Any consistent and sufficiently strong system of first-order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first-order extensions of such systems. Our purpose is to discover what is minimally required by way of such extension in order to be able to prove the Gödel sentence in a non-trivial fashion. The extended methods of formal proof must capture the essentials of the so-called 'semantical argument' for the truth of the Gödel sentence. We are concerned to show that (...) the deflationist has at his disposal such extended methods-methods which make no use or mention of a truth-predicate. This consideration leads us to reassess arguments recently advanced-one by Shapiro and another by Ketland-against the deflationist's account of truth. Their main point of agreement is this: they both adduce the Gödel phenomena as motivating a 'thick' notion of truth, rather than the deflationist's 'thin' notion. But the so-called 'semantical argument', which appears to involve a 'thick' notion of truth, does not really have to be semantical at all. It is, rather, a reflective argument. And the reflections upon a system that are contained therein are deflationarily licit, expressible without explicit use or mention of a truth-predicate. Thus it would appear that this anti-deflationist objection fails to establish that there has to be more to truth than mere conformity to the disquotational T-schema. (shrink)

N. Tennant; XI*—Entailment and Proofs, Proceedings of the Aristotelian Society, Volume 79, Issue 1, 1 June 1979, Pages 167–190, https://doi.org/10.1093/aristote.

I propose an anti-realist account of truth and paradox according to which the logico-semantic paradoxes are not genuine inconsistencies. The ‘global’ proofs of absurdity associated with these paradoxes cannot be brought into normal form. The account combines epistemicism about truth with a proof-theoretic diagnosis of paradoxicality. The aim is to combine a substantive philosophical account of truth with a more rigorous and technical diagnosis of the source of paradox for further consideration by logicians. Core Logic plays a central role in (...) the account on offer. It is shown that the account is not prey to the problem of revenge paradox. (shrink)

Using a quantified propositional logic involving the operators it is known that and it is possible to know that, we formalize various interesting philosophical claims involved in the realism debate. We set out inferential rules for the epistemic modalities, ranging from ones that are obviously analytic, to ones that are epistemologically more substantive or even controversial. Then we investigate various aporias for the realism debate. These are constructively inconsistent triads of claims from our list: a claim expressing some sort of (...) common ground in the debate; a characteristically anti-realist thesis about truth and knowability; and a characteristically realist thesis about determinacy of truth value. Various patterns of reductio proof for these inconsistent triads are generated, so as to display their variety. The reductio proofs use only the inferential rules set out earlier. The philosophical utility of each aporia for the anti-realist is then assessed. This involves consideration of the acceptability of the premiss expressing common ground; the strength and plausibility of the anti-realist premiss; the strength of the realist premiss that is the ultimate target of the reductio; and the analytic status of the inferential rules applied within the proof. (shrink)

N. Tennant; XI*—Entailment and Proofs, Proceedings of the Aristotelian Society, Volume 79, Issue 1, 1 June 1979, Pages 167–190, https://doi.org/10.1093/aristote.

I clarify how the requirement of conservative extension features in the thinking of various deflationists, and how this relates to another litmus claim, that the truth-predicate stands for a real, substantial property. I discuss how the deflationist can accommodate the result, to which Cieslinski draws attention, that non-conservativeness attends even the generalization that all logical theorems in the language of arithmetic are true. Finally I provide a four-fold categorization of various forms of deflationism, by reference to the two claims of (...) conservativeness and substantiality. This helps to clarify the various possible positions in the deflationism debate. (shrink)

The over-arching theme is that we can redeem Frege's key philosophical insights concerning (natural and real) numbers and our knowledge of them, despite Russell's famous discovery of paradox in Frege's own theory of classes. That paradox notwithstanding, numbers are still logical objects, in some sense created or generated by methods or principles of abstraction— which of course cannot be as ambitious as Frege's Basic Law U. These principles not only bring numbers into existence, as it were, but also afford a (...) distinctive form of epistemic access to them. The usual mathematical axioms governing the two kinds of numbers are to be derived as results in (higher-order) logic. These derivations will exploit appropriate definitions of the primitive constants, functions, and predicates of the brand of number theory concerned. (For example: 0, 1; s, +, x; (; N(z); R(z).) No supplementation by intuition or sensory experience will be needed in the derivations of these axioms. The trains of reasoning involved will depend only on our grasp of logical validities, supplemented by appropriate definitions. Result: logicism is vindicated; and the mathematical knowledge derived in this way is revealed to be analytic, not synthetic. (shrink)

In his paper 'Changing the Theory of Theory Change: Reply to My Critics', N. Tennant (1997b) reacts to the critical reception of an earlier article of his. The present note rectifies some of the most serious misrepresentations in Tennant's reply.

In his paper ‘Changing the Theory of Theory Change: Reply to My Critics’, N. Tennant (1997b) reacts to the critical reception of an earlier article of his. The present note rectifies some of the most serious misrepresentations in Tennant's reply.

The rules for Core Logic are stated, and various important results about the system are summarized. We describe its relationship to other systems, such as Classical Logic, Intuitionistic Logic, Minimal Logic, and the Anderson–Belnap relevance logicR. A precise, positive explication is offered of what it is for the premises of a proof to connect relevantly with its conclusion. This characterization exploits the notion of positive and negative occurrences of atoms in sentences. It is shown that all Core proofs are relevant (...) in this precisely defined sense. We survey extant results about variable-sharing in rival systems of relevance logic, and find that the variable-sharing conditions established for them are weaker than the one established here for Core Logic (and for its classical extension). Proponents of other systems of relevance logic (such asRand its subsystems) are challenged to formulate a stronger variable-sharing condition, and to prove thatRor any of its subsystems satisfies it, but that Core Logic does not. We give reasons for pessimism about the prospects for meeting this challenge. (shrink)

A general method is provided whereby bizarre revisions of consistent theories with respect to contingent sentences that they refute can be delivered by revision-functions satisfying both the basic and the supplementary postulates of the AGM-theory of theory-revision.

This paper rebuts criticisms by Hintikka of the author's account of game-theoretic semantics for classical logic. At issue are (i) the role of the axiom of choice in proving the equivalence of the game-theoretic account with the standard truth-theoretic account; (ii) the alleged need for quantification over strategies when providing a game-theoretic semantics; and (iii) the role of Tarski's Convention T. As a result of the ideas marshalled in response to Hintikka, the author puts forward a new conjecture concerning the (...) relationship among truth, meaning and translation. (shrink)

This book develops Tennant's Natural Logicist account of the foundations of the natural, rational, and real numbers. Tennant uses this framework to distinguish the logical from the intuitive aspects of the basic elements of arithmetic.

The naive anti-realist holds the following principle: (◊K) All truths are knowable. This unrestricted generalization (◊K), as is now well known, falls prey to Fitch’s Paradox (Fitch 1963: 38, Theorem 1). It can be used as the only suspect principle, alongside others that cannot be impugned, to prove quite generally, and constructively, that the set {p, ¬Kp} is inconsistent (Tennant 1997: 261). From this it would follow, intuitionistically, that any proposition that is never actually known to be true (by (...) anyone, at any time) is false; and it would follow, classically, that every truth is known (by someone, at some time). (shrink)

Kalrnaric. We set out a system T, consisting of normal proofs constructed by means of elegantly symmetrical introduction and elimination rules. In the system T there are two requirements, called ( ) and ()), on applications of discharge rules. T is sound and complete for Kalmaric arguments. ( ) requires nonvacuous discharge of assumptions; ()) requires that the assumption discharged be the sole one available of highest degree. We then consider a 'Duhemian' extension T*, obtained simply by dropping the requirement (...) ()). T* is a proper subsystem of intuitionistic relevant logic. Our main result is that T* is a double negation consistency companion to classical logic. Thus all one needs to add to T* to obtain classical logic is the (intuitionistic) absurdity rule, and the (classical) rule of double nega-. (shrink)

I examine Paul Boghossian's recent attempt to argue for scepticism about logical rules. I argue that certain rule- and proof-theoretic considerations can avert such scepticism. Boghossian's 'Tonk Argument' seeks to justify the rule of tonk-introduction by using the rule itself. The argument is subjected here to more detailed proof-theoretic scrutiny than Boghossian undertook. Its sole axiom, the so-called Meaning Postulate for tonk, is shown to be false or devoid of content. It is also shown that the rules of Disquotation and (...) of Semantic Ascent cannot be derived for sentences with tonk dominant. These considerations deprive Boghossian's scepticism of its support. (shrink)

The Taming of the True continues the project Neil Tennant began in Anti-realism and Logic of investigating and defending anti-realism. Tennant’s earlier book anticipated a second volume, in which issues related to empirical discourse would be addressed in greater detail. The Taming of the True provides this sequel. It also attempts a ground-clearing project, by addressing challenges to some of the presuppositions and implications of Tennant’s anti-realist position. Finally, it takes an opportunity to revisit some of the (...) issues examined in the earlier book, in light of the evolving debate over realism and anti-realism. (shrink)

The system of natural deduction that originated with Gentzen , and for which Prawitz proved a normalization theorem, is re-cast so that all elimination rules are in parallel form. This enables one to prove a very exigent normalization theorem. The normal forms that it provides have all disjunction-eliminations as low as possible, and have no major premisses for eliminations standing as conclusions of any rules. Normal natural deductions are isomorphic to cut-free, weakening-free sequent proofs. This form of normalization theorem renders (...) unnecessary Gentzen's resort to sequent calculi in order to establish the desired metalogical properties of his logical system.Ultimate normal forms are well-adapted to the needs of the computational logician, affording valuable constraints on proof-search. They also provide an analysis of deductive relevance. There is a deep isomorphism between natural deductions and sequent proofs in the relevantized system. (shrink)

This is a reply to Timothy Williamson ’s paper ‘Tennant’s Troubles’. It defends against Williamson ’s objections the anti-realist’s knowability principle based on the author’s ‘local’ restriction strategy involving Cartesian propositions, set out in The Taming of the True. Williamson ’s purported Fitchian reductio, involving the unknown number of books on his table, is analyzed in detail and shown to be fallacious. Williamson ’s attempt to cause problems for the anti-realist by means of a supposed rigid designator generates a (...) contradiction with arithmetic right away, upon instantiating the obviously relevant theorem that every natural number is provably odd or provably even. The paper also explains and formulates a globally restricted knowability principle, which likewise blocks the attempted reductio. (shrink)