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)

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 systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program that could (...) be called Natural Logicism, an exposition of which will build on the (meta)logical ideas explained here. (shrink)

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 light of our recent work on truthmakers. We then examine the role of apriority in the realism/anti-realism debate. We raise a (...) hitherto unnoticed problem for any Orthodox Realist who attempts to explain the a priori. (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.

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)

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)

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)

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 Thinning and Cut and has Reflexivity as its only structural rule. Yet it suffices for classical mathematics, and it furnishes all the proofs and disproofs one needs for the (...) hypothetico-deductive method in science. We explain exactly what Classical Core Logic is, why it is a substructural logic par excellence, and what the basic requirements would be for a logic to be ‘suited to acting as [a] background logic for science’. We also explain how Classical Core Logic meets all these requirements. We end by examining Williamson’s argument in order to expose where its error lies. (shrink)

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–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is (...) a fragment of my system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. (shrink)

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. (shrink)

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.. (shrink)

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. 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 appeals to excluded middle when applying, to the problem of truthmaker-transmission, the already classical metalinguistic theory of model-relative evaluations.

Shows how to program on a computer (in Prolog) the effective skills taught in introductory and intermediate logic courses. The topics include the relevance of relevance, representing formulae and proofs, avoiding loops and blind alleys, and other aspects. Of interest to computational logicians, proof-theorists, cognitive scientists, and workers in artificial intelligence. Distributed by Columbia U. Press. Annotation copyright by Book News, Inc., Portland, OR.

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)

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)

This paper treats entailment as a subrelation of classical consequence and deducibility. Working with a Gentzen set-sequent system, we define an entailment as a substitution instance of a valid sequent all of whose premisses and conclusions are necessary for its classical validity. We also define a sequent Proof as one in which there are no applications of cut or dilution. The main result is that the entailments are exactly the Provable sequents. There are several important corollaries. Every unsatisfiable set is (...) Provably inconsistent. Every logical consequence of a satisfiable set is Provable therefrom. Thus our system is adequate for ordinary mathematical practice. Moreover, transitivity of Proof fails upon accumulation of Proofs only when the newly combined premisses are inconsistent anyway, or the conclusion is a logical truth. In either case Proofs that show this can be effectively determined from the Proofs given. Thus transitivity fails where it least matters — arguably, where it ought to fail! We show also that entailments hold by virtue of logical form insufficient either to render the premisses inconsistent or to render the conclusion logically true. The Lewis paradoxes are not Provable. Our system is distinct from Anderson and Belnap''s system of first degree entailments, and Johansson''s minimal logic. Although the Curry set paradox is still Provable within naive set theory, our system offers the prospect of a more sensitive paraconsistent reconstruction of mathematics. It may also find applications within the logic of knowledge and belief. (shrink)

This is a reply to the considerations advanced by Schroeder-Heister and Tranchini as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant and subsequently amended in Tennant. Countering these considerations lends new importance to the parallelized forms of elimination rules in natural deduction.

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. (shrink)

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? (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)

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)

1Certain developments in recent philosophy of mind that contemporary philosophers would regard as both novel and important were fully anticipated by writers in (or reacting to) the tradition of Nat...

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. (shrink)

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 generalizations—Variability, Heritability and Differential Reproduction—along with an innocuous assumption of Causal Efficacy, to derive Adaptive Evolution as a necessary consequence. Adaptive Evolution in turn, with one assumption of scale (‘Deep Time’), implies (...) the observational generalization of Adaptation. It is a fascinating methodological task to regiment the premises and make the reasoning both rigorous and clear. Doing so reveals how surprisingly small an amount of mathematics is needed in order to carry out the argument. The investigation also reveals the crucial role played by heritability, and how heritability itself admits of Darwinian explanation. (shrink)

Given the harmony principle for logical operators, compositionality ought to ensure that harmony should obtain at the level of whole contents. That is, the role of a content qua premise ought to be balanced exactly by its role as a conclusion. Frege's contextual definition of propositional content happens to exploit this balance, and one appeals to the Cut rule to show that the definition is adequate. We show here that Frege's definition remains adequate even when one relevantizes logic by abandoning (...) an unrestricted Cut rule. The proof exploits the fact that in the relevantized logic, which abandons the unrestricted rule of Cut, any failure of the transitivity of deduction is offset by the epistemic gain involved in learning that a stronger-than-expected result holds. (shrink)

This volume is a tribute by his peers, and by younger scholars of the next generation, to Harvey M. Friedman, perhaps the most profound foundationalist since Kurt Godel. Friedman's researches, beginning precociously in his mid-teens, have fundamentally shaped our contemporary understanding of set theory, recursion theory, model theory, proof theory and metamathematics. His achievements in concept formation and theory formulation have also renewed the standard set by Godel and Alfred Tarski for the general intellectual interest and importance of technical work (...) in foundations. Friedman pioneered the now well-established and flourishing field of Reverse Mathematics, whose aim is to calibrate the intrinsic logico-mathematical consistency-strength of all the important theorems of mathematics. He has relentlessly pursued the full extent of the incompleteness phenomena into which Godel provided the first revealing glimpse. The Godel--Friedman program, as it is now deservingly called, seeks to find simple, natural and elegant mathematical statements of a combinatorial nature, that can be proved to be independent of set theory even when extended by powerful large-cardinal existence axioms. (shrink)

We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, in (...) the case of set theory. These are defined expressions (such as ‘subset of’, or ‘power set of’) that are treated as basic in the lingua franca of informal set theory. Employing pasigraphs in accordance with their own natural-deduction rules enables one to ‘atomicize’ rigorous mathematical reasoning. (shrink)

This study takes a careful inferentialist look at Graham Priest’s Logic of Paradox. 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.

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? (shrink)

Written for any readers interested in better harnessing philosophy’s real value, this book covers a broad range of fundamental philosophical problems and certain intellectual techniques for addressing those problems. In Introducing Philosophy: God, Mind, World, and Logic , Neil Tennant helps any student in pursuit of a ‘big picture’ to think independently, question received dogma, and analyse problems incisively. It also connects philosophy to other areas of study at the university, enabling all students to employ the concepts and (...) techniques of this millennia-old discipline throughout their college careers – and beyond. KEY FEATURES AND BENEFITS: -- Investigates the philosophy of various subjects , helping students contextualize philosophy and view it as an interdisciplinary pursuit; also helps students with majors outside of philosophy to see the relationship between philosophy and their own focused academic pursuits -- Author comes from a distinguished background in Logic and Philosophy of Language, which gives the book a level of rigor, balance, and analytic focus sometimes missing from primers to philosophy -- Introduces students to various important philosophical distinctions providing skills that are important for undergraduates to develop in order to inform their study at higher levels. They are essential for further work in philosophy but they are also very beneficial for students pursuing most other disciplines -- Is much more methodologically comprehensive than competing introductions, giving the student the ability to address a wide range of philosophical problems – and not just the ones reviewed in the book -- Offers a companion website with links to apt primary sources, organized chapter-by-chapter, making unnecessary a separate Reader/Anthology of primary sources – thus providing students with all reading material necessary for the course -- Provides five to ten discussion questions for each chapter, helping instructors and students better interact with the ideas and concepts in the text. (shrink)

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. (shrink)

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 …. (shrink)

Storrs McCall appeals to a particular true but improvable sentence of formal arithmetic to argue, by appeal to its irrefutability, that human minds transcend Turing machines. Metamathematical oversights in McCall's discussion of the Godel phenomena, however, render invalid his philosophical argument for this transcendentalist conclusion.

Written for any readers interested in better harnessing philosophy's real value, this book covers a broad range of fundamental philosophical problems and certain intellectual techniques for addressing those problems. In Introducing Philosophy: God, Mind, World, and Logic, Neil Tennant helps any student in pursuit of a 'big picture' to think independently, question received dogma, and analyse problems incisively. It also connects philosophy to other areas of study at the university, enabling all students to employ the concepts and techniques (...) of this millennia-old discipline throughout their college careers - and beyond. KEY FEATURES AND BENEFITS: -- Investigates the philosophy of various subjects, helping students contextualize philosophy and view it as an interdisciplinary pursuit; also helps students with majors outside of philosophy to see the relationship between philosophy and their own focused academic pursuits -- Author comes from a distinguished background in Logic and Philosophy of Language, which gives the book a level of rigor, balance, and analytic focus sometimes missing from primers to philosophy -- Introduces students to various important philosophical distinctions providing skills that are important for undergraduates to develop in order to inform their study at higher levels. They are essential for further work in philosophy but they are also very beneficial for students pursuing most other disciplines -- Is much more methodologically comprehensive than competing introductions, giving the student the ability to address a wide range of philosophical problems - and not just the ones reviewed in the book -- Offers a companion website with links to apt primary sources, organized chapter-by-chapter, making unnecessary a separate Reader/Anthology of primary sources - thus providing students with all reading material necessary for the course -- Provides five to ten discussion questions for each chapter, helping instructors and students better interact with the ideas and concepts in the text. (shrink)

We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inferenceis, orconsists 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 essentiallyself-referring. That is, any rule$\rho $is to be understood via a specification that involves, embedded within it, reference to rule$\rho $itself. Just how we arrive at this position is explained by (...) reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them. (shrink)

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)

In his book Bayes or Bust?, John Earman (1992: 63–65) seeks to set out the Bayesian reasoning that would vindicate the pre-theoretic intuition that a theory receives confirmation from having its observational predictions borne out by experience.

Peter G¨ ardenfors 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 G¨.

Peacocke argues for a ‘generalized rationalism’, holding that ‘all entitlement has a fundamentally a priori component.’ (2) But his rationalism ‘differs from those of Frege and Gödel, just as theirs differ from that of Leibniz.’ He requires both substantive theories of intentional content and of understanding, and systematic formal theories of referential semantics and truth. We need an externalist theory of content: ‘Only mental states with externally individuated contents can make judgements about the external, mind-independent world rational.’ (123) Purely evidential (...) conceptions of meaning and content are inadequate. (34-49) They cannot account for the following: a thinker often has to work out what would be evidence for a content; contents cannot depend, for their identity, on all of the infinitely ramifying evidential connections among them; and thinkers conceive, however tacitly, of (at least some) observed properties as categorical. By contrast with an evidential theory, a truthconditional theory of content can account for all these problematic facts. Peacocke states, develops and defends three principles of rationalism which collectively ‘relate entitlement to truth, to the identity of states and their intentional contents, and to the a priori.’ (3-4) He does not thoroughly explain his central notion of entitlement, but this much is clear: any thinker is entitled to various transitions in, or into, thought. An example of a transition into thought would be that from one’s perceptual experience to an observational judgment. An example of a transition in thought would be a logical inference from certain premises to a conclusion. A transition is rational just in case the thinker is entitled to it. (Note that this aims to explain rationality in terms of entitlement, not the other way round.) It is clear from 28 that Peacocke needs an abstract ontology of entitlements (such as proofs, in the case of mathematics). Yet he does not endorse ‘Gödel’s obscure quasi-perceptual and quasi-causal epistemology of mathematics and the abstract sciences.’ (54.. (shrink)

Christopher Peacocke, in A Study of Concepts, motivates his account of possession conditions for concepts by means of an alleged parallel with the conditions under which numbers are abshacted to give the numerosity of a predicate. There are, however, logical mistakes in Peacocke's treatment of numbers, which undermine his intended analogy. Nevertheless Peacocke's account of possession conditions for concepts is not rendered inadequate simply by virtue of being deprived of the intended analogy and the motivation it was supposed to afford. (...) His account of concepts deserves still to be considered on its own merits, even if it is more idiosyncratic for being isolated from the paradigm case of numerical abstraction. Peacocke's own account of concepts as abstract objects turns out, though, not to have the logical form that he himself was seeking for it. We show how to re-cast it in an equivalent form of the kind he requires. Then we re-formulate it so as to achieve complete generality. This exercise helps to clarify the central theses in Peacocke's account of concepts. It invites the conclusion that his account of content-determination is rather platitudinous and unoriginal — except for a claim of 'doxastic sufficiency' for content-determination, which emerges at the end of the discussion. (shrink)

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 is an application of a primitive rule; and the natural deductions themselves can be checked effectively for formal correctness.

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 is an application of a primitive rule; and the natural deductions themselves can be checked effectively for formal correctness.

Christopher Peacocke, in A Study of Concepts, motivates his account of possession conditions for concepts by means of an alleged parallel with the conditions under which numbers are abstracted to give the numerosity of a predicate. There are, however, logical mistakes in Peacocke.

This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which ‘respects (...) certain important intuitions’ 1 : apriorism, realism, and empiricism. The book contains some clarification of these ‘isms’, and some thoughtful critiques of major positions regarding them, as espoused by such representative figures as Boghossian, Bealer, Peacocke, Field, Bostock, Maddy, Locke, Kant, C.I. Lewis, Ayer, Quine, Fodor, and McDowell. The philosophical reader will find some interest and value in these wider-ranging discussions. Our concern in this review, however, is to examine closely the original positive proposal on offer.Arithmetical truths, the author maintains, are conceptual truths. Knowing truths like 7+5=12 involves no ‘epistemic reliance on any empirical evidence’; but that, she says, is not to claim ‘epistemic independence of the senses altogether’. She wants to show that "experience grounds our concepts … and then mere conceptual examination enables us to learn arithmetical truths ." Concepts that are ‘appropriately sensitive’ to ‘the nature of [an independent] reality’ she calls grounded. Because of the role of grounded concepts, ‘arithmetical truths explain our arithmetical beliefs in the right sort of way for those beliefs to count as knowledge’ .In the context of her concentration on the special nature of arithmetical knowledge, the author offers what could strike some bystanders as an unnecessarily over-ambitious account of knowledge tout court. Knowledge, for the author, is "true belief which … ". (shrink)

Neil Tennant (Tennant, 2005) has offered an important observation about the AGM theory of belief revision (G¨ardenfors, 1988). We attempt to restate and demonstrate his result in a slightly different way. Fix a formal language L that embeds sentential logic. Given K ⊆ L and ϕ ∈ L, K ⊥ ϕ denotes the class of maximally consistent subsets of K that do not imply ϕ. That is, A ∈ K ⊥ ϕ iff A ⊆ K, A |= (...) ϕ, and there is no B ⊆ K such that B ⊃ A and.. (shrink)

The Taming of the True poses a broad challenge to realist views of meaning and truth that have been prominent in recent philosophy. Neil Tennant argues compellingly that every truth is knowable, and that an effective logical system can be based on this principle. He lays the foundations for global semantic anti-realism and extends its consequences from the philosophy of mathematics and logic to the theory of meaning, metaphysics, and epistemology.

An account of how a rational agent should revise beliefs in the light of new evidence.