Showing papers by "Neil Tennant published in 2009"

Revamping the Restriction Strategy

Abstract: This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘φ, therefore 3Kφ’ should be formulated as a consistency requirement not on the premise φ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of φ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘3Kφ therefore φ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK -thesis in order to get around the Cartesian restriction on applications of the knowability principle. ∗To appear in Joseph Salerno, ed., All Truths are Known: New Essays on the Knowability Paradox, Oxford University Press. This paper would not have been written without the stimulation, encouragement and criticism that I have enjoyed from Joseph Salerno, Salvatore Florio, Christina Moisa, Nicholaos Jones, and Patrick Reeder.

Natural Logicism via the Logic of Orderly Pairing

Abstract: The aim here is to describe how to complete the constructive logicist program, in the author’s book Anti-Realism and Logic, of deriving all the Peano–Dedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neo-Fregean done so. These outstanding axioms need to be derived in a way fully in keeping with the spirit and the letter of Frege’s logicism and his doctrine of definition. To that end this study develops a logic, in the Gentzen-Prawitz style of natural deduction, for the operation of orderly pairing. The logic is an extension of free first-order logic with identity. Orderly pairing is treated as a primitive. No notion of set is presupposed, nor any set-theoretic notion of membership. The formation of ordered pairs, and the two projection operations yielding their left and right coordinates, form a coeval family of logical notions. The challenge is to furnish them with introduction and elimination rules that capture their exact meanings, and no more. Orderly pairing as a logical primitive is then used in order to introduce addition and multiplication in a conceptually satisfying way within a constructive logicist theory of the natural numbers. Because of its reliance, throughout, on sense-constituting rules of natural deduction, the completed account can be described as ‘natural logicism’.

Inferential Semantics for First-Order Logic: Motivating Rules of Inference from Rules of Evaluation

Abstract: ’ is a theorem’ and ‘’ is logically true’ are special cases. In the early 1970s Tim gave a formative series of lectures emphasizing how proofs are to be understood as perfected arguments, in Aristotle’s sense. The present discussion of verication and falsication is fully in the inferentialist spirit of Tim’s emphases. The aim is to render even the notions ‘’ is true’ and ‘’ is false’ as essentially relational and inferential. A sentence’s truth-value is determined relative to collections of rules of inference that constitute an interpretation. Moreover, truthmakers and falsity-makers are themselves proof-like objects, encoding the inferential process of evaluation involved. The inference rules involved in the determination of truth-value are almost identical to those involved in securing the transmission of truth from premises to conclusion of a valid argument. We shall see how smoothly one can ‘morph’ the former into the latter.