Richard Whately, answered from the texts and cited to the page.
The syllogism is the form to which every valid argument may be reduced, so that its structure becomes visible and its defects, if any, may be detected. Strip any piece of reasoning down to its bare bones and you will find three terms, three propositions, and a single governing principle. Those three terms stand in fixed relations to one another.1
The term taken as the subject of the conclusion is the minor-term; the predicate of the conclusion is the major-term; and the term used to establish the connexion between those two — the one that appears in both premises but drops out of the conclusion — is the middle-term, the medium of proof.1 Each term occurs exactly twice across the three propositions, which is what makes the inference possible at all.
The whole machine rests on two canons: if two terms agree with one and the same third, they agree with each other; if one term agrees and another disagrees with one and the same third, those two disagree with each other.2 On the first canon rests the validity of every affirmative conclusion; on the second, of every negative one. A syllogism that violates neither canon is sound; one that violates either is not, whatever appearance of argument it may wear.2
One caution deserves particular emphasis. If the middle-term is ambiguous — used in different senses across the two premises — there are in reality two middle-terms, though but one in sound.3 The extremes are then not being compared to the same third thing, and no conclusion follows. More equivocation shelters under this single defect than under any other.
Now, what can the syllogism prove? That depends on the figure — the arrangement of the middle-term across the premises. Where the middle-term is the subject of each premise (the third figure), only particular conclusions can be drawn; to infer a universal would always involve an illicit process of the minor-term.4 This third figure is, as I have called it, the exceptive or refutatory figure — the natural form for arguments that go to establish the contradictory of some universal proposition that another party has maintained.4
And the whole, every figure, every mood, reduces ultimately to Aristotle's dictum: what is predicated, affirmatively or negatively, of a term distributed, may be predicated in like manner of anything contained under that term.5 There is no sound argument that cannot be brought to this form without departing from its real meaning and drift.5
The form is more prolix than ordinary use requires — no one needs to write out a syllogism to buy bread — but it is the most perspicuous form in which an argument can be exhibited for inspection, and inspection is exactly what most bad reasoning cannot survive.
Of these three terms, then, that which is taken as the Subject of the Conclusion ('Z') is called the 'Minor-term;' the Predicate of the Conclusion ['Y'] is called the 'Major-term;' (from its being usually of more extensive signification than the 'Minor,' of which it is predicated) and the Term ['X'] which is used for establishing the connexion between those two, is thence called the 'Middle-term,' [or 'medium of proof.']Easy Lessons on Reasoning
first, if two terms agree with one and the same third, they agree with each other: secondly, if one term agrees, and another disagrees with one and the same third, these two disagree with each other. On the former of these Canons rests the validity of affirmative conclusions; on the latter, of negative: for no categorical syllogism can be faulty which does not violate these Canons; none correct which does not.Elements of Logic
if the middle-term is ambiguous, there are in reality two middle-terms, in sense, though but one in sound. An ambiguous Middle-term is either an equivocal term used in different senses in the two premisses.Elements of Logic
In this third Figure you will find that none but Particular Conclusions can be drawn. To infer a Universal would always, you will find, involve an 'illicit-process of the Minor-term.'" and "The third Figure might be called the 'exceptive' or the 'refutatory' Figure; (or, agreeably to the expression of the Greek writers, the 'enstatic;') as being a very natural form of expressing arguments which go to establish the contradictory of some Universal Proposition that any one may have maintained.Easy Lessons on Reasoning
All Reasoning whatever, then, rests on the one simple Principle laid down by Aristotle, that 'what is predicated, either affirmatively or negatively, of a term distributed, may be predicated in like manner, (i. e. affirmatively or negatively) of any thing contained under that term'... there is no sound argument but what can be reduced into this form, without at all departing from the real meaning and drift of it; and the form will be found (though more prolix than is needed for ordinary use) the most perspicuous in which an argument can be exhibited.Elements of Logic