Mind 116 (461):23-53 (2007)

Peter Milne
University of Stirling
From the point of view of proof-theoretic semantics, we examine the logical background invoked by Neil Tennant's abstractionist realist account of mathematical existence. To prepare the way, we must first look closely at the rule of existential elimination familiar from classical and intuitionist logics and at rules governing identity. We then examine how well free logics meet the harmony and uniqueness constraints familiar from the proof-theoretic semantics project. Tennant assigns a special role to atomic formulas containing singular terms. This, we find, secures harmony and uniqueness but militates against the putative realism.
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DOI 10.1093/mind/fzm023
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The Harmony of Identity.Ansten Klev - 2019 - Journal of Philosophical Logic 48 (5):867-884.
Isomorphism Invariance and Overgeneration.Owen Griffiths & A. C. Paseau - 2016 - Bulletin of Symbolic Logic 22 (4):482-503.
Harmonious Rules for Identity.Owen Griffiths - 2014 - Review of Symbolic Logic 7 (3):499-510.
Necessity of Thought.Cesare Cozzo - 2015 - In Heinrich Wansing (ed.), Dag Prawitz on Proofs and Meaning. Springer. pp. 101-20.

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