*Neil Tennant*

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780198777892
- eISBN:
- 9780191823367
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198777892.003.0010
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

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 ...
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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.Less

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.

*Neil Tennant*

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780198777892
- eISBN:
- 9780191823367
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198777892.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Core Logic avoids the Lewis First Paradox, even though it contains ∨-Introduction, and a form of ∨-Elimination that permits core proof of Disjunctive Syllogism. The reason for this is that the method ...
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Core Logic avoids the Lewis First Paradox, even though it contains ∨-Introduction, and a form of ∨-Elimination that permits core proof of Disjunctive Syllogism. The reason for this is that the method of cut-elimination will unearth the fact that the newly combined premises form an inconsistent set. A new formal-semantical relation of logical consequence, according to which B is not a consequence of A,¬A, is available as an alternative to the conventionally defined relation of logical consequence. Nevertheless we can make do with the conventional definition, and still show that (Classical) Core Logic is adequate unto it. Although Core Logic eschews unrestricted Cut, nevertheless (i) Core Logic is adequate for all intuitionistic mathematical deduction; (ii) Classical Core Logic is adequate for all classical mathematical deduction; and (iii) Core Logic is adequate for all the deduction involved in the empirical testing of scientific theories.Less

Core Logic avoids the Lewis First Paradox, even though it contains ∨-Introduction, and a form of ∨-Elimination that permits core proof of Disjunctive Syllogism. The reason for this is that the method of cut-elimination will unearth the fact that the newly combined premises form an inconsistent set. A new formal-semantical relation of logical consequence, according to which B is not a consequence of A,¬A, is available as an alternative to the conventionally defined relation of logical consequence. Nevertheless we can make do with the conventional definition, and still show that (Classical) Core Logic is adequate unto it. Although Core Logic eschews unrestricted Cut, nevertheless (i) Core Logic is adequate for all intuitionistic mathematical deduction; (ii) Classical Core Logic is adequate for all classical mathematical deduction; and (iii) Core Logic is adequate for all the deduction involved in the empirical testing of scientific theories.