[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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