NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/logic/2111

Text File | 1992-11-19 | 1.6 KB | 37 lines

Newsgroups: sci.logic
Path: sparky!uunet!stanford.edu!CSD-NewsHost.Stanford.EDU!Sunburn.Stanford.EDU!pratt
From: [e-mail] (Vaughan R. Pratt)
Subject: Re: implication truth table
Message-ID: <[e-mail]>
Sender: [e-mail]
Organization: Computer Science Department,  Stanford University.
References: <[e-mail]> <[e-mail]>
Date: Thu, 19 Nov 1992 22:30:28 GMT
Lines: 25

In article <[e-mail]> [e-mail] (Gavan Tredoux) writes:
>[e-mail] (John McLean) writes:
>: From [e-mail] ()
>: >My question is: WHY is the implication relation defined as it is?
>
>[some reasons]
>
>There are logics that don't allow this paradox of
>material implication - relevance logics, where the antecedent
>and consequent have to be relevant to each other for the
>implication to hold. Of course there are a million and one
>ways of doing this.
>
>Primary strength of classical logic: there's only one.

This is true for intuitionistic logic as well, which has only one
implication up to logical equivalence.  A->B is the weakest
interpretation of this implication for which modus ponens remains
sound, that is, the weakest C such that A&C |- B.  Assuming only the
cut rule and A|-A, this uniquely determines A->B up to logical
equivalence, where logical equivalence of A and B means A|-B and B|-A.
This definition may seem less paradoxical than the classical definition
of A->B as "~A or B".
--
Vaughan Pratt              A fallacy is worth a thousand steps.