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Nicolai, Carlo; Rossi, Lorenzo (2018): Principles for Object-Linguistic Consequence: from Logical to Irreflexive. In: Journal of Philosophical Logic, Vol. 47, No. 3: pp. 549-577
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We discuss the principles for a primitive, object-linguistic notion of consequence proposed by (Beall and Murzi, Journal of Philosophy, 3 pp. 143-65 (2013)) 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 (and some of its variants) 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 (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naive, self-applicable notion of consequence.