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Oxford dictionary definition of the word syllogism:
a deductive inference consisting of two premises and a conclusion, all of which are categorial propositions. The subject of the conclusion is the minor term and its predicate the major term; the middle term occurs in both premises but not the conclusion. There are 256 such arguments but only 24 are valid. Some men are mortal; some men are angelic; so some mortals are angelic is invalid, while some temples are in ruins; all ruins are fascinating; so some temples are fascinating is valid. Here fascinating, in ruins, and temples are respectively major, middle, and minor terms
a deductive inference of certain other forms with two premises, such as the hypothetical syllogism, if P then Q; if Q then R; so if P then R
a piece of deductive reasoning from the general to the particular
a subtle or deceptive piece of reasoning