Categorical Logic

3.2 Deductive Logics: Categorical and Propositional.

When one of the company said to him, "Convince me that logic is necessary," "Would you have me demonstrate it to you?" he said. "Yes." Then I must use a demonstrative form of argument. "Granted." "And how will you know, then whether my arguments mislead you?" On this, the man being silent, Epictetus said, "You see that even by your own confession, logic is necessary; since without it, you cannot even learn whether it is necessary or not." from Epictetus, Enchiridion

You already have an intuitive idea of what logic is -- it is the study of reasoning that depends upon the way ideas are connected in patterns. But to go further, we need a more precise definition. Consider the following two arguments:

1. All Italians are human.

2. All humans are mortal.

C. All Italians are mortal.

1. If Socrates influenced Plato, then Plato's philosophy has some features of Socrate' thinking.

2. Socrates influenced Plato.

C. Plato's philosophy has some features of Socrates' thinking.

Notice that both of these arguments use patterns and that it is the patterns themselves that make the arguments work. Logic begins with the realization that the connections among ideas that lead to worthwhile inferences 102

can be generalized so that we can focus on the connections or patterns of ideas in isolation from the content. To see this, we will rewrite the two arguments above replacing their content with placeholders. 1. All S are M. 2. All M are P. C. All S are P. 1. If A, then B. 2. A. C. B. In the first case we replaced every instance of "Italians" with S, "humans" with M, and "mortal" with P. In the second case we replaced every independent clause with letters A and B, making sure that the same clause was replaced with the same letter. The remarkable thing to notice now -- and you should try to imagine what it must have been like to be one of the first humans to notice it -¬is that the resulting patterns could be filled by any content (collections of things in the first argument, true/false claims in the second) and the reasoning would be sound. Historically, logicians were the first people who noticed this and wrote about it. We can generalize patterns of thought by abstracting content and studying the relationships among the patterns. Notice one more thing about the argument patterns above. The first one is filled with terms, while the second one is filled with true/false propositions. The terms in the first argument refer to categories of things (concrete or abstract) and this logic is called "categorical logic." It was first codified by Aristotle. The second is about logical relationship among propostions and is called "propositional logic." The Stoics were the first thinkers in the West to write about these patterns, but propositional logic developed its modern form in the 19'" and 20"' centuries. Taking categorical logic first, let's look at the basic set of relationships Aristotle discovered. He distinguished categorical propositions according to two distinctions: 2

1. Whether they predicate something (assert something) about everything in a category -- in which case they are universal -- or some particulars in a category and; 2. Whether they predicate something about a category -- in which case they are affirmative -- or deny it, and therefore are called "negative". With these two distinctions we can sort categorical propositions into four groups: Universal Affirmative: All S are P. Universal Negative: No S are P. Particular Affirmative: Some S are P. Particular Negative: Some S are not P. Even before we look at arguments with categorical logic, you can see that there are truth relationships among different types of propositions. For instance, if "All Humans are Mortal" is true (and it is) then "No Humans are Mortal" is false. We call these relationships "truth functional" because the truth of the proposition is a function (can be determined according to a rule) of the other. The "Square of Opposition" is a traditional schema for summarizing the primary truth functional relationships among the four types of categorical propositions. Notice that we refer to each type of categorical proposition with a unique letter A, E,1, or O. The Square of Opposition A: All S are P. N Contraries 1 E: No S are P. Subalterns Contradic£ories Subalterns

2: Some S are P. Subcontraries L 0: Some S are not P.

The following relationships hold between categorical statements in the square of opposition: Contradictories: Contradictories have opposite truth values. If one is true, the other must be false. Contraries: Contraries cannot both be true, but can both be false. Subcontraries: Subcontraries cannot both be false, but can both be true. Subalterns: From a true A or E proposition, you can infer a true I or 0 proposition.' There are some additional truth functional relationships within categorical propositions, but they involve logical operations that can be carried out on particular propositions. For instance, you can "take the converse" of a proposition by switching its subject and predicate. So "All S are P" becomes "All P are S," "No S are P," becomes "No P are S," and so one. Conversion only works ("works" here means "preserves the truth value of the original proposition in the resulting one) on E and I propositions. It is worth noting that in everyday reasoning people sometimes reason from an A proposition to its converse or from an O proposition to its converse. Consider the following examples for each statement type: 1. All bachelors are unmarried males. All unmarried males are bachelors. 2. All students are members of the university community. All members of the university community are students. 3. Some professors are not philosophers. Some philosophers are not professors. 4. Some students are not physics majors. Some physics majors are not students. ' The relationship between subalterns only holds if you assume that the subject and predicate terms actually exist. This is called the existential interpretation of categorical logic. Aristotle made this assumption because he felt that when we assert something we should always be talking about something real. Modern logics often make different assumptions about existential import, but we will stick with Aristotle's approach. 4

Examples 1 and 3 both have pairs of true claims, but examples 2 and 4 show that for A and O propositions you can have a true claim whose conversion is not true. This means that the inference from and A or O proposition to its converse is not valid. It may seem like a technical point, but there is a very specific practical lesson in it. Whenever someone argues that it "just follows" from a statement of one logical form (A, E, 1, O) to another, you should think a moment about whether the conclusion they are advancing really follows as a matter of logic. To test this, try to come up with a counterexample in which the premise is true and the conclusion is false. If you can find such a counterexample, then the inference pattern is not valid. You can also take the obverse of a proposition by changing it from affirmative to negative (or negative to affirmative) and then changing the predicate to its complement. The complement of P is non-P, the complement of "dogs" is "non-dogs" (everything in the universe that isn't a dog), and so on. Now we can ask the same question as with conversion: Which of the following inferences are valid? 1. All S are P. Therefore, no S are non-P. 2. No S are P. Therefore, all S are non-P. 3. Some S are P. Therefore, some S are not non-P. 4. Some S are not P. Therefore, some S are non-P. The obverse of a true A, E, I, or O statement is always true. You will not find a counter-example. Let's consider one final operation: contraposition. In this case we would reason, for example, from "All skiers are thrill seekers" to "All non-thrill seekers are non-skiers." With contraposition we switch the subject and predict within each proposition and put in their complements. This produces valid inferences for A and O propositions but not for E and I propositions. Very few people -- even very few philosophers -- remember which propositions conversion, obversion, or contraposition work with or even the exact operation of each by name. It is still valuable to think through the operations, however. The real lesson here may not be obvious, so perhaps we should underscore it. In actual argument contexts, you may want to argue that an immediate inference can be made from a proposition with one logical form 5

to another. To evaluate the validity of that inference, you have two choices. You can hope that you remember the operation and which propositions it is valid for or you can try to figure out, on the spot, whether the inference is valid. To test the inference, you can try to come up with a counter-example in which you find a true premise which produces a false conclusion, as we did with obversion. Or, you can try to picture the class relationship itself by trying to think about whether the premise creates divisions and connections among the categories of things that necessarily includes the situation pictured in the conclusion. Try this with the example above about skiers and thrill seekers and you may find that you can "see" the logical relationship in the premise in such a way that makes it immediately apparent that the conclusion must be true. Exercise Set: Practice with immediate inference. In the first part of the exercise below, determine whether the immediate inference being made is converse, obverse, or contraposition and whether it is a valid inference. In the second part, symbolize the sentence given and perform the indicated operation. Then determine whether the resulting inference is valid. A. 1. All S is P. Therefore, All P is S. 2. Some S is P. Therefore, Some non-P is non-S. 3. No S is P. Therefore, all P is non-S. 4. Some S is not P. Therefore, some S is non-P. 5. Some S is P. Therefore, some P is S. B. 1. All students are scholars. Converse. 2. Some melons are ripe. Obverse. 3. No children allowed near the bar. Contraposition. 4. Some people are dangerous. Obverse. 5. All garage bands are loud. Obverse. 3.3 Categorical Syllogisms One goal for developing a logical system is to understand better the logical structure of particular statements and how changing those statements affects their truth value. But we also want our logical system to tell us about 6

the validity of complete arguments, not just immediate inferences from individual statements. In categorical logic, the argument form is typically a syllogism, an argument with two premises and, of course, a conclusion. One way to distinguish different syllogisms is by the combination of the four proposition types used. For example, you could represent a syllogism with all "A" propositions as "AAA" and one with "A" proposition premises and a particular affirmative conclusion as "AAI". The combination of proposition types in a categorical syllogism is called the mood of the syllogism and there are 64 different moods. Another way syllogisms differ from one another is in the pattern of their subject and predicate terms. Consider the following two "AII" syllogisms: 1. 1. All P is M. 2. Some S are M. C. Some S are P. 2. 1. All M is P. 2. Some Mare S. C. Some S are P. Notice that while these syllogisms have the same mood, their middle terms -- the term or category that is repeated -- are in different places in the premises. There are two more positions they could have been in as well -- The middle terms could have been split between the subject and predicate terms of the major and minor premises as follows: 3. 1. All MisP. 2. Some S are M. C. Some S are P. 4. 1. All PisM. 2. Some Mare S. C. Some S are P. 7

These four possible configurations of the middle term relative to the subject and predicate terms are called the four figures. Since there are four figures for each of the 64 moods of categorical syllogism, there are are 256 syllogist forms in classical syllogistic logic. The obvious question is, "Which of the 256 syllogistic forms are valid?" There are several approaches we could use to determine this. We could inspect each form and try to develop a counter-example for it or a reason for assuring ourselves that it is valid. We could use a method of picturing the relationships in the premises using Venn Diagrams .6 Or, we could rely on the work of 2,000 years of logicians who have developed three handy rules for evaluating a syllogism. Since our goal in learning a bit about categorical logic is primarily to pick up some critical awareness of logical form and validity, we will use the third method, though your instructor may choose to introduce Venn Diagrams instead or in addition. If you were studying logic in the West at any time up through the 18 '" century, you probably would have been required to memorize the 15 valid forms of the the categorical syllogism. While logic is no less important a discipline than it ever (indeed, given the reliance of human beings on circuits and computer programs, logic is a matter of life and death for millions of people every day!), our development of the art of reflection depends less upon memorizing forms and rules than upon understanding the way logical structures create valid patterns. To understand the three rules for evaluating categorical syllogisms, we need to introduce one more concept, the idea of "distribution." A term in a categorical proposition is distributed if the proposition says something about every member of the class named by that term. So, for example, in "All S is P," something is said about every member of the category named in the subject term, S, but not every "P". Even if Pi: "All male park rangers are handsome" is true, it would not follow that "All handsome people are park rangers. Nothing is being said about all "handsome people" in P,. However, in "No S is P" I Venn Diagrams date from the work of John Venn, a British logician who introduced the Venn Diagram as a pictorial method for representing logical relationships. For more resources on Venn Diagrams, visit the textbook website. 8

something is said about every member of both classes. Can you see why? In the case of an "I" proposition, we are only asserting that some S's are P, so the subject term is not distributed. Further, we are not saying anything about all of the P's. Determining whether the terms in the last propositon type, the "O" proposition, are distributed is a little trickier. We are not saying anything about all of the members of the category "S" in "Some S are not-P", so the subject term is not distributed, but we are actually saying something about all of the members of the category "P" -- every member of P is not one of the S's designated by "Some S". For example, if I say, "Some cars are not gas-powered," then I am asserting of all gas powered vehicles that they do not include those particular cars mentioned in the subject term. To summarize, in the following diagram the distributed terms are underlined: Summary of Distributed Terms in Categorical Propositions A: All S is P. E: No S is P. I: Some S is P. 0: Some S is not P. (Distributed terms are underlined.) With this in mind, the three rules for evaluating categorical syllogisms for validity can be stated as follows: 1. The number of negative claims in the conclusion must equal the number of negative claims in the premises. 2. At least one premise must distribute the middle term. 3. Any term that is distributed in the conclusion of the syllogism must be distributed in its premises. As you can see, the concept of distribution plays a significant role in these rules. Let's look at a couple of valid and invalid syllogisms using these rules. First, consider this syllogism:

1. All S are M. 2. Some M are not P. C. Some S are not P. This syllogism complies with the first rule (it has one negative proposition in each part) and fits the third rule (P is distributed in both the conclusion and P2' but it does not distribute it's middle term, that is, the premises do not say something about all of the members of the category named by "M" in either premise. The argument pattern is therefore invalid. Notice that we were able to determine distribution and validity without knowing anything about the actual content of the categories S, M, and P. This is an essential value of logic -- the ability to know from the form of an expression that it can or cannot guarantee true conclusions. You could also determine the invalidity of this syllogism by thinking carefully about its form without the rules. Here an example will help. Suppose you used the following instance of the argument pattern: 1. All crew team members are my friends. 2. Some of my friends are people who are not good at math. C. Some crew team members are people who are not good at math. Intuitively, you might notice that there's no reason to think that the friends refered to in premise 2 are crew team members. After all, the first premise does not say that "All my friends are crew team members." So when you go to say something about crew team members in the conclusion, you can be sure you're talking about friends, but not necessarily those friends mentioned in premise 2. Let's try changing the first premise so the argument reads as follows: 1. All my friends are crew team members. 2. Some of my friends are people who are not good at math. C. Some crew team members are people who are not good at math. The person who gives this argument may have a more limited social life, but at least they have a valid argument. Now that we know that all of this person's friends are crew team members, anything he says about his friends is also true of the crew team. Notice too that the rules are satisfied by this syllogism.

Puzzling out which of the 256 standard form categorical syllogisms follow the rules of validity is probably not a worthwhile use of your time even if your goal is to become a better thinker.' The essential skill to take away from our brief survey of categorical logic is the ability to recognize how immediate inferences (obversion, conversion, and contraposition) and syllogistic inferences can be valid or invalid. By picturing the logical relationships among categories, you should become more confident of finding reliable and unreliable patterns of reasoning. Try out your skills on the following exercise set. Exercise: Determine the validity or invalidity of the following arguments. With the arguments in part A (given in syllogistic form), apply the rules for determining validity. In part B, you are given arguments that could be translated into categorical form, but you should try to determine validity simply by picturing the logical relationships in the premises.