The syllogism is a logical system that was invented by Aristotle which deduces valid inferences from given premises. It is categorical in nature because each of two premises and the conclusion has an internal relationship of belonging or inclusion. Specifically, there is a major premise of a general nature and a minor premise that is usually specific, or of reduced generality. Both are combined deductively to reach or prove the conclusion.
Both premises and the conclusion deal with three categories two at a time, a subject term (S), a middle term (M), and a predicate term (P), joined by one of four binary inclusion relations. The major premise deals with M and P, the minor premise deals with S and M, and the conclusion with S and P. The four types of relations are denoted by the letters A, E, I, O (also a, e, i, o) and are described below. The premises may have M first or second, but the conclusion always has the S first and the P second.
S = Subject
M = Middle
P = Predicate
A = a = XaY = All X are Y
E = e = XeY = All X are not Y
I = i = XiY = Some X are Y
O = o = XoY = Some X are not Y
Major premise: MxP or PxM, x = a, e, i, or o
Minor premise: SxM or MxS
The distinction between the four Figures concerns the placement of the middle term M in each of the premises. In order to highlight this order, I’ve written them with ( and ) on the side of the relation where the M is.
Figure 1: MxP, SyM, SzP: (xy)z
Figure 2: PxM, SyM, SzP: x(y)z
Figure 3: MxP, MyS, SzP: (x)yz
Figure 4: PxM, MyS, SzP: x()yz
There are only 24 valid inferences out of all possible combinations, six for each of the four Figures (and some of these may be erroneous sometimes due to the existential fallacy). In addition, they were given mnemonic names in the Middle Ages by adding consonants around the vowels of the relations. And so the valid inferences and their names (or something close to it) are as follows (by my notation and in no special order):
For example, (aa)a, or Barbara, is a syllogism of the form: All Y are Z; All X are Y; thus All X are Z.
Vaughan Pratt / Aristotle, Boole, and Categories (PDF, October 12, 2015)
Vaughan Pratt / Aristotle, Boole, and Chu: Duality since 350 BC (Slides, August 12, 2015)