Neil Tennant

The Ohio State University | OSU · Department of Philosophy

This book defines and develops the program of Natural Logicism for the natural, rational, and real numbers. The central method is to formulate rules of natural deduction governing variable-binding number-abstraction operators and other logico-mathematical expressions such as zero and successor. The introduction and elimination rules for a number-ab...

Williamson argues for the contention that substructural logics are ‘ill-suited to acting as background logics for science’. That contention, if true, would be very important, but it is refutable, given what is already known about certain substructural logics. Classical Core Logic is a substructural logic, for it eschews the structural rules of Thin...

This is a reply to the considerations advanced by Schroeder-Heister and Tranchini (Ekman’s paradox, Unpublished typescript) as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant (Dialectica 36:265–296, 1982) and subsequently amended in Tennant (Analysis 55:199–207, 1995). Countering these...

We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inference is , or consists in . We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentially self-referring . That is, any...

This paper clarifies, revises, and extends the account of the transmission of truthmakers by core proofs that was set out in chap. 9 of Tennant (2017). Brauer provided two kinds of example making clear the need for this. Unlike Brouwer’s counterexamples to excluded middle, the examples of Brauer that we are dealing with here establish the need for...

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 inferentialis...

The one-page 1978 informal proof of Goodman and Myhill is regimented in a weak constructive set theory in free logic. The decidability of identities in general ($a\!=\!b\vee\neg a\!=\!b$) is derived; then, of sentences in general ($\psi\vee\neg\psi$). Martin-Löf’s and Bell’s receptions of the latter result are discussed. Regimentation reveals the f...

Our regimentation of Goodman and Myhill’s proof of Excluded Middle revealed among its premises a form of Choice and an instance of Separation. Here we revisit Zermelo’s requirement that the separating property be definite. The instance that Goodman and Myhill used is not constructively warranted. It is that principle, and not Choice alone, that pre...

This study takes a careful inferentialist look at Graham Priest’s Logic of Paradox (LP). I conclude that it is sorely in need of a proof-system that could furnish formal proofs that would regiment faithfully the “naïve logical” reasoning that could be undertaken by a rational thinker within LP (if indeed such reasoning could ever take place).

This is in part a reply to a recent work of Vidal-Rosset, which expresses various mistaken beliefs about Core Logic. Rebutting these leads us further to identify, and argue against, some mistaken core beliefs about logic.

It is shown how Tarski’s 1929 axiomatization of mereology secures the reflexivity of the ‘part of’ relation. This is done with a fusion-abstraction principle that is constructively weaker than that of Tarski; and by means of constructive and relevant reasoning throughout. We place a premium on complete formal rigor of proof. Every step of reasoning...

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...

We explore the consequences, for logical system-building, of taking seriously (i) the aim of having irredundant rules of inference, and (ii) a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of Reflexivity, Thinning, and Cut. Reflexivity survives in the minimally necessary form...

In his very first publication, Gentzen introduced the structural rules of thinning and cut on sequents. He did not consider rules for logical operators. Gentzen provided a most interesting ‘structural completeness proof’, which it is the concern of this study to explain and clarify. We provide an improved (because more detailed) proof of Gentzen’s...

In an earlier paper in this journal I provided detailed motivation for the constructive and relevant system of Core Logic ; explained its main features; and established that, even though Cut is not a rule of the system, nevertheless Cut is admissible for it, and indeed with potential epistemic gain. If Π is a proof of the sequent Δ : A , and Σ is a...

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...

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 logic R. A precise, positive explication is offered of what it is for the premises of a proof to connect re...

This is Part I of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. Here we explain what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguish grounds for knowledge from grounds of truth, in...

We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical s...

GrandyRichard E.. Advanced logic for applications. Synthese library, vol. 110. D. Reidel Publishing Company, Dordrecht and Boston1977, xiii + 168 pp. - Volume 47 Issue 3 - Neil Tennant

We present a logically detailed case-study of Darwinian evolutionary explanation. Special features of Darwin's explanatory schema made it an unusual theoretical breakthrough, from the point of view of the philosophy of science. The schema employs no theoretical terms, and puts forward no theoretical hypotheses. Instead, it uses three observational...

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Praw...

This study is in two parts. In the first part, various important principles of classical extensional mereology are derived on the basis of a nice axiomatization involving ‘part of’ and fusion. All results are proved here with full Fregean (and Gentzenian) rigor. They are chosen because they are needed for the second part. In the second part, this n...

The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain.

This account of rational belief revision explains how a rational agent ought to proceed when adopting a new belief - a difficult matter if the new belief contradicts the agent's old beliefs. Belief systems are modeled as finite dependency networks. So one can attend not only to what the agent believes, but also to the variety of reasons the agent h...

The aim here is to set out a philosophy, foundations and methodol-ogy for mathematics that can best be described as 'natural logicist'. It is what logicists might have arrived at, had they enjoyed the benefits of Gentzen's methods of natural deduction. Mathematical content, we argue, is best captured by rules set out in a natural deduction format,...

What would a satisfying neo-Kantian and neo-Quinean account of logic be like? How should we account for its role as an instrument for acquiring knowledge, and as an instrument for the criticism of theories? How should we account for its special status in the epistemic scheme of things? I intend below to re-apply the distinctions between analytic an...

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 Cieśliński draws attention, that non-conservativeness attends e...

We present a logically detailed case-study of explanation and prediction in Newtonian mechanics. The case in question is that of a planet’s elliptical orbit in the Sun’s gravitational field. Care is taken to distinguish the respective contributions of the mathematics that is being applied, and of the empirical hypotheses that receive a mathematical...

In response to a problem pointed out by Steinberger (2009), the Harmony Principle of Tennant forthcoming (b) is here given a sequent formulation in order better to illustrate its application to the existential quantifier. 1. The Harmony of the rules for the standard existential quantifier The usual definition of the notion 'ϕ is a (logically) stron...

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 book...

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 e...

I criticize dialetheism from the point of view of an anti-realist with sympathy for relevantism in logical reasoning. I argue 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 th...

This chapter argues that thinkers have cognitive homes with the following minimal chattels: when they are in a state of understanding, with respect to any sentence, that it is meaningful for them (as a representation of how things are), then they indeed know (or at least are in a position to know) that they are in that state. That is to say, the co...

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 '', therefore 3K'' should be formulated as a consistency requirement not on the premise ' of an application of the rule, but rather on the set of assumptions on which...

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...

This article advances an unabashedly partisan view of how best to "relevantize" a logic. The view is laid out as informally as possible, given the technical nature of the subject matter. Here, "relevantizing" is understood as the project of formulating a decent system of logic that does not endorse Lewis's First Paradox: A, {reversed not sign}A:B....

Peter Gärdenfors proved a theorem purporting to show that it is impossible to adjoin to the AGM-postulates for belief-revision a principle of monotonicity for revisions. The principle of monotonicity in question is implied by the Ramsey test for conditionals. So Gärdenfors’ result has been interpreted as demonstrating that it is impossible to combi...

This chapter examines the relationship between logic, mathematics, and the natural sciences. A constructivist version of a mathematical theory is adequate for all the applications to be made of the theory within natural science. In the overall context of the hypothetico-deductive method in natural science, constructivist logical reasoning is adequa...

Peter Milne (2007) poses two challenges to the inferential theorist of meaning. This study responds to both. First, it argues that the method of natural deduction idealizes the essential details of correct informal deductive reasoning. Secondly, it explains how rules of inference in free logic can determine unique senses for the existential quantif...

Michael Friedman maintains that Carnap did not fully appreciate the impact of Gödel's first incompleteness theorem on the prospect for a purely syntactic definition of analyticity that would render arithmetic analytically true. This paper argues against this claim. It also challenges a common presumption on the part of defenders of Carnap, in their...

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 inferentialis...

I defend a conventionalist view of logical and (some) mathematical truths against the criticisms of Quine and Stroud. Conventionalism is best formulated by appealing to sense-conferring rules governing important logical and mathematical expressions. Conventional necessity can be understood as arising from these rules in a way that is immune to Quin...

Why is semantics important? More precisely, why is formal semantics important? What is so special about the subject matter of formal semantics, and its methods, that sets it apart as a necessary component in any account of language? By formal semantics I understand those broadly algebraic endeavours that set out to model the relationship between la...

AGM-theory, named after its founders Carlos Alchourrn, Peter Grdenfors and David Makinson, is the leading contemporary paradigm in the theory of belief-revision. The theory is reformulated here so as to deal with the central relational notions ‘J is a contraction of K with respect to A’ and ‘J is a revision of K with respect to A’. The new theory i...

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.

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...

I reformulate the AGM-account of contraction (which would yield an account also of revision). The reformulation involves using introduction and elimination rules for relational notions. Then I investigate the extent to which the two main methods of partial meet contraction and safe contraction can be employed for theories closed under intuitionisti...

This chapter explains an approach to relevantization of logical reasoning that seeks to maximize epistemic gain. It does so by retaining Disjunctive Syllogism and making admissible only a restricted rule of Cut. The virtue of this approach is that one can show that the resulting relevant logic is adequate for mathematics and science. In the course...

I present a general theory of abstraction operators which treats them as variable-binding term-forming operators, and provides a reasonably uniform treatment for definite descriptions, set abstracts, natural number abstraction, and real number abstraction. This minimizing, extensional and relational theory reveals a striking similarity between defi...