Logic: terms, propositions and syllogism

Philosophy · Logic, ethics and special topics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Traditional logic sorts categorical propositions into four forms by quantity (universal or particular) and quality (affirmative or negative): A, E, I and O. A term is distributed when the proposition speaks of every member of its class. The square of opposition shows how the four forms with the same subject and predicate are related.

A syllogism draws a conclusion from two premises. It has three terms: the major term (predicate of the conclusion), the minor term (subject of the conclusion) and the middle term, which appears in both premises but not in the conclusion.

Deduction gives a conclusion that follows necessarily. Induction goes from observed cases to a general claim that is only probable.

Rules to remember

A: All S are P. E: No S is P. I: Some S are P. O: Some S are not P.
Distribution: A distributes the subject only; E both terms; I neither; O the predicate only.
Square of opposition: A and O, and E and I, are contradictories (one true, one false). A and E are contraries (cannot both be true, may both be false). I and O are subcontraries (cannot both be false, may both be true). I follows from A, and O from E (subalternation).
Rules of the syllogism: exactly three terms; the middle term distributed at least once; no term distributed in the conclusion unless distributed in its premise; no conclusion from two negative premises; a negative premise gives a negative conclusion; no conclusion from two particular premises.
Fallacies: undistributed middle; illicit major; illicit minor; four terms.
Figure = position of the middle term. First: M–P, S–M. Second: P–M, S–M. Third: M–P, M–S. Fourth: P–M, M–S.
Mood = the letters of major premise, minor premise and conclusion in that order. Valid moods of the first figure: AAA, EAE, AII, EIO.

Common mistake

Many candidates confuse contraries with contradictories. If 'All S are P' is false, only 'Some S are not P' is certainly true; 'No S is P' may still be false.

Worked example

'All squares are rectangles; no circles are rectangles; therefore no circles are squares.' Give the mood and figure and say whether the syllogism is valid.

The conclusion is 'No circles are squares', so the minor term is 'circles', the major term is 'squares' and the middle term is 'rectangles'.
Major premise 'All squares are rectangles' is A; minor premise 'No circles are rectangles' is E; the conclusion is E. The mood is AEE.
The middle term is the predicate of both premises, so this is the second figure.
The middle term is distributed in the E premise; 'squares' and 'circles' are distributed in their premises; one premise and the conclusion are negative. No rule is broken.

Answer: Mood AEE in the second figure; the syllogism is valid.

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