

Some non-P is not non-S.

Figure 3.9 shows the diagrams for the four statement types and their contrapositives, colored so that you can see which regions represent the same groups.

Figure 3.9: Contrapositive Venn diagrams

Using the converse and contrapositive diagrams, you can infer the original statement.

Eight Venn diagrams shown in pairs of two. The diagrams of the first pair are identical, with the left circle (S) shaded. The diagram on the left is labeled “All S is P,” and the diagram on the right is labeled “All non-P is non-S.” The second pair shows the overlapping section of the left diagram shaded (No S is P) and the outside area of the right diagram shaded (No non-P is non-S). The third pair shows a red x in the overlapping section of the left diagram (Some S is P) and a red X in the outside area of the right diagram (Some non-P is non-S). The fourth pair shows two identical diagrams with a red x in the left circle (S). The left diagram is labeled “Some S is not P,” and the right diagram is labeled “Some non-P is not non-S.”

As you can see, contraposition preserves meaning in universal affirmative and particular negative statements. So from either of these types of statements, we can immediately infer their contrapositive, and from the contrapositive, we can infer the original statement. In other words, these statements are equivalent; therefore, in those two cases, the contrapositive is valid.

In the other cases, particular affirmative and universal negative, we can see that there is information in the conclusion that is not present in diagram of the premise; these immediate inferences are invalid.

There are more immediate inferences that can be made, but our main focus in this chapter is on arguments with multiple premises, which tend to be more interesting, so we are going to move on to syllogisms.

3.6 Categorical Logic: Categorical Syllogisms

Whereas contraposition and conversion can be seen as arguments with only one premise, a syllogism is a deductive argument with two premises. The categorical syllogism, in which a conclusion is derived from two categorical premises, is perhaps the most famous—and certainly one of the oldest—forms of deductive argument. The categorical syllogism—which we will refer to here as just “syllogism”—presented by Aristotle in his Prior Analytics (350 BCE/1994), is a very specific kind of deductive argument and was subsequently studied and developed extensively by logicians, mathematicians, and philosophers.

Terms

We will first discuss the syllogism’s basic outline, following Aristotle’s insistence that syllogisms are arguments that have two premises and a conclusion. Let us look again at our standard example:

All S are M.

All M are P.

Therefore, all S are P.

There are three total terms here: S, M, and P. The term that occurs in the predicate position in the conclusion (in this case, P) is the major term. The term that occurs in the subject position in the conclusion (in this case, S) is the minor term. The other term, the one that occurs in both premises but not the conclusion, is the middle term (in this case, M).

The premise that includes the major term is called the major premise. In this case it is the first premise. The premise that includes the minor term, the second one here, is called the minor premise. The conclusion will present the relationship between the predicate term of the major premise (P) and the subject term of the minor premise (S) (Smith, 2014).

There are 256 possible different forms of syllogisms, but only a small fraction of those are valid, which can be shown by testing syllogisms through the traditional rules of the syllogism or by using Venn diagrams, both of which we will look at later in this section.

Distribution

As Aristotle understood logical propositions, they referred to classes, or groups: sets of things. So a universal affirmative (type A) proposition that states “All Clydesdales are horses” refers to the group of Clydesdales and says something about the relationship between all of the members of that group and the members of the group “horses.” However, nothing at all is said about those horses that might not be Clydesdales, so not all members of the group of horses are referred to. The idea of referring to members of such groups is the basic idea behind distribution: If all of the members of a group are referred to, the term that refers to that group is said to be distributed.

Using our example, then, we can see that the proposition “All Clydesdales are horses” refers to all the members of that group, so the term Clydesdales is said to be distributed. Universal affirmatives like this one distribute the term that is in the first, or subject, position.

However, what if the proposition were a universal negative (type E) proposition, such as “No koala bears are carnivores”? Here all the members of the group “koala bears” (the subject term) are referred to, but all the members of the group “carnivores” (the predicate term) are also referred to. When we say that no koala bears are carnivores, we have said something about all koala bears (that they are not carnivores) and also something about all carnivores (that they are not koala bears). So in this universal negative proposition, both of its terms are distributed.

To sum up distribution for the universal propositions, then: Universal affirmative (A) propositions distribute only the first (subject) term, and universal negative (E) propositions distribute both the first (subject) term and the second (predicate) term.

The distribution pattern follows the same basic idea for particular propositions. A particular affirmative (type I) proposition, such as “Some students are football players,” refers only to at least one member of the subject class (“students”) and only to at least one member of the predicate class (“football players”). Thus, remembering that some is interpreted as meaning “at least one,” the particular affirmative proposition distributes neither term, for this proposition does not refer to all the members of either group.

Finally, a particular negative (type O) proposition, such as “Some Floridians are not surfers,” only refers to at least one Floridian—but says that at least one Floridian does not belong to the entire class of surfers or is excluded from the entire class of surfers. In this way, the particular negative proposition distributes only the term that refers to surfers, or the predicate term.

To sum up distribution for the particular propositions, then: particular affirmative (I) propositions distribute neither the first (subject) nor the second (predicate) term, and particular negative (O) propositions distribute only the second (predicate) term. This is a lot of detail, to be sure, but it is summarized in Table 3.4.

Proposition

Subject

Predicate

A

Distributed

Not

E

Distributed

Distributed

I

Not

Not

O

Not

Distributed

Table 3.4: Distribution

Once you understand how distribution works, the rules for determining the validity of syllogisms are fairly straightforward. You just need to see that in any given syllogism, there are three terms: a subject term, a predicate term, and a middle term. But there are only two positions, or “slots,” a term can appear in, and distribution relates to those positions.

Rules for Validity

Once we know how to determine whether a term is distributed, it is relatively easy to learn the rules for determining whether a categorical syllogism is valid. The traditional rules of the syllogism are given in various ways, but here is one standard way:

Rule 1 : The middle term must be distributed at least once.

Rule 2 : Any term distributed in the conclusion must be distributed in its corresponding premise.

Rule 3 : If the syllogism has a negative premise, it must have a negative conclusion, and if the syllogism has a negative conclusion, it must have a negative premise.

Rule 4 : The syllogism cannot have two negative premises.

Rule 5 : If the syllogism has a particular premise, it must have a particular conclusion, and if the syllogism has a particular conclusion, it must have a particular premise.

A syllogism that satisfies all five of these rules will be valid; a syllogism that does not will be invalid. Perhaps the easiest way of seeing how the rules work is to go through a few examples. We can start with our standard syllogism with all universal affirmatives:

All M are P.

All S are M.

Therefore, all S are P.

The Origins of Logic

The text describes five rules for determining a syllogism's validity, but Aristotle's fundamental rules were far more basic.

Critical Thinking Questions

1. The law of noncontradiction and the excluded middle establish that a proposition cannot be both true and false and must be either true or false. Can you think of a proposition that violates either of these rules?

2. Aristotle's syllogism form, or the standard argument form, allows us to condense arguments into their fundamental pieces for easier evaluation. Try putting an argument you have heard into the standard form.

Rule 1 is satisfied: The middle term is distributed by the first premise; a universal affirmative (A) proposition distributes the term in the first (subject) position, which here is M. Rule 2 is satisfied because the subject term that is distributed by the conclusion is also distributed by the second premise. In both the conclusion and the second premise, the universal affirmative proposition distributes the term in the first position. Rule 3 is also satisfied because there is not a negative premise without a negative conclusion, or a negative conclusion without a negative premise (all the propositions in this syllogism are affirmative). Rule 4 is passed because both premises are affirmative. Finally, Rule 5 is passed as well because there is a universal conclusion. Since this syllogism passes all five rules, it is valid.

These get easier with practice, so we can try another example:

Some M are not P.

All M are S.

Therefore, some S are not P.

Rule 1 is passed because the second premise distributes the middle term, M, since it is the subject in the universal affirmative (A) proposition. Rule 2 is passed because the major term, P, that is distributed in the O conclusion is also distributed in the corresponding O premise (the first premise) that includes that term. Rule 3 is passed because there is a negative conclusion to go with the negative premise. Rule 4 is passed because there is only one negative premise. Rule 5 is passed because the first premise is a particular premise (O). Since this syllogism passes all five rules, it is valid; there is no way that all of its premises could be true and its conclusion false.

Both of these have been valid; however, out of the 256 possible syllogisms, most are invalid. Let us take a look at one that violates one or more of the rules:

No P are M.

Some S are not M.

Therefore, all S are P.

Rule 1 is passed. The middle term is distributed in the first (major) premise. However, Rule 2 is violated. The subject term is distributed in the conclusion, but not in the corresponding second (minor) premise. It is not necessary to check the other rules; once we know that one of the rules is violated, we know that the argument is invalid. (However, for the curious, Rule 3 is violated as well, but Rules 4 and 5 are passed).

Venn Diagram Tests for Validity

Another value of Venn diagrams is that they provide a nice method for evaluating the validity of a syllogism. Because every valid syllogism has three categorical terms, the diagrams we use must have three circles:

Three interlocking circles creating a total of seven distinct areas.

The idea in diagramming a syllogism is that we diagram each premise and then check to see if the conclusion has been automatically diagrammed. In other words, we determine whether the conclusion must be true, according to the diagram of the premises.

It is important to remember that we never draw a diagram of the conclusion. If the argument is valid, diagramming the premises will automatically provide a diagram of the conclusion. If the argument is invalid, diagramming the premises will not provide a diagram of the conclusions.

Diagramming Syllogisms With Universal Statements

Particular statements are slightly more difficult in these diagrams, so we will start by looking at a syllogism with only universal statements. Consider the following syllogism:

All S is M.