PHI 102 - Reasoning and Critical Thinking

Categorical Syllogisms

Categorical Syllogisms: Are simply syllogisms crucially involving categorical propositions

Here are some examples: What follows from the following pairs of premises?

1. All men are mortal. Socrates is a man.	2. No man is a fish. Nemo is a fish.
3. All whales are mammals. No mammal is a fish.	4. No mammals are fish. Some fish are good to eat.

In 1-4 above, the first premise is called a universal categorical statement. In effect, such universal categorical statements are generalized inference licenses, specifying that certain kinds of statements follow from certain other kinds of statements. In 3, both of the premises turn out to be universal, while in 1,2, and 4, the second premise is called a particular statement (an instance of a kind). Now what respective kinds of hypothetical reasoning do you suppose the above allowable inferences resemble?

Evaluating The Validity of Categorical Syllogisms

As the name suggests, categorical syllogisms are short deductive arguments, where the premises (typically there are two) and conclusion are categorical statements. One can readily assess their validity by thinking of universal categorical statements as expressing conditionals. This handout outlines a way to convert categorical syllogisms into hypothetical syllogisms, the validity of which you should already know how to determine.

The procedure is as follows:

1. Identify premises and conclusion. This is the first step in evaluating the validity of any argument. If the syllogism is "mixed" (that is, it contains a universal premise and a particular premise, it is also good practice to list the universal premise first. If both of the premises are particular (they talk about particular individuals or "some" members inside or outside a particular class, and so can’t be converted into conditionals), then the syllogism will be invalid.

2. Convert all universal categorical propositions into conditional statements.

a. "All A’s are B’s." becomes "If something is an A, then it is a B."

b. "No A’s are B’s." becomes "If something is an A, then it is not a B."

c. "Only A’s are B’s." becomes "If something is a B, then it is an A."

3. If the syllogism is "mixed" (that is, it has a single conditional premise), then go ahead and determine whether its form is valid (AA or DC) or invalid (DA or AC).

Hint: If the syllogism’s "middle term" appears in the antecedent of the conditional premise, then it either affirms or denies the antecedent. If it appears in the consequent of the conditional premise, then it either affirms or denies the consequent.

For clarity’s sake, it might also be helpful to reverse the particular premise:

a. "Some A’s are B’s." can be expressed as "Some B’s are A’s."

b. "Some A’s are not B’s" can be expressed as "Some non-B’s are A’s"

Finally, if the syllogism has a valid form, check to make sure that the stated conclusion really is (or is equivalent to) the conclusion that follows from applying modus ponens (AA) or modus tollens (DC). Once again, you might need to reverse a particular statement.

4. If the resulting syllogism is "pure" (that is, both of the premises and the conclusion are conditional statements), then identify its middle term. The middle term is the one that occurs in both of the premises.

5. Use the law of contraposition to diagonalize the middle term. If the middle term occurs in the antecedent of both of the premises or in the consequent of both premises, then apply the law of contraposition upon one (not both!) of the premises (it will not matter which). The result should be a syllogism in which the middle term appears in the antecedent of one premise and the consequent of the other.

According to the Law of Contraposition: "If p, then q." is equivalent to "If not q, then not p." So….

a. "If something is an A, then it is a B." (or "All A’s are B’s.") becomes "If something is not a B, then it is not an A."

b. "If something is an A, then it is not a B" (or "No A’s are B’s.") becomes "If something is a B, then it is not an A."

6. Check to see whether the premises are in proper shape for hypothetical syllogism. If the middle term is negative in one premise but positive in the other, then the syllogism is not in proper shape, and the syllogism is invalid.

7. If the premises are in proper shape, identify their "natural conclusion." The natural conclusion is the conditional statement that is the result of "canceling out" the middle term in the premises.

8. If the actual conclusion of the syllogism is equivalent to the natural conclusion or its contraposition, then the syllogism is valid. Otherwise, it is invalid.

Comments:

1. Here’s a short cut: In step 4. above, if the middle term occurs in both of the antecedents of the premises or in both of their consequents AND the middle term is affirmative in both instances or negative in both, then the syllogism is invalid.

The reason: In these cases diagonalization is bound to produce a syllogism that isn’t in proper form for hypothetical syllogism.

2. Some syllogisms might have both categorical and conditional statements. Indeed, categorical statements will sometimes be embedded within conditional statements, in which case they might play a role in an extended argument with subconclusions. Note finally that it is indeed possible to chain syllogisms together (especially those with several universal or conditional premises), and to apply several of our inference rules (especially hypothetical syllogism) all at once.