Aristotelian Syllogisms

Aristotelian Syllogisms

after Raymond McCall, Basic Logic (Barnes & Noble, 1967); symbolic apparatus from Elementary Logic, by Benson Mates (Oxford, 1972)

Copyright (c) 1998, 1999, 2002 Kelley L. Ross, Ph.D. All Rights Reserved

via: http://www.friesian.com/aristotl.htm

Parts of a syllogism:

A: a universal affirmative proposition—All S is P [(x)(Sx -> Px)].
E: a universal negative proposition—No S is P [(x)(Sx -> -Px)].
I: a particular affirmative proposition—Some S is P [(x)(Sx & Px)].
O: a particular negative proposition—Some S is not P [(x)(Sx & -Px)].

The predicate of an affirmative proposition is regarded as having particular quantification, the predicate of a negative proposition, universal.

S: subject of the conclusion.
P: predicate of the conclusion.
M: the middle term.

The Major Premise of a syllogism contains the predicate of the conclusion and the middle term.

The Minor Premise contains the subject of the conclusion and the middle term.

The four figures, possible combinations of middle terms as subjects or predicates of major or minor premises, are:

1st 2nd 3rd 4th
M P P M M P P M
S M S M M S M S
—— —— —— ——
S P S P S P S P

All the possible moods, or kinds of propositions in the two premises (the moods that turn out to be valid in some figure are in bold face):

Major Premise: AAAA IIII EEEE OOOO
Minor Premise: AEIO AEIO AEIO AEIO

Rules of the syllogism:

1) There are only three terms in a syllogism (by definition).
2) The middle term is not in the conclusion (by definition).
3) The quantity of a term cannot become greater in the conclusion.
4) The middle term must be universally quantified in at least one premise.
5) At least one premise must be affirmative.
6) If one premise is negative, the conclusion is negative.
7) If both premises are affirmative, the conclusion is affirmative.
8) At least one premise must be universal.
9) If one premise is particular, the conclusion is particular.
10) In extensional logic, if both premises are universal, the conclusion
is universal. (See DARAPTI, etc., and “In Defense of Bramantip”)

These moods have premises that are both particular or both negative and so do not produce valid syllogisms:

Major Premise: II EE OOO
Minor Premise: IO EO EIO

One more mood is always invalid:

Major Premise: I In this mood the major term would have particular
Minor Premise: E quantification in the major premise; but, since the
– conclusion would have to be negative, it would have
Conclusion: O universal quantification there, violating rule 3.

Other moods are eliminated in each figure. The vowels in the names for the moods give the types of propositions in the major premise, the minor premise, and then the conclusion, respectively.

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