In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth- in-content. These notions are explained, exegetical documentation provided, and semi-formal recursive definitions are given.

A general definition theory should serve as a foundation for the mathematical study of definitional structures. The central notion of such a theory is a precise explication of the intuitively given notion of a definitional structure. The purpose of this paper is to discuss the proof theory of partial inductive definitions as a foundation for this kind of a more general definition theory. Among the examples discussed is a suggestion for a more abstract definition of lambda-terms (derivations in natural deduction) (...) that could provide a basis for a more systematic definitional approach to general proof theory. (shrink)

Logical inferentialists contend that the meanings of the logical constants are given by their inference rules. Not just any rules are acceptable, however: inferentialists should demand that inference rules must reflect reasoning in natural language. By this standard, I argue, the inferentialist treatment of quantification fails. In particular, the inference rules for the universal quantifier contain free variables, which find no answer in natural language. I consider the most plausible natural language correlate to free variables—the use of variables in the (...) language of informal mathematics—and argue that it lends inferentialism no support. (shrink)

The paper is concerned with Quine's substitutional account of logical truth. The critique of Quine's definition tends to focus on miscellaneous odds and ends, such as problems with identity. However, in an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level. In the article he shows that Quine's substitutional account of logical truth cannot be generalized properly to the general concept of (...) logical consequence. The purpose of this paper is threefold: first, to introduce the reader to the metamathematics of Quine's substitutional definition of logical truth; second, to make Boolos' result accessible to a broader audience by giving a detailed and self-contained presentation of his proof; and, finally, to discuss some of the possible implications and how a defender of the Quinean concepts might react to the challenge posed by Boolos' result. (shrink)

A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: 1. If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich (...) will enroll. There are many different things one can say about this argument, but many agree that if we do not equivocate (if the terms mean the same thing in the premises and the conclusion) then the argument is valid, that is, the conclusion follows deductively from the premises. This does not mean that the conclusion is true. Perhaps the premises are not true. However, if the premises are true, then the conclusion is also true, as a matter of logic. This entry is about the relation between premises and conclusions in valid arguments. (shrink)

It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is in fact not even disagreement in any interesting sense, let alone a faultless one, in spite of appearances and claims to the contrary. A position I call classical pluralism is sketched, (...) intended to provide a coherent methodological stance towards the issue. Some reasons to recommend this stance are given, as well as some speculations as to why not everyone might want to follow the recommendation. (shrink)

The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...) In sections 5 and 6 I discuss some problems arising with regard to Prawitz’s views. (shrink)

In this dissertation, we shall investigate whether Tennant's criterion for paradoxicality(TCP) can be a correct criterion for genuine paradoxes and whether the requirement of a normal derivation(RND) can be a proof-theoretic solution to the paradoxes. Tennant’s criterion has two types of counterexamples. The one is a case which raises the problem of overgeneration that TCP makes a paradoxical derivation non-paradoxical. The other is one which generates the problem of undergeneration that TCP renders a non-paradoxical derivation paradoxical. Chapter 2 deals with (...) the problem of undergeneration and Chapter 3 concerns the problem of overgeneration. Chapter 2 discusses that Tenant’s diagnosis of the counterexample which applies CR−rule and causes the undergeneration problem is not correct and presents a solution to the problem of undergeneration. Chapter 3 argues that Tennant’s diagnosis of the counterexample raising the overgeneration problem is wrong and provides a solution to the problem. Finally, Chapter 4 addresses what should be explicated in order for RND to be a proof-theoretic solution to the paradoxes. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

This article offers an overview of inferential role semantics. We aim to provide a map of the terrain as well as challenging some of the inferentialist’s standard commitments. We begin by introducing inferentialism and placing it into the wider context of contemporary philosophy of language. §2 focuses on what is standardly considered both the most important test case for and the most natural application of inferential role semantics: the case of the logical constants. We discuss some of the (alleged) benefits (...) of logical inferentialism, chiefly with regards to the epistemology of logic, and consider a number of objections. §3 introduces and critically examines the most influential and most fully developed form of global inferentialism: Robert Brandom’s inferentialism about linguistic and conceptual content in general. Finally, in §4 we consider a number of general objections to IRS and consider possible responses on the inferentialist’s behalf. (shrink)

I argue that there are two ways of construing Wittgenstein’s slogan that meaning is use. One accepts the view that the notion of meaning must be explained in terms of truth-theoretic notions and is committed to the epistemic conception of truth. The other keeps the notion of meaning and the truth-theoretic notions apart and is not committed to the epistemic conception of truth. I argue that Dummett endorses the first way of construing Wittgenstein’s slogan. I address the issue by discussing (...) two of Dummett’s arguments against the realist truth-theoretic conception of meaning: the manifestation argument and the argument for the unintelligibility of classical logic. I examine the dialectic of those arguments and show that they rest on the assumption that meaning needs to be explained in terms of truth-theoretic notions.Tvrdim da postoje dva načina shvaćanja Wittgensteinova slogana da je značenje upotreba. Jedno prihvaća gledište da se pojam značenja mora objasniti na osnovi istinitosnoteoretskih pojmova te je obvezano na epistemičku koncepciju istine. Drugo pojam značenja i istinitosnoteoretske pojmove drži odvojenima te nije obvezano na epistemičku koncepciju istine. Tvrdim da Dummett prihvaća prvo shvaćanje Wittgensteinova slogana. Problemu pristupam tako što raspravljam o dvama Dummettovim argumentima protiv realističke istinitosnoteoretske koncepcije značenja: o argumentu iz manifestacije i argumentu za neshvatljivost klasične logike. Ispitujem dijalektiku tih argumenata i pokazujem da počivaju na pretpostavci da se značenje mora objasniti na osnovi istinitosnoteoretskih pojmova. (shrink)

This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field’s contention that such objections (...) show consequenceto be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We suggest that the accountsin question have the resources to meet the objections standardly thought to herald their demise and make two main claims: (i) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (ii) that consequence is a paradoxical notion if truth is. (shrink)

I give an account of proof terms for derivations in a sequent calculus for classical propositional logic. The term for a derivation δ of a sequent Σ≻Δ encodes how the premises Σ and conclusions Δ are related in δ. This encoding is many–to–one in the sense that different derivations can have the same proof term, since different derivations may be different ways of representing the same underlying connection between premises and conclusions. However, not all proof terms for a sequent Σ≻Δ (...) are the same. There may be different ways to connect those premises and conclusions. -/- Proof terms can be simplified in a process corresponding to the elimination of cut inferences in sequent derivations. However, unlike cut elimination in the sequent calculus, each proof term has a unique normal form (from which all cuts have been eliminated) and it is straightforward to show that term reduction is strongly normalising—every reduction process terminates in that unique normal form. Further- more, proof terms are invariants for sequent derivations in a strong sense—two derivations δ1 and δ2 have the same proof term if and only if some permutation of derivation steps sends δ1 to δ2 (given a relatively natural class of permutations of derivations in the sequent calculus). Since not every derivation of a sequent can be permuted into every other derivation of that sequent, proof terms provide a non-trivial account of the identity of proofs, independent of the syntactic representation of those proofs. (shrink)

Homotopy Type theory and its Model theory provide a novel formal semantic framework for representing scientific theories. This framework supports a constructive view of theories according to which a theory is essentially characterised by its methods. The constructive view of theories was earlier defended by Ernest Nagel and a number of other philosophers of the past but available logical means did not allow these people to build formal representational frameworks that implement this view.

In his seminal address delivered in 1945 to the Royal Society Gilbert Ryle considers a special case of knowing-how, viz., knowing how to reason according to logical rules. He argues that knowing how to use logical rules cannot be reduced to a propositional knowledge. We evaluate this argument in the context of two different types of formal systems capable to represent knowledge and support logical reasoning: Hilbert-style systems, which mainly rely on axioms, and Gentzen-style systems, which mainly rely on rules. (...) We build a canonical syntactic translation between appropriate classes of such systems and demonstrate the crucial role of Deduction Theorem in this construction. This analysis suggests that one's knowledge of axioms and one's knowledge of rules under appropriate conditions are also mutually translatable. However our further analysis shows that the epistemic status of logical knowing-how ultimately depends on one's conception of logical consequence: if one construes the logical consequence after Tarski in model-theoretic terms then the reduction of knowing-how to knowing-that is in a certain sense possible but if one thinks about the logical consequence after Prawitz in proof-theoretic terms then the logical knowledge-how gets an independent status. Finally we extend our analysis to the case of extra-logical knowledge-how representable with Gentzen-style formal systems, which admit constructive meaning explanations. For this end we build a typed sequential calculus and prove for it a ``constructive'' Deduction Theorem interpretable in extra-logical terms. We conclude with a number of open questions, which concern translations between knowledge-how and knowledge-that in this more general semantic setting. (shrink)