Neil Tennant

We define a system IR of first-order intuitionistic relevant logic. We show that intuitionistic mathematics (on the assumption that it is consistent) can be relevantized, by virtue of the following metatheorem: any intuitionistic proof of A from a setX of premisses can be converted into a proof in IR of eitherA or absurdity from some subset ofX. Thus IR establishes the same inconsistencies and theorems as intuitionistic logic, and allows one to prove every intuitionistic consequence of any consistent set of premisses.

SummaryI defend a conventionalist view of logical and 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 Quine's and Stroud's criticisms of the earlier formulation of conventionalism, in which stress was incorrectly laid on axiomatic systems of logic.RésuméJe soutiens, en dépit des critiques de Quine et de Stroud, une conception conventionnaliste des vérités logiques et de vérités mathématiques. La meilleure façon de formuler le conventionnalisme est de faire appel à des règies conférant un sens à?on;importantes expressions logiques et mathématiques. La nécessité conventionnelle peut être comprise comme provenant de ces règies par des voies qui ne donnent pas prise aux critiques de Quine et de Stroud ag ľégard de ľancienne formulation du conventionnalisme, dans laquelle ľaccent était mis, a tort, sur les systèmes axiomatiques de la logique.ZusammenfassungGegen Quines und Strouds Kritiken verteidige ich eine konventionalistische Auffassung der logischen und mathematischer Wahrheiten. Der Konventionalismus wird am besten durch Berufung auf bedeutungsstiftende Regeln, die den Gebrauch wichtiger logischer und mathematischer Ausdrücke bestimmen, formuliert. Konventionelle Notwendigkeit kann als von diesen Regeln hervorgebracht verstanden werden – und das zwar auf eine Art, die sie vor Quines und Strouds Anfechtungen schützen. In der von diesen kritisierten, früheren Fassung wurde zu unrecht das Gewicht auf axiomatische Systeme der Logik gelegt

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

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

AGM-theory, named after its founders Carlos Alchourrón, Peter Gärdenfors 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 is based on a principal-case analysis of the domains of definition of the three main kinds of theory-change (expansion, contraction and revision). The new theory is stated by means of introduction and elimination rules for the relational notions. In this new setting one can re-examine the relationship between contraction and revision, using the appropriate versions of the so-called Levi and Harper identities. Among the positive results are the following. One can derive the extensionality of contraction and revision, rather than merely postulating it. Moreover, one can demonstrate the existence of revision-functions satisfying a principle of monotonicity. The full set of AGM-postulates for revision-functions allow for completely bizarre revisions. This motivates a Principle of Minimal Bloating, which needs to be stated as a separate postulate for revision. Moreover, contractions obtained in the usual way from the bizarre revisions, by using the Harper identity, satisfy Recovery. This provides a new reason (in addition to several others already adduced in the literature) for thinking that the contraction postulate of Recovery fails to capture the Principle of Minimal Mutilation. So the search is still on for a proper explication of the notion of minimal mutilation, to do service in both the theory of contraction and the theory of revision. The new relational formulation of AGM-theory, based on principal-case analysis, shares with the original, functional form of AGM-theory the idealizing assumption that the belief-sets of rational agents are to be modelled as consistent, logically closed sets of sentences. The upshot of the results presented here is that the new relational theory does a better job of making important matters clear than does the original functional theory. A new setting has been provided within which one can profitably address two pressing questions for A GM-theory: (1) how is the notion of minimal mutilation (by both contractions and revisions) best analyzed? and (2) how is one to rule out unnecessary bloating by revisions?

Kant writes: If … the only aim of Nature regarding some creature possessed of reason and a will were its preservation, its well-being, in a word its happiness, then she would have come to a very bad arrangement in choosing its reason as executor of that aim. For all actions that it had to execute in this her intention, and the whole regulation of its behaviour would have been able to be prescribed to it much more precisely by instinct, and that aim thereby much more certainly maintained, than ever could happen through reason …

Brian Skyrms has studied the evolutionary dynamics of a simple bargaining game. Fair-dealing is the strategy 'demand 1/2', competing with the more modest strategy 'demand 1/3' and the greedier strategy 'demand 2/3'. Individuals leave offspring in proportion to their accumulated payoffs. The rules for payoffs from encounters penalize low- and high-demanders. The result is a significant basin of attraction for fair-dealing as an evolutionarily stable strategy. From these considerations Skyrms concludes that a disposition to fair-dealing could have evolved. A very different picture emerges, however, when one considers genetic bases for the dispositions involved. A simple two-allele sexual model produces very different stable equilibria in the distribution of behavioral phenotypes. The equilibria for Skyrms's purely phenotypic selection process will not in general be attainable once one enters some simple genetic considerations

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

The Theory of theory change has contraction and revision as its central notions. Of these, contraction is the more fundamental. The best-known theory, due to Alchourrón, Gärdenfors, and Makinson, is based on a few central postulates. The most fundamental of these is the principle of recovery: if one contracts a theory with respect to a sentence, and then adds that sentence back again, one recovers the whole theory. Recovery is demonstrably false. This paper shows why, and investigates how one can nevertheless characterize contraction in a theoretically fruitful way. The theory proposed lends itself to implementation, which in turn could yield new theoretical insights. The Main proposal is a ‘staining algorithm’ which identifies which sentences to reject when contracting a theory. The algorithm requires one to be clear about the structure of reasons one has for including sentences within one's theory.

This paper addresses an objection raised by Timothy Williamson to the ‘restriction strategy’ that I proposed, in The Taming of The True, in order to deal with the Fitch paradox. Williamson provides a new version of a Fitch-style argument that purports to show that even the restricted principle of knowability suffers the same fate as the unrestricted one. I show here that the new argument is fallacious. The source of the fallacy is a misunderstanding of the condition used in stating the restricted knowability principle. I also rebut Williamson’s criticism of my argument for the claim that any proposition of the form ‘it is known that ϕ’ is decidable if ϕ is decidable.

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.

We examine the arguments on both sides of the recent debate (Hale and Wright v. Field) on the existence, and modal status, of the natural numbers. We formulate precisely, with proper attention to denotational commitments, the analytic conditionals that link talk of numbers with talk of numerosity and with counting. These provide conceptual controls on the concept of number. We argue, against Field, that there is a serious disanalogy between the existence of God and the existence of numbers. We give stronger reasons than those advanced by Wright for resisting Field's analogy. We argue that the rules governing the basic numerical notions commit us to the natural numbers as necessary existents. We also show that the latest twist in the debate involving 'surdons' leaves both sides in a stalemate

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 diagnosis of the force of Gödel's own critique of Carnap in his Gibbs Lecture. The author is grateful to Michael Friedman for valuable comments. Part of the research towards this paper was carried out while the author was a Visiting Fellow at the Center for Philosophy of Science at the University of Pittsburgh. The paper was presented to the Center's Fellowship Reunion Conference in Athens in 1992. It was committed for publication in the Proceedings of that conference, but those Proceedings never appeared. By the time it became evident that they would never appear, both the hard copy and the source file had been mislaid. The hard copy re-surfaced in 2007. The literature on this topic since 1992 appears to leave some space for the ideas and arguments presented here. Although the paper has been updated in light of the more recent literature, its basic thesis, presented in 1992, remains the same. Only 3 is new, questioning a basic presumption made by more recent commentators in their presentation of Gödel's criticism of Carnap in his Gibbs Lecture. For helpful comments on the current version, the author is indebted to Robert Kraut, Stewart Shapiro, and Adam Podlaskowski. CiteULike    Connotea    Del.icio.us    What's this?