This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...) equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

This paper aims to contribute to the analysis of the nature of mathematical modality and hyperintensionality, and to the applications of the latter to absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority (...) and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance a topic-sensitive epistemic two-dimensional truthmaker semantics, if hyperintensional approaches are to be preferred to possible worlds semantics. I examine the relation between two-dimensional hyperintensional states and epistemic set theory, providing two-dimensional hyperintensional formalizations of the modal logic of ZFC, large cardinal axioms, $\Omega$-logic, and the Epistemic Church-Turing Thesis. (shrink)

It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen argues that this view—call it logical inferentialism—is undermined by some “very little known” considerations by Carnap (1943) to the effect that “in a definite sense, it is not true that the standard rules of inference” themselves suffice (...) to “determine the meanings of [the] logical constants” (p. 2). In a nutshell, Carnap showed that the rules allow for non-normal interpretations of negation and disjunction. Raatikainen concludes that “no ordinary formalization of logic [... ] is sufficient to ‘fully formalize’ all the essential properties of the logical constants” (ibid.). We suggest that this is a mistake. Pace Raatikainen, intuitionists like Dummett and Prawitz need not worry about Carnap’s problem. And although bilateral solutions for classical inferentialists—as proposed by Timothy Smiley and Ian Rumfitt—seem inadequate, it is not excluded that classical inferentialists may be in a position to address the problem too. (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)

If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical (...) constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch''s paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett''s meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost. (shrink)

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 form of Choice used in deriving Excluded Middle. It also reveals an abstraction principle that the proof employs. It will be argued that the Goodman–Myhill result does not provide the constructive set (...) theorist with a dispositive reason for not adopting (full) Choice. (shrink)

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to languages containing (...) the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results. (shrink)

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

The historical consensus is that logical evidence is special. Whereas empirical evidence is used to support theories within both the natural and social sciences, logic answers solely to a priori evidence. Further, unlike other areas of research that rely upon a priori evidence, such as mathematics, logical evidence is basic. While we can assume the validity of certain inferences in order to establish truths within mathematics and test scientifi c theories, logicians cannot use results from mathematics or the empirical sciences (...) without seemingly begging the question. Appeals to rational intuition and analyticity in order to account for logical knowledge are symptomatic of these commitments to the apriority and basicness of logical evidence. This chapter argues that these historically prevalent accounts of logical evidence are mistaken, and that if we take logical practice seriously we fi nd that logical evidence is rather unexceptional, sharing many similarities to the types of evidence appealed to within other research areas. (shrink)

This dissertation looks at a set of interconnected questions concerning the foundations of probability, and gives a series of interconnected answers. At its core is a piece of old-fashioned philosophical analysis, working out what probability is. Or equivalently, investigating the semantic question of what is the meaning of ‘probability’? Like Keynes and Carnap, I say that probability is degree of reasonable belief. This immediately raises an epistemological question, which degrees count as reasonable? To solve that in its full generality would (...) mean the end of human inquiry, so that won’t be attempted here. Rather I will follow tradition and merely investigate which sets of partial beliefs are coherent. -/- The standard answer to this question, what is commonly called the Bayesian answer, says that degrees of belief are coherent iff they form a probability function. I disagree with the way this is usually justified, but subject to an important qualification I accept the answer. The important qualification is that degrees of belief may be imprecise, or vague. Part one of the dissertation, chapters 1 to 6, looks largely at the consequences of this qualification for the semantic and epistemological questions already mentioned. It turns out that when we allow degrees of belief to be imprecise, we can discharge potentially fatal objections to some philosophically attractive theses. Two of these, that probability is degree of reasonable belief and that the probability calculus provides coherence constraints on partial beliefs, have been mentioned. Others include the claim, defended in chapter 4, that chance is probability given total history. -/- As well as these semantic and epistemological questions, studies of the foundations of probability usually include a detailed discussion of decision theory. For reasons set out in chapter 2, I deny we can gain epistemological insights from decision theory. Nevertheless, it is an interesting field to study on its own, and it might be expected that there would be decision theoretic consequences of allowing imprecise degrees of belief. As I show in part two, this expectation seems to be mistaken. Chapter 9 shows that there aren’t interesting consequences of this theory for decision theory proper, and chapters 10 and 11 show that Keynes’s attempt to use imprecision in degrees of belief to derive a distinctive theory of interest rates is unsound. -/- Chapters 7 and 8 provide a link between these two parts. In chapter 7 I look at some previous philosophical investigations into the effects of imprecision. In chapter 8 I develop what I take to be the best competitor to the theory defended here – a constructivist theory of probability. On this view degrees of belief are precise, but the relevant coherence constraint is a constructivist probability calculus. This view is, I think, mistaken, but the calculus has some intrinsic interest, and there are at least enough arguments for it to warrant a chapter-length examination. (shrink)

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

This volume examines the concept of falsification as a central notion of semantic theories and its effects on logical laws. The point of departure is the general constructivist line of argument that Michael Dummett has offered over the last decades. From there, the author examines the ways in which falsifications can enter into a constructivist semantics, displays the full spectrum of options, and discusses the logical systems most suitable to each one of them. While the idea of introducing falsifications into (...) the semantic account is Dummett's own, the many ways in which falsificationism departs quite radically from verificationism are here spelled out in detail for the first time. The volume is divided into three large parts. The first part provides important background information about Dummett’s program, intuitionism and logics with gaps and gluts. The second part is devoted to the introduction of falsifications into the constructive account and shows that there is more than one way in which one can do this. The third part details the logical effects of these various moves. In the end, the book shows that the constructive path may branch in different directions: towards intuitionistic logic, dual intuitionistic logic and several variations of Nelson logics. The author argues that, on balance, the latter are the more promising routes to take. "Kapsner’s book is the first detailed investigation of how to incorporate the notion of falsification into formal logic. This is a fascinating logico-philosophical investigation, which will interest non-classical logicians of all stripes." Graham Priest, Graduate Center, City University of New York and University of Melbourne. (shrink)

"Necessity is the mother of invention. " Part I: What is in this book - details. There are several different types of formal proof procedures that logicians have invented. The ones we consider are: 1) tableau systems, 2) Gentzen sequent calculi, 3) natural deduction systems, and 4) axiom systems. We present proof procedures of each of these types for the most common normal modal logics: S5, S4, B, T, D, K, K4, D4, KB, DB, and also G, the logic that (...) has become important in applications of modal logic to the proof theory of Peano arithmetic. Further, we present a similar variety of proof procedures for an even larger number of regular, non-normal modal logics. We also consider some quasi-regular logics, including S2 and S3. Virtually all of these proof procedures are studied in both propositional and first-order versions. Finally, we present the full variety of proof methods for Intuitionistic logic. We actually give two quite different kinds of tableau systems for the logics we consider, two kinds of Gentzen sequent calculi, and two kinds of natural deduction systems. Each of the two tableau systems has its own uses; each provides us with different information about the logics involved. They complement each other more than they overlap. Of the two Gentzen systems, one is of the conventional sort, common in the literature. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...) developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic. (shrink)

Este livro marca o início da Série Investigação Filosófica. Uma série de livros de traduções de textos de plataformas internacionalmente reconhecidas, que possa servir tanto como material didático para os professores das diferentes subáreas e níveis da Filosofia quanto como material de estudo para o desenvolvimento pesquisas relevantes na área. Nós, professores, sabemos o quão difícil é encontrar bons materiais em português para indicarmos. E há uma certa deficiência na graduação brasileira de filosofia, principalmente em localizações menos favorecidas, com relação (...) ao conhecimento de outras línguas, como o inglês e o francês. Tentamos, então, suprir essa deficiência, ao introduzirmos traduções de textos importantes ao público de língua portuguesa, sem nenhuma finalidade comercial e meramente pela glória da filosofia. O presente volume é constituído de três traduções de verbetes importantes sobre lógica, da Enciclopédia de Filosofia da Stanford: (1) A Lógica de Aristóteles, (2) Lógica Clássica, (3) Lógica Modal. (shrink)

The Manifestation Requirement, advanced by Dummett in his critique of semantic realism, has been criticized for being behavioristic, and the responses have been made that the critics are mistaken. However, the dispute has failed to exhibit the point of the Requirement. In this paper, I shall argue (1) that, in the light of Anscombe's theory of practical knowledge, knowledge of linguistic meaning is to be seen as the knowledge-how that forms the basis of the practical knowledge that an agent has (...) of his own intentional actions, (2) that such knowledge of linguistic meaning is to be based on mutual agreements in a community, and (3) that the Manifestation Requirement is a condition for the possibility of such mutual agreements. As a result, the Manifestation Requirement finds its basis on the idea that language and linguistic acts are tools that we use while continually improving them. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)

Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal anomalies (...) that Brouwer has invited us to introduce into mathematics. (shrink)

This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective N satisfying nucleus property, called here substructural nuclear logics, and its subclass, called here substructural nuclear image-based logics, where N further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce operational Kripke-style semantics for those logics and provide two sorts of completeness results for (...) them, one of which is based on algebraic completeness for all the logics and the other of which is based on set-theoretic one for the nuclear image-based logics. (shrink)

The general aim of this book is to provide an elementary exposition of some basic concepts in terms of which both classical and non-dassicallogirs may be studied and appraised. Although quantificational logic is dealt with briefly in the last chapter, the discussion is chiefly concemed with propo gjtional cakuli. Still, the subject, as it stands today, cannot br covered in one book of reasonable length. Rather than to try to include in the volume as much as possible, I have put (...) emphasis on some selected topics. Even these could not be roverrd completely, but for each topic I have attempted to present a detailed and precise t’Xposition of several basic results including some which are non-trivial. The roots of some of the central ideas in the volume go back to J. Luka siewicz’s seminar on mathematicallogi. (shrink)

The paper is a critique of the widespread conception of logic as a neutral arbiter between metaphysical theories, one that makes no `substantive’ claims of its own (David Kaplan and John Etchemendy are two recent examples). A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds (however mistaken), with a consequent proliferation of alternative logics. However, this apparent contentiousness of logic is often treated as though it were (...) neutralized by the possibility of studying all these alternative logics within an agreed metalogical framework, typically that of first-order logic with set theory. In effect, metalogic is given the role of neutral arbiter. The paper will consider a variety of examples in which deep logical disputes re-emerge at the meta-level. One case is quantified modal logic, where some varieties of actualism require a modal meta-language (as opposed to the usual non-modal language of possible worlds model theory) in order not to make their denial of the Barcan formula self-defeating. Similarly, on some views the intended model theory for second-order logic can only be given in a second-order metalanguage—this may be needed to avoid versions of Russell’s paradox when the first-order quantifiers are read as absolutely unrestricted. It can be shown that the phenomenon of higher-order vagueness eventually forces fuzzy logical treatments of vagueness to use a fuzzy metalanguage, with consequent repercussions for what first-order principles are validated. The difficulty of proving the completeness of first-order intuitionistic logic on its intended interpretation by intuitionistically rather than just classically valid means is a more familiar example. These case studies will be discussed in some detail to reveal a variety of ways in which even metalogic is metaphysically contested, substantial and non-neutral. (shrink)

Writers such as Stalnaker and Dummett have argued that specific features of subjunctive conditional statements undermine the principle of bivalence. This, paper is concerned with rebutting such claims. 1. It is shown how subjective conditionals pose a prima facie threat to bivalence, and how this threat can be dissolved by a distinction between the results of negating a subjective conditional and of negating its consequent. To make this distinction is to side with Lewis against Stalnaker in a dispute about possible (...) worlds semantics for such conditionals, and reasons are given for doing so. 2. These arguments are extended to answer Dummett''s claim that behaviourist and phenomenalist analyses in terms of subjunctive conditions violate bivalence. This answer is shown to be compatible with the principle that hypothetical statements are true only in virtue of categorical facts. (shrink)

Many advocates of non-classical logic for reasons external to mathematics claim that their proposed revisions are consistent with the use of classical logic within pure mathematics. Doubts are raised about such claims, concerning the applicability of pure mathematics to natural and social science. -/- .

Certain anti-realisms about mathematics are distinguished by their taking proof rather than truth as the central concept in the account of the meaning of mathematical statements. This notion of proof which is meaning determining or canonical must be distinguished from a notion of demonstration as more generally conceived. This paper raises a set of objections to Dummett's characterisation of the notion via the notion of a normalised natural deduction proof. The main complaint is that Dummett's use of normalised natural deduction (...) proofs relies on formalisation playing a role for which it is unfit. Instead I offer an alternative account which does not rely on formalisation and go on to examine the relation of proof to canonical proof, arguing that rather than requiring an explicit characterisation of canonical proofs we need to be more aware of the complexities of that relation. (shrink)

Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas’ important classical system CK. I give ICK both worlds (...) semantics and algebraic semantics, and prove that these are equivalent. I give a Gödel-type embedding of ICK into CK and a Glivenko-type embedding of CK into ICK. I axiomatize ICK and prove soundness, completeness, and decidability results. Finally, I discuss extending ICK. (shrink)

This paper discusses the relevance of supertask computation for the determinacy of arithmetic. Recent work in the philosophy of physics has made plausible the possibility of supertask computers, capable of running through infinitely many individual computations in a finite time. A natural thought is that, if supertask computers are possible, this implies that arithmetical truth is determinate. In this paper we argue, via a careful analysis of putative arguments from supertask computations to determinacy, that this natural thought is mistaken: supertasks (...) are of no help in explaining arithmetical determinacy. (shrink)

Throughout his philosophical career, Michael Dummett held firmly two theses: the theory of meaning has a central position in philosophy and all other forms of philosophical inquiry rest upon semantic analysis, in particular semantic issues replace traditional metaphysical issues; the theory of meaning is a theory of understanding. I will defend neither of them. However, I will argue that there is an important lesson we can learn by reflecting on the link between linguistic competence and semantics, which I take to (...) be an important part of Dummett’s legacy in philosophy of language. I discuss this point in relation to Cappelen and Lepore’s criticism of Incompleteness Arguments. (shrink)

The paradox of phase transitions raises the problem of how to reconcile the fact that we see phase transitions happen in concrete, finite systems around us, with the fact that our best theories—i.e. statistical-mechanical theories of phase transitions—tell us that phase transitions occur only in infinite systems. In this paper we aim to clarify to which extent this paradox is relative to the mathematical framework which is used in these theories, i.e. classical mathematics. To this aim, we will explore the (...) philosophical consequences of adopting constructive instead of classical mathematics in a statistical-mechanical theory of phase transitions. It will be shown that constructive mathematics forces certain ‘de-idealizations’ of such theories: talk of actually infinite systems is meaningless, there are no discontinuous functions, and—in a sense which will be clarified—constructive real numbers reflect our imperfect methods of determining the values of physical quantities. As such, so it will be argued, constructive mathematics offers a means to gain insight in the idealizations introduced in classical theories and the philosophical issues surrounding them. (shrink)

I will be concerned with the following question: are there compelling arguments for postulating a distinction between the truth of a statement and the recognition of its truth, when truth is conceived along the lines of a suitable generalization of the intuitionistic idea that it should be characterized as the existence of a proof? I will argue that the distinction is not necessary within the conceptual framework of intuitionism by replying to two arguments to the contrary, one based on the (...) paradox of inference, the other on considerations concerning the content of a statement. (shrink)

In his paper “Explaining Deductive Inference” Prawitz states what he calls «a fundamental problem of logic and the philosophy of logic»: the problem of explaining «Why do certain inferences have the epistemic power to confer evidence on the conclusion when applied to premisses for which there is evidence already?». In this paper I suggest a way of articulating, and partly modifying, the intuitionistic answer to this problem in such a way as to both answer Prawitz’s problem and satisfy a requirement (...) I argue to be crucial for any epistemic theory of the meaning of the logical constants: the requirement that evidence is epistemically transparent. (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 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 rigor. They are chosen because they are needed for the second part. In the second part, this natural-deduction framework is used in order to regiment David Lewis’s justification of his Division Thesis, which features prominently in his combination of mereology with class (...) theory. The Division Thesis plays a crucial role in Lewis’s informal argument for his Second Thesis in his book Parts of Classes. In order to present Lewis’s argument in rigorous detail, an elegant new principle is offered for the theory that combines class theory and mereology. The new principle is called the Canonical Decomposition Thesis. It secures Lewis’s Division Thesis on the strong construal required in order for his argument to go through. The exercise illustrates how careful one has to be when setting up the details of an adequate foundational theory of parts and classes. The main aim behind this investigation is to determine whether an anti-realist, inferentialist theorist of meaning has the resources to exhibit Lewis’s argument for his Second Thesis—which is central to his marriage of class theory with mereology—as a purely conceptual one. The formal analysis shows that Lewis’s argument, despite its striking appearance to the contrary, can be given in the constructive, relevant logic IR. This is the logic that the author has argued, elsewhere, to be the correct logic from an anti-realist point of view. The anti-realist is therefore in a position to regard Lewis’s argument as purely conceptual. (shrink)

In his famous paper, An Unsolvable Problem of Elementary Number Theory, Alonzo Church identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár's An Argument Against the Plausibility of Church's Thesis from 1959. The aim of this paper is to present Kalmár's argument and to fill in missing details based on his (...) general philosophical thoughts on mathematics. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)

Mark Jago argues for truthmaker maximalism in some recent papers based on a key premise: that every truth could have a truthmaker. Jago contends that many would pretheoretically accept this principle and that counterexamples to it would be difficult to find. In this note, I show how truthmaker non-maximalists can use a modified version of Peter Milne's argument against maximalism to provide a counterexample to this key premise.

The notion of harmony has played a pivotal role in a number of debates in the philosophy of logic. Yet there is little agreement as to how the requirement of harmony should be spelled out in detail or even what purpose it is to serve. Most, if not all, conceptions of harmony can already be found in Michael Dummett's seminal discussion of the matter in The Logical Basis of Metaphysics. Hence, if we wish to gain a better understanding of the (...) notion of harmony, we do well to start here. Unfortunately, however, Dummett's discussion is not always easy to follow. The following is an attempt to disentangle the main strands of Dummett's treatment of harmony. The different variants of harmony as well as their interrelations are clarified and their individual shortcomings qua interpretations of harmony are demonstrated. Though no attempt is made to give a detailed alternative account of harmony here, it is hoped that our discussion will lay the ground for an adequate rigorous treatment of this central notion. (shrink)

A novel solution to the knowability paradox is proposed based on Kant’s transcendental epistemology. The ‘paradox’ refers to a simple argument from the moderate claim that all truths are knowable to the extreme claim that all truths are known. It is significant because anti-realists have wanted to maintain knowability but reject omniscience. The core of the proposed solution is to concede realism about epistemic statements while maintaining anti-realism about non-epistemic statements. Transcendental epistemology supports such a view by providing for a (...) sharp distinction between how we come to understand and apply epistemic versus non-epistemic concepts, the former through our capacity for a special kind of reflective self-knowledge Kant calls ‘transcendental apperception’. The proposal is a version of restriction strategy: it solves the paradox by restricting the anti-realist’s knowability principle. Restriction strategies have been a common response to the paradox but previous versions face serious difficulties: either they result in a knowability principle too weak to do the work anti-realists want it to, or they succumb to modified forms of the paradox, or they are ad hoc. It is argued that restricting knowability to non-epistemic statements by conceding realism about epistemic statements avoids all versions of the paradox, leaves enough for the anti-realist attack on classical logic, and, with the help of transcendental epistemology, is principled in a way that remains compatible with a thoroughly anti-realist outlook. (shrink)

Different formal tools are useful for different purposes. For example, when it comes to modelling degrees of belief, probability theory is a better tool than classical logic; when it comes to modelling the truth of mathematical claims, classical logic is a better tool than probability theory. In this paper I focus on a widely used formal tool and argue that it does not provide a good model of a phenomenon of which many think it does provide a good model: I (...) shall argue that while supervaluationism may provide a model of probability of truth, or of assertability, it cannot provide a good model of truth -- supertruth cannot be truth. The core of the argument is that an adequate model of truth must render certain connectives truth-functional (at least in certain circumstances) -- and supervaluationism does not do so (in those circumstances). (shrink)

Famously, Michael Dummett argues that considerations concerning the role of language in communication lead to the rejection of classical logic in favor of intuitionistic logic. Potentially, this results in massive revisions of established mathematics. Recently, Neil Tennant (“The law of excluded middle is synthetic a priori, if valid”, Philosophical Topics 24 (1996), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle grounded on a metaphysical principle of determinacy. This article shows (...) that the for the anti-realist, the law of excluded middle entails that humans have wildly implausible abilities. The proposed synthesis between anti-realism and classical mathematics thus fails. (shrink)

Should weak forms of the axiom of choice really be accepted within constructive mathematics? A critical view of the Brouwer-Heyting-Kolmogorov interpretation, accompanied by the intention to include nondeterministic algorithms, leads us to subscribe to Richman's appeal for dropping countable choice. As an alternative interpretation of intuitionistic logic, we propose to renew dialogue semantics.

We introduce a sequent calculusBfor a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic. quantum and non-modal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterizeBpositively: reflection, symmetry and visibility.A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, (...) and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection.To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cut-elimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube. (shrink)

Michael Dummett’s realism debate is a semantic dispute about the kind of truth conditions had by a given class of sentences. According to his semantic realist, the truth conditions are potentially verification-transcendent in that they may obtain (or not) despite the fact that we may be forever unable to recognize whether they obtain. According to Dummett’s semantic anti-realist, the truth conditions are of a different sort. Essentially, for the anti-realist, that the truth conditions obtain (whenever they do) is a matter (...) that is always recognizable by us in principle. On this view, truth cannot outrun all possible human knowledge. Unsurprisingly, the outcome of the debate is sometimes said to hinge on whether all truths are knowable.1 More carefully the point of contention is the following.. (shrink)

ABSTRACTIn reply to Linnebo, I defend my analysis of Tait's argument against the use of classical logic in set theory, and make some preliminary comments on Linnebo's new argument for the same conclusion. I then turn to Shapiro's discussion of intuitionistic analysis and of Smooth Infinitesimal Analysis. I contend that we can make sense of intuitionistic analysis, but only by attaching deviant meanings to the connectives. Whether anyone can make sense of SIA is open to doubt: doing so would involve (...) making sense of mathematical quantities whose relationship to zero and to one another is inherently indeterminate. (shrink)

According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute (...) between classicists and intuitionists. I argue for this by presenting an account of consequence, and a cognate semantic theory for the language of the propositional calculus, which respects the meanings of the connectives as embodied in the familiar classical truth-tables, does not presuppose Bivalence, and with respect to which the rules of the intuitionist propositional calculus are sound and complete. Thus the disagreement between classicists and intuitionists, at least, need not stem from their attaching different senses to the connectives; one may deny the doctrine without changing the subject. The basic notion of my semantic theory is truth at a possibility, where a possibility is a way that things might be, but which differs from a possible world in that the way in question need not be fully specific or determinate. I compare my approach with a previous theory of truth at a possibility due to Lloyd Humberstone, and with a previous attempt to refute Quine’s thesis due to John McDowell. (shrink)