Neil Tennant

The two main titular results are (I) no finite matrix of values characterizes propositional Core Logic $$\mathbb {C}$$, and (II) the connectives of propositional Core Logic are independent. These results echo the corresponding ones for propositional Intuitionistic Logic. Our reason for expounding them for Core Logic is that their metaproofs are easier and more accessible in that setting; and the result for Intuitionistic Logic corresponding to (II) is then a corollary to the Admissibility of Cut in Core Logic.

Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truth-conditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is contrasted with the more recent quietist inferentialism of Brandom. Various other issues are highlighted for inferentialism in general, by reference to which different kinds of inferentialism can be characterized. Inferentialism for the logical operators is explained, with special reference to the Principle of Harmony. The statement of that principle in the author’s book _Natural Logic_ is fine-tuned here in the way obviously required in order to bar an interesting would-be counterexample furnished by Crispin Wright, and to stave off any more of the same.

This chapter continues the anti-realist's quest for a principled way to avoid Fitch's paradox. It proposes that the Cartesian restriction on the anti-realist's knowability principle ‘φ, therefore ◇_K_φ’ 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 ‘◇_K_φ 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.

This chapter criticizes dialetheism from the point of view of an anti-realist with sympathy for relevantism in logical reasoning. It argues that the view that there are true contradictions suffers both from an improper understanding of the interrelations among absurdity, contrariety, falsity, and negation, and from an incorrect diagnosis of what gives rise to the well-known contradictions in semantics and mathematical foundations. Anti-realism emerges as a better reflective equilibrium than dialetheism in confrontation with all these phenomena. Both positions require logical revisions of classical logic. But anti-realism's logical reforms are better motivated than those of the dialetheist, and the resulting logic is more adequate for the methodological demands of mathematics and empirical science. Priest's prospect of an ‘intuitionist dialetheism’ is unconvincing, both because of important features of intuitionistic logic, such as the independence of the logical operators and the normalizability of proof, and because the intuitionist (or anti-realist) disagrees so strongly on the actual alleged examples of dialetheias in logic and foundations.

This study focuses on certain combinations of rules or conditions involving a would-be ‘provability’ or ‘truth’ predicate that would render a system of arithmetic containing them either straightforwardly inconsistent (if those predicates were assumed to be definable) or logico-semantically paradoxical (if those predicates were taken as primitive and governed by the rules in question). These two negative properties are not to be conflated; we conjecture, however, that they are complementary. Logico-semantic paradoxicality, we contend, admits of proof-theoretic analysis: the ‘disproofs’ involved do not reveal straightforward inconsistency. This is because, unlike the disproofs involved in establishing straightforward inconsistencies, these paradox-revealing ‘disproofs’ cannot be brought into normal form.The border between metamathematical proofs of certain (constructive) impossibility results and the non-normalizable (and always constructive) disproofs engendered by semantic paradoxicality is not fully understood. The respective strategies of reasoning on each side—genuine proofs of inconsistency versus whatever kind of ‘disproof’ uncovers semantic paradoxicality—seem somehow similar. They seem to involve the same ‘lines of reasoning’. But there is an important and principled difference between them.This difference will be emphasized throughout our discussion of certain arithmetical impossibility results, and closely related semantic paradoxes. The proof-theoretic criterion for paradoxicality is that in the case of paradoxes (as opposed to genuine inconsistencies) the apparent ‘disproofs’ that use the rules stipulated for the primitive predicates in question cannot be brought into normal form. In proof-theoretic terminology: their reduction sequences do not terminate. This means that cut fails for languages generating paradox. But cut holds for the language of arithmetic. It follows that the paradox-generating primitive predicates of a semantically closed language cannot be defined in arithmetical terms. For, if they could be, then they could be replaced by their definitions within the paradoxical disproofs, and the resulting disproofs would be normalizable.

This study seeks to reveal the proper source of the (correct) rules of natural deduction (and their associated rules of the sequent calculus). Perhaps surprisingly, this source consists of just the familiar truth tables (deriving from Frege). These tables can be construed inferentially. The primitive steps of value-computation correspond to primitive steps of ‘inference’. We shall call them, however, primitive steps (or rules) of evaluation. These can be steps of verification or of falsification. The rules of evaluation constitute the inductive clauses in a metalinguistic co-inductive definition of model-relative verifications and falsifications.We then show how the rules of evaluation can be ‘morphed’ into rules of natural deduction. Rules of verification thereby become introduction rules, and rules of falsification become elimination rules. The morphing produces model-invariant rules in the simplest way possible. It preserves, for natural deduction, the feature of relevance that is involved in truth-tabular computation. This makes for a system of natural deduction (and a directly corresponding sequent calculus) that is relevant.

Alongside the sequent calculus for Classical Core Logic $${\mathbb {C}}^+$$ we set forth some new sequent calculi that we call $${\mathbb {T}}$$, $${\mathbb {C}}^{++}$$, and $${\mathbb {K}}$$. $${\mathbb {T}}$$ encodes truth-tabular reasoning; $${\mathbb {C}}^{++}$$ classicizes Core Logic $${\mathbb {C}}$$ by having multiple succedents; and $${\mathbb {K}}$$ is a cut-free sequent calculus inspired by insights of Ketonen. Our aim is to establish that $${\mathbb {C}}^{++}$$ is weakly complete, and to do so in a way that makes no use of standard semantic notions _or_ the rule of Cut. All the concepts defined and applied in the course of this study will be proof-theoretic, turning on rules of inference as meaning-constitutive. It is intended to be a contribution to the program of proof-theoretic semantics, from the distinctive vantage point of the Core logician.

This chapter presents an ‘absolutist’ view about logic—Core Logic. Core Logic is relevant, in a sense heretofore not satisfactorily explicated. The so-called loss of unrestricted transitivity of deduction in Core Logic brings with it epistemic gain. Core Logic suffices for Intuitionistic Mathematics, Classical Mathematics, the hypothetico-deductive testing of scientific theories against empirical evidence, and the reasoning involved in the logical and semantic paradoxes. Core Logic is the minimal inviolable core of logic that is needed, and suffices, for rational belief revision. Core Logic is the logic of ‘conceptual constitution’: it suffices for neo-logicist derivation of number-theoretic axioms from deeper logical principles governing the operator #xΦ(x) (the number of Φs). Core Logic has the nice property that its natural deductions and sequent proofs are structurally isomorphic, and it is obtained by smoothly generalizing an inferential reading of the truth-tables. There is an automated proof-finder for (propositional) Core Logic, whose decision problem is PSPACE-complete (like that of Intuitionistic Logic). Any classical core proof provides a means of ‘quasi-’effectively transforming any verifications of its premises, relative to a model M, into a verification of its conclusion, relative to M.

Berry’s Paradox, like Russell’s Paradox, is a ‘paradox’ in name only. It differs from genuine logico-semantic paradoxes such as the Liar Paradox, Grelling’s Paradox, the Postcard Paradox, Yablo’s Paradox, the Knower Paradox, Prior’s Intensional Paradoxes, and their ilk. These latter arise from semantic closure. Their genuine paradoxicality manifests itself as the non-normalizability of the formal proofs or disproofs associated with them. The Russell, the Berry, and the Burali-Forti ‘paradoxes’, by contrast, simply reveal the straightforward inconsistency of their respective existential claims—that the Russell set exists; that the Berry number exists; and that the ordinal of the well-ordering of all ordinals exists. The disproofs of these existential claims are in free logic and are in normal form. They show that certain complex singular terms do not—indeed, cannot—denote. All this counsels reconsideration of Ramsey’s famous division of paradoxes and contradictions into his Group A and Group B. The proof-theoretic criterion of genuine paradoxicality formally explicates an informal and occasionally confused notion. The criterion should be allowed to reform our intuitions about what makes for genuine paradoxicality, as opposed to straightforward (albeit surprising) inconsistency.

We furnish a core-logical development of the Gödel numbering framework that allows metamathematicians to attain limitative results about arithmetical truth without incorporating a genuine truth predicate into the language in a way that would lead to semantic closure. We show how Tarski’s celebrated theorem on the arithmetical undefinability of arithmetical truth can be established using only core logic in both the object language and the metalanguage. We do so at a high level of abstraction, by augmenting the usual first-order language of arithmetic with a primitive predicate Tr and then showing how it cannot be a truth predicate for the augmented language. McGee established an important result about consistent theories that are in the language of arithmetic augmented by such a “truth predicate” Tr and that use Gödel numbering to refer to expressions of the augmented language. Given the nature of his sought result, he was forced to use classical reasoning at the meta level. He did so, however, on the additional and tacit presupposition that the arithmetical theories in question (in the object language) would be closed under classical logic. That left open the dialectical possibility that a constructivist (or intuitionist) could claim not to be discomfited by the results, even if they were to “give a pass” on the unavoidably classical reasoning at the meta level. In this study we “constructivize” McGee’s result, by presuming only core logic for the object language. This shows that the perplexity induced by McGee’s result will confront the constructivist (or intuitionist) as well.

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.