Syllogisms

“All whales are mammals. All mammals breathe air. So all whales breathe air.” Medieval students called this pattern Barbara, and its three a’s are a code for the three sentences.

Four kinds of sentence

A syllogism is an argument with two premises and a conclusion, each of which links two groups of things. Traditional logic works with four kinds of these categorical sentences, named by letters. A: “All S are P” (universal affirmative). E: “No S are P” (universal negative). I: “Some S are P” (particular affirmative). O: “Some S are not P” (particular negative). Here some means at least one, so “Some students passed” is true even if every student passed.

The four types pair up as negations. A and O contradict each other: “All S are P” is false exactly when at least one S is not P. E and I also contradict each other. This is the same lesson as the first unit of this course: the negation of “all” is “not all”, which is the O sentence, not the E sentence.

Three terms, one of them in the middle

Every syllogism uses exactly three terms. In the whale example they are whales, mammals and things that breathe air. The term that appears in both premises but not in the conclusion is the middle term, here mammals. It does the work of connecting the other two. The subject of the conclusion (whales) is the minor term and the predicate of the conclusion (things that breathe air) is the major term. The premise containing the major term is called the major premise, and the one with the minor term is the minor premise.

Medieval teachers gave each valid pattern a name whose vowels list the types of its three sentences. Barbara is A, A, A: all M are P, all S are M, so all S are P. Celarent is E, A, E: “No reptiles are warm-blooded. All snakes are reptiles. So no snakes are warm-blooded.” The Stanford Encyclopedia of Philosophy notes that these names are still widely used and that each one also encodes how Aristotle proved the pattern.

The undistributed middle

A syllogism can look right and still fail. “All cats are mammals. All dogs are mammals. So all dogs are cats.” Both premises are true and the conclusion is plainly false, so the form must be invalid. The middle term, mammals, never covers its whole group in either premise: the premises only put cats somewhere inside mammals and dogs somewhere inside mammals, and they may be in different places. Logicians call this the fallacy of the undistributed middle, and the Stanford Encyclopedia lists it among the formal fallacies. A quick test for any syllogism is to swap in groups where you know the answer, as with cats and dogs, and see whether true premises can lead to a false conclusion.