Term logic/Danielsavoiu's summary

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[edit] 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.
  • Universal negative: No S is a P.
  • Particular affirmative: Some S are P.
  • Particular negative: Not every S is a P.
  • Univ. Affirm: Written SaP (a comes from a-fir-mo, latin for to affirm, we take the first vowel, since it's universal)
  • Univ. Neg: Written SeP (e comes from ne-go, latin for to deny, we take the first vowel, since it's universal)
  • Part. Affirm: Written SiP (i comes from a-fir-mo, latin for to affirm, we take the second vowel, since it's particular)
  • Part. Neg Written SoP (o comes from ne-go, latin for to deny, we take the second vowel, since it's 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 in 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, we 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 obverted or converted to fall to a specific valid case. Conversion is generally achieved by switching terms.

Sap = PiS

SiP = PiS

SeP = PoS

SoP = Ø

Obvertion 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.