Foundations of Mathematics

Foundations and Logic

What do the terms propositions and syllogisms mean in Aristotelian logic?

There is a great deal to Aristotelian logic—too much to mention in this text. But overall, there are some general terms and properties. For example, propositions are sentences with two terms—a grammatical subject and predicate. A proposition has the properties of quality and quantity only. They can either be a negative or positive proposition in terms of quality; in terms of quantity, they are either universal or particular propositions. That means the four types of propositions are: universal affirmative (for example, “all men are mortal”); universal negative (for example, “no men are mortal”); particular affirmative (for example, “some men are strong”); and particular negative (for example, “some men are not mortal”). In most texts, these four proposition types are denoted by the letters A, E, I, and O, respectively.

Syllogisms are composed of two premises and a conclusion; the conclusion comes from the two premises in a certain way. Apparently, Aristotle liked to name his logical offerings, so each word in the syllogism has a label. He said that each premise must have a term in common, which he called the middle term. The other terms he called extreme terms, divided into the major term, or the predicate of the conclusion and minor term, or subject of the conclusion. And logically, of course, the premise with the minor term is called the minor premise, and the one with the major term is called the major premise. Aristotle further divided the syllogisms into perfect syllogisms (“…which needs nothing other than what has been stated to make plain what necessarily follows…”) and imperfect syllogisms (“…needs either one or more propositions, which are indeed the necessary consequence of the terms set down, but have not been expressly stated as premises…”).


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