Michael Dummett has interpreted and expounded upon intuitionism under the influence of Wittgensteinian views on language, meaning and cognition. I argue against the application of some of these views to intuitionism and point to shortcomings in Dummett's approach. The alternative I propose makes use of recent, post-Wittgensteinian views in the philosophy of mind, meaning and language. These views are associated with the claim that human cognition exhibits intentionality and with related ideas in philosophical psychology. Intuitionism holds that mathematical constructions are (...) mental processes or objects. Constructions are, in the first instance, forms of consciousness or possible experience of a particular type. As such, they must be understood in terms of the concept of intentionality. This view has a historical basis in the literature on intuitionism. In a famous 1931 lecture Heyting in fact identifies constructions with fulfilled or fulfillable mathematical intentions. I consider some of the consequences of this identification and contrast them with Dummett's views on intuitionism. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

SummaryMany contemporary philosophers of language believe that realist metaphysics and a beha‐viouristic approach to language are incompatible, debate centring on which is to be given up. In this paper I argue that no incompatibility has been shown to exist. In the first section I attempt to give both a characterization of, and an argument for, behaviourism. Then I attempt to characterize realism more generally than is often done, evaluating the work of Dummett, Quine, Putnam and Wittgenstein, as recently interpreted, in (...) that light. Finally I attempt to sketch a behaviouristic justification of classical predicate logic and offer some programmatic remarks on behaviouristic accounts of mathematics and natural science.RésuméBien des philosophes contemporains du langage, estimant qu'une métaphysique réaliste et une approche behavioriste sont incompatibles, centrent le débat sur la question «lequel des deux doit‐il être sacrifité?» Je montre dans ce papier qu'on n'a jamais pu prouver l'existence d'une incompatibilité. Dans la première partie, je tente de caractériser et de défendre le behaviorisme. Puis j'essaie de caractériser le réalisme plus généralement que cela n'a été souvent fait et évalue dans cette perspective les oeuvres de Dummett, de Quine, de Putnam et de Wittgenstein à la lumière de leurs interprétations récentes. Finalement, je propose une justification behavioriste de la classique logique des prédicats et quelques remarques programmatiques quant à une approche behavioriste des mathématiques et des sciences naturelles.ZusammenfassungManche zeitgenössische Philosophen glauben, dass eine realistische Metaphysik nicht mit einer behavioristischen Betrachtungsweise der Sprache vereinbar sei, wobei sich der Streit um die Frage dreht, welche der beiden Haltungen aufgegeben werden müsse. Ich argurnentiere, dass die Unvereinbarkeit nicht wirklich nachgewiesen wurde. Im ersten Abschnitt versuche ich, den Beha‐viorismus genauer zu charakterisieren und zu seinen Gunsten zu argumentieren. Ich versuche dann, eine allgemeinere Charakterisierung des Realismus zu geben, als das üblich ist, indem ich die Werke von Dummett, Quine, Putnam und Wittgenstein, wie sie neuerdings unter diesen Gesichts‐punkt interpretiert werden, beurteile. lch versuche schliesslich, eine behavioristische Rechtferti‐gung der klassischen Prädikatenlogik zu skizzieren, und lege auch einige programmatische Bemerkungen zur behavioristischen Behandlung von Mathematik und Naturwissenschaft vor. (shrink)

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.

This paper argues that logical inferentialists should reject multiple-conclusion logics. Logical inferentialism is the position that the meanings of the logical constants are determined by the rules of inference they obey. As such, logical inferentialism requires a proof-theoretic framework within which to operate. However, in order to fulfil its semantic duties, a deductive system has to be suitably connected to our inferential practices. I argue that, contrary to an established tradition, multiple-conclusion systems are ill-suited for this purpose because they fail (...) to provide a 'natural' representation of our ordinary modes of inference. Moreover, the two most plausible attempts at bringing multiple conclusions into line with our ordinary forms of reasoning, the disjunctive reading and the bilateralist denial interpretation, are unacceptable by inferentialist standards. (shrink)

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)

In this paper I argue that there can be genuine (as opposed to merely verbal) disputes about whether a sentence form is logically true or an argument form is valid. I call such disputes ?cases of deviance?, of which I distinguish a weak and a strong form. Weak deviance holds if one disputant is right and the other wrong, but the available evidence is insufficient to determine which is which. Strong deviance holds if there is no fact of the matter. (...) In section 2 I argue that weak deviance need not be trivial and may even be interesting. Section 3 considers what it could mean to say that logic is determined by a theory, especially a theory of meaning, an idea that arises in section 2. In section 4 I discuss the dispute between classical and relevance logicians over entailment and argue that it is a case of strong deviance. Finally, in section 5 I show that the result of the previous section is not absolute but relative to the background logic used in reaching it. (shrink)

I. An argument is presented for the conclusion that the hypothesis that no one will ever decide a given proposition is intuitionistically inconsistent. II. A distinction between sentences and statements blocks a similar argument for the stronger conclusion that the hypothesis that I have not yet decided a given proposition is intuitionistically inconsistent, but does not block the original argument. III. A distinction between empirical and mathematical negation might block the original argument, and empirical negation might be modelled on Nelson''s (...) strong negation, but only on intuitionistically unacceptable assumptions. IV. Intuitionists may have to accept the original argument, and therefore be committed to a dubious view of time on which there cannot be merely inductive evidence for statements about the infinite future. (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)

Suppose we want to take seriously the neoverificationist idea that an intuitionistic theory of meaning can be generalized in such a way as to be applicable not only to mathematical but also to empirical sentences. The paper explores some consequences of this attitude and takes some steps towards the realization of this program. The general idea is to develop a meaning theory, and consequently a formal semantics, based on the idea that knowing the meaning of a sentence is tantamount to (...) having a criterion for establishing what is a justification for it. Section 1 motivates a requirement of epistemic transparency imposed onto justifications conceived as mental states. In Section 2, the formal notion of justification for an atomic formula is defined, in terms of the notion of cognitive state. In Section 3, the definition is extended to logically complex formulas. In Section 4, the notion of truth-ground is introduced and is used to give a definition of logical validity. (shrink)

Proof-theoretic semantics is an approach to logical semantics based on two ideas, of which the first is that the meaning of a logical connective can be explained by stipulating that some mode of inference, e.g., a natural deduction introduction or elimination rule, is permissible. The second idea is that the soundness of rules which are not stipulated outright may be deduced by some proof-theoretic argument from properties of the rules which are stipulated outright. I examine the first idea. My main (...) conclusion is that the idea is more problematic, and requires more discussion, than has been generally realized. I mention five problems which will have to be overcome before the idea can be accepted as definitely viable. (shrink)

The requirement of proof-theoretic harmony has played a pivotal role in a number of debates in the philosophy of logic. Different authors have attempted to precisify the notion in different ways. Among these, three proposals have been prominent in the literature: harmony–as–conservative extension, harmony–as–leveling procedure, and Tennant’s harmony–as–deductive equilibrium. In this paper I propose to clarify the logical relationships between these accounts. In particular, I demonstrate that what I call the equivalence conjecture —that these three notions essentially come to the (...) same thing—is erroneous. (shrink)

Without violating the spirit of Game-Theoretical semantics, its results can be re-worked in Martin-Löf''s Constructive Type Theory by interpreting games as types of Myself''s winning strategies. The philosophical ideas behind Game-Theoretical Semantics in fact highly recommend restricting strategies to effective ones, which is the only controversial step in our interpretation. What is gained, then, is a direct connection between linguistic semantics and computer programming.

The intention here is that of giving a formal underpinning to the idea of 'meaning-is-use' which, even if based on proofs, it is rather different from proof-theoretic semantics as in the Dummett-Prawitz tradition. Instead, it is based on the idea that the meaning of logical constants are given by the explanation of immediate consequences, which in formalistic terms means the effect of elimination rules on the result of introduction rules, i. e. the so-called reduction rules. For that we suggest an (...) extension to the Curry-Howard interpretation which draws on the idea of labelled deduction, and brings back Frege's device of variable-abstraction to operate on the labels (i. e., proof-terms) alongside formulas of predicate logic. (shrink)

The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a ‘theory of meaning’.