Axiomathes

, Volume 25, Issue 4, pp 495–513 | Cite as

Duality and Inferential Semantics

Original Paper
First Online:
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Abstract

It is well known that classical inferentialist semantics runs into problems regarding abnormal valuations (Carnap in Formalization of logic. Harvard University Press, Cambridge, 1943; Hjortland in Notre Dame J Formal Logic 55(4):445–467, 2014; Peregrin in J Philos Logic 39(3):255–274, 2010). It is equally well known that the issues can be resolved if we construct the inference relation in a multiple-conclusion sequent calculus. The latter has been prominently developed in recent work by Restall (Logic, methodology and philosophy of science: proceedings of the twelfth international congress. Kings College Publications, pp 189–205, 2005), with the guiding interpretation that the valid sequent Open image in new window says that the simultaneous assertion of all of Γ with the denial of all of Δ is incoherent. However, such structures face significant interpretive challenges (Rumfitt in Grazer Philos Stud 77(1):61–84, 2008; Steinberger in J Philos Logic 40(3):333–355, 2011; Tennant in The taming of the true. Oxford University Press, Oxford, 1997), and they do not provide an adequate grasp on the machinery of the duality of assertions and denials that could (a) provide an abstract account of inferential semantics; (b) show why the dual treatment is semantically superior. This paper explores a slightly different tack by considering a dual-calculus framework consisting of two, single-conclusion, inference relations dealing with the preservation of assertion and the preservation of denial, respectively. In this context, I develop an abstract inferentialist semantics, before going on to show that the framework is equivalent to Restall’s, whilst providing a better grasp on the underlying proof-structure.

Keywords

Inferentialism Proof Refutation Multiple-conclusion Semantics Sequent 
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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.UCASurreyUK

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