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Core Logic$
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Neil Tennant

Print publication date: 2017

Print ISBN-13: 9780198777892

Published to Oxford Scholarship Online: October 2017

DOI: 10.1093/oso/9780198777892.001.0001

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The Relevance Properties of Core Logic

The Relevance Properties of Core Logic

Chapter:
(p.262) Chapter 10 The Relevance Properties of Core Logic
Source:
Core Logic
Author(s):

Neil Tennant

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198777892.003.0010

Ironically Anderson and Belnap argue for the rejection of Disjunctive Syllogism by means of an argument that appears to employ it. We aim to establish a ‘variable-sharing’ result for Classical Core Logic that is stronger than any such result for any other system. We define an exigent relevance condition R(X,A) on the premise-set X and the conclusion A of any proof, exploiting positive and negative occurrences of subformulae. This treatment includes first-order proofs. Our main result on relevance is that for every proof of A from X in Classical Core Logic, we have R(X,A). R(X,A) is a best possible explication of the sought notion of relevance. Our result is optimal, and challenges relevantists in the Anderson–Belnap tradition to identify any strengthening of the relation R(X,A) that can be shown to hold for some subsystem of Anderson–Belnap R but that can be shown to fail for Classical Core Logic.

Keywords:   Disjunctive Syllogism, variable sharing, relevance condition, Anderson–Belnap R

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