Aristotle defined the syllogism as, "...a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so." Despite this very general definition, in Aristotle's work Prior Analytics, he limits himself to categorical syllogisms that consist of three categorical propositions. These include categorical modal syllogisms.
From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably. The syllogism was at the core of traditional deductive reasoning, where facts are determined by combining existing statements, in contrast to inductive reasoning where facts are determined by repeated observations.
Within academic contexts, the syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift (Concept Script) (1879), but syllogisms remain useful in some circumstances, and for general-audience introductions to logic.
There are infinitely possible syllogisms, but they're basic structure consists of three parts: a major premise, a minor premise and a conclusion. Each part is a categorical proposition, and each categorical proposition contains two categorical terms. In Aristotle, each of the premises is in the form "All A are B," "Some A are B", "No A are B" or "Some A are not B", where "A" is one term and "B" is another. "All A are B," and "No A are B" are termed universal propositions; "Some A are B" and "Some A are not B" are termed particular propositions. More modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate of the conclusion); in a minor premise, it is the minor term (the subject) of the conclusion. For example:
Major premise: All humans are mortal.
Minor premise: All Greeks are humans.
Conclusion: All Greeks are mortal.
Each of the three distinct terms represents a category. In the above example, humans, mortal, and Greeks. Mortal is the major term, Greeks the minor term. The premises also have one term in common with each other, which is known as the middle term; in this example, humans. Both of the premises are universal, as is the conclusion.
Uff, I feel a little knot in my brain. Don't you? Just kidding! Or am I? Wait, what?