User:Danielsavoiu/Term logic

Aristotle's logical system
Aristotle recognised four kinds of quantified sentences, each of which contain a subject and a predicate:


 * Universal affirmative: Every S is a P.
 * Written SaP (a comes from af-fir-mo, Latin for "I affirm"; the first vowel is taken, since it is universal)


 * Universal negative: No S is a P.
 * Written SeP (e comes from ne-go, Latin for "I deny"; the first vowel is taken, since it is universal)


 * Particular affirmative: Some S are P.
 * Written SiP (i comes from af-fir-mo, Latin for "I affirm"; the second vowel is taken, since it is particular)


 * Particular negative: Not every S is a P.
 * Written SoP (o comes from ne-go, Latin for "I deny"; the second vowel is taken, since it is particular)

There are various ways to combine such sentences into syllogisms, both valid and invalid. In Mediaeval times, students of Aristotelian logic classified every possibility and gave them names. For example, the Barbara syllogism is as follows:


 * Every Y is a Z.
 * Every X is a Y.
 * Therefore, every X is a Z.

Barbara comes from the three sentences used:

MaP SaM - SaP

At first glance, this may seem the same as:

SaM MaP - SaP

However, in Aristotelian logic this is not so. One logical law states that the predicate must be given by the first premise, the subject by the second. It would however be correct to write:

SaM MaP - PaS

A syllogism can furthermore fall into one of the following patterns: I   I I   III   I V M?P | P?M | M?P | P?M S?M | S?M | M?S | M?S

For each there are several valid modes. To check for validity, see if the terms are distributed. To be distributed means to be either:
 * 1) Subject of a Universal Premise (SaP; SeP)
 * 2) Predicate of a Negative Premise (Sep; SoP)

Lastly, a premise can be converted or obverted to fall into a specific valid case. Conversion is generally achieved by switching terms.

Sap = PiS

SiP = PiS

SeP = PoS

SoP = Ø

Obversion is generally achieved by negating the predicate.

SaP = Se-P

SiP = So-P

SeP = Sa-P

SoP = Ø

Due to technical limitations, the negated term cannot be displayed as it should be. It is not S?-P, but rather S?P with a line over the P.

Aristotle also recognised the various immediate entailments that each type of sentence has. For example, the truth of a universal affirmative entails the truth of the corresponding particular affirmative, as well as the falsity of the corresponding universal negative and particular negative. The square of opposition or square of Boethius lists all these logical entailments.

Famously, Aristotelian logic runs into trouble when one or more of the terms involved is empty (has no members). For example, under Aristotelian logic, "all trespassers will be prosecuted" implies the existence of at least one trespasser.