Principles for Object-Linguistic Consequence: from Logical to Irreflexive

Journal of Philosophical Logic 47 (3):549-577 (2018)
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Abstract

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by ) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth. Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence.

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Author Profiles

Carlo Nicolai
King's College London
Lorenzo Rossi
Università di Torino

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References found in this work

Outline of a theory of truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Saving truth from paradox.Hartry Field - 2008 - New York: Oxford University Press.
Structural Proof Theory.Sara Negri, Jan von Plato & Aarne Ranta - 2001 - New York: Cambridge University Press. Edited by Jan Von Plato.

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