[FOM] Intuitionists and excluded-middle
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Thu Oct 27 23:19:19 EDT 2005
Discussion on this issue has seemed to accept the Intuitionists' own
self-assessment that they were denying the Law of the Excluded
Middle. A similar question arises in the dual case about whether to
accept Paraconsistentists' own self-assessment that they are denying
the Law of Non-Contradiction. For if the 'negation' in
Intuitionistic Logic is not a contradiction forming operator, but a
contrary forming one, then its 'p v -p' does not say that either a
proposition or its contradictory is true; and if the 'negation' in
Paraconsistent Logic is not a contradiction forming operator but a
subcontrary forming one, then its 'p.-p' does not say that a
proposition and its contradictory are both true.
Formal proofs that contradiction is not involved in either case have
recently been given, see J.-Y.Béziau, "Paraconsistent logic from a
modal viewpoint", Journal of Applied Logic, 3 (2005), pp.7-14. The
issue has a history, since, for example, in 'What not? A Defense of
Dialetheic Theory of Negation' in D.Gabbay and H.Wansing (eds) What
is Negation (Kluwer, Dordrecht 1999), p110, Graham Priest agreed
about Intuitionism, but disagreed about Dialetheism. And I myself
took Priest up on the subcontrariety issue in 'Paraconsistent
Logics?', Journal of Philosophical Logic 24 (1995) 1-4.
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