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)

ABSTRACT In our response Field's ‘Properties, Propositions and Conditionals’, we explore the methodology of Field's program. We begin by contrasting it with a proof-theoretic approach and then commenting on some of the particular choices made in the development of Field's theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field's response to Restall’s [2010] ‘What Are We to Accept, and What Are We to Reject, (...) While Saving Truth from Paradox?’. (shrink)

In order to explicate Gentzen’s famous remark that the introduction-rules for logical constants give their meaning, the elimination-rules being simply consequences of the meaning so given, we develop natural deduction rules for Sheffer’s stroke, alternative denial. The first system turns out to lack Double Negation. Strengthening the introduction-rules by allowing the introduction of Sheffer’s stroke into a disjunctive context produces a complete system of classical logic, one which preserves the harmony between the rules which Gentzen wanted: all indirect proof reduces (...) to direct proof. (shrink)

Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may (...) be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions. (shrink)

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett gives a mistaken elaboration (...) of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen's. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic. (shrink)

The connection between theories of truth and meaning is explored. Theories of truth and meaning are connected in a way such that differences in the conception of what it is for a sentence to be true are engendered by differences in the conception of how meanings depend on each other, and on a base of underlying facts. It is argued that this view is common ground between Davidson and Dummett, and that their dispute over realism is really a dispute in (...) the theory of meaning over holism and molecularism. The view offered is contrasted with influential pragmatist and modest views. (shrink)

This essay investigates whether or not we should think that the things we say are identical to the things our sentences mean. It is argued that these theoretical notions should be distinguished, since assertoric content does not respect the compositionality principle. As a paradigmatic example, Kaplan's formal language LD is shown to exemplify a failure of compositionality. It is demonstrated that by respecting the theoretical distinction between the objects of assertion and compositional values certain conflicts between compositionality and contextualism are (...) avoided. This includes the conflict between eternalism and the semantics of tense, the embedding problems for contextualism about epistemic modals and taste claims, and the conflict between direct reference and the semantics of bound pronouns (and monstrous operators). After presenting the theoretical picture which distinguishes assertoric content from compositional semantic value, some objections to the picture are addressed. In so doing, the objection from King (2003) stemming from apparent complications with the interaction of temporal expressions and attitude reports is assessed and shown to be non-threatening. (shrink)

This paper addresses the problem of upgrading functional information to knowledge. Functional information is defined as syntactically well-formed, meaningful and collectively opaque data. Its use in the formal epistemology of information theories is crucial to solve the debate on the veridical nature of information, and it represents the companion notion to standard strongly semantic information, defined as well-formed, meaningful and true data. The formal framework, on which the definitions are based, uses a contextual version of the verificationist principle of truth (...) in order to connect functional to semantic information, avoiding Gettierization and decoupling from true informational contents. The upgrade operation from functional information uses the machinery of epistemic modalities in order to add data localization and accessibility as its main properties. We show in this way the conceptual worthiness of this notion for issues in contemporary epistemology debates, such as the explanation of knowledge process acquisition from information retrieval systems, and open data repositories. (shrink)

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this circle we may (...) consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting’s view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference. (shrink)

What is the appropriate notion of truth for sentences whose meanings are understood in epistemic terms such as proof or ground for an assertion? It seems that the truth of such sentences has to be identified with the existence of proofs or grounds, and the main issue is whether this existence is to be understood in a temporal sense as meaning that we have actually found a proof or a ground, or if it could be taken in an abstract, tenseless (...) sense. Would the latter alternative amount to realism with respect to proofs or grounds in a way that would be contrary to the supposedly anti-realistic standpoint underlying the epistemic understanding of linguistic expressions? Before discussing this question, I shall consider reasons for construing linguistic meaning epistemically and relations between such reasons and reasons for taking an anti-realist point of view towards the discourse in question. (shrink)

According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of (...) this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence. (shrink)

Several proof-theoretic notions of validity have been proposed in the literature, for which completeness of intuitionistic logic has been conjectured. We define validity for intuitionistic propositional logic in a way which is common to many of these notions, emphasizing that an appropriate notion of validity must be closed under substitution. In this definition we consider atomic systems whose rules are not only production rules, but may include rules that allow one to discharge assumptions. Our central result shows that Harrop’s rule (...) is valid under substitution, which refutes the completeness conjecture for intuitionistic logic. (shrink)

Rorty's assessment of the origins of the analytic/continental divide is discussed and criticized on several grounds. Rorty's plea in favour of the philosophical quietism implicit in the pragmatists' dismissal of metaphysics rests on an uncritical faith in scientific progress. To emphasize the tensions implicit in the ideal of progress a novel interpretation of the significance of Faust's encounter with Sorge (Care) is offered and compared with Heidegger's construal of Sorge in Sein und Zeit . A better understanding of the analytic/continental (...) divide is gained if we concentrate on the impact of Nazism and Fascism on the philosophical tradition with which philosophers living on the Continent had to come to terms after the Second World War. In this respect Adorno's attitude to progress in general and to philosophy in particular is very instructive and can be usefully contrasted with Rorty's dismissal of metaphysics. (shrink)

The main objective of this paper is to sketch unifying conceptual and formal framework for inference that is able to explain various proof techniques without implicitly changing the underlying notion of inference rules. We base this framework upon the so-called two-dimensional, i.e., deduction to deduction, account of inference introduced by Tichý in his seminal work The Foundation’s of Frege’s Logic (1988). Consequently, it will be argued that sequent calculus provides suitable basis for such general concept of inference and therefore should (...) not be seen just as technical tool, but philosophically well-founded system that can rival natural deduction in terms of its “naturalness”. (shrink)

This dialogue between Achilles and the Tortoise – in the spirit of those of Carroll and Hofstadter – argues against the idea, identified with the “traditional” interpretation of Aristotle’s “sea battle argument”, that future contingents are an exception to the Principle of Bivalence. It presents examples of correct everyday predictions, without which one would not be able to decide and to act; however, doing this is incompatible with the belief that the content of these predictions lacks a truth-value. The cost (...) of using a non-classical logic to cope with that may be too high for Stagirite’s defenders, and they would still need to explain why our ordinary predictions seem to have a binary truth-value. In the end, the paper suggests that the problem of future contingents – and of free will – is not a logical problem at all, but rather a limit on what an agent can believe before taking a decision. (shrink)

The topic of this paper is the question whether there is a logic which could be justly called the logic of inference. It may seem that at least since Prawitz, Dummett and others demonstrated the proof-theoretical prominency of intuitionistic logic, the forthcoming answer is that it is this logic that is the obvious choice for the accolade. Though there is little doubt that this choice is correct (provided that inference is construed as inherently single-conclusion and complying with the Gentzenian structural (...) rules), I do not think that the usual justification of it is satisfactory. Therefore, I will first try to clarify what exactly is meant by the question, and then sketch a conceptual framework in which it can be reasonably handled. I will introduce the concept of 'inferentially native' logical operators (those which explicate inferential properties) and I will show that the axiomatization of these operators leads to the axiomatic system of intuitionistic logic. Finally, I will discuss what modifications of this answer enter the picture when more general notions of inference are considered. (shrink)

What is a minimal proof-theoretical foundation of logic? Two different ways to answer this question may appear to offer themselves: reduce the whole of logic either to the relation of inference, or else to the property of incompatibility. The first way would involve defining logical operators in terms of the algebraic properties of the relation of inference—with conjunction $$\hbox {A}\wedge \hbox {B}$$ A ∧ B as the infimum of A and B, negation $$\lnot \hbox {A}$$ ¬ A as the minimal (...) incompatible of A, etc. The second way involves introducing logical operators in terms of the relation of incompatibility, such that X is incompatible with $$\{\lnot \hbox {A}\}$$ { ¬ A } iff every Y incompatible with X is incompatible with {A}; and X is incompatible with $$\{\hbox {A}\!\wedge \!\hbox {B}\}$$ { A ∧ B } iff X is incompatible with {A,B}; etc. Whereas the first route leads us naturally to intuitionistic logic, the second leads us to classical logic. The aim of this paper is threefold: to investigate the relationship of the two approaches within a very general framework, to discuss the viability of erecting logic on such austere foundations, and to find out whether choosing one of the ways we are inevitably led to a specific logical system. (shrink)

This paper gives attention to a special point in Brandom.

The paper "Does Epistemological Holism lead to Meaning – Holism" (Cozzo, 2002) touches one of the main problems of a molecularist theory of meaning: how to restrict the class of inferences connected with a word, in order to define the sense of the word. I will discuss the starting point of this approach, mainly the pre-theoretical criterion against meaning holism: meaning holism, following a well-known argument by Dummett, reduces communication to a mystery. However there is a strong background assumption of (...) this argument: communication is sharing the same meanings. Accepting this assumption without acknowledging it makes the entire proposal more problematic than it appears at first sight. In what follows I will try to clarify the possibility of a different reaction to meaning holism, putting forward some distinctions which come to light when the above stated assumption is made explicit. Then, some other comments will follow on the difficulty of avoiding extreme meaning holism, even within Cozzo's elegant attempt to implement a molecularist vision; in the end, his notion of the sense of a word will appear to be difficult to define, if these holistic aspects of language take the space they deserve even in his theory. I conclude with two remarks regarding two relevant requirements for a theory of meaning (conservativeness and harmony). (shrink)

One particular topic in the literature on Frege’s conception of sense relates to two apparently contradictory theses held by Frege: the isomorphism of thought and language on one hand and the expressibility of a thought by different sentences on the other. I will divide the paper into five sections. In (1) I introduce the problem of the tension in Frege’s thought. In (2) I discuss the main attempts to resolve the conflict between Frege’s two contradictory claims, showing what is wrong (...) with some of them. In (3), I analyze where, in Frege’s writings and discussions on sense identity, one can find grounds for two different conceptions of sense. In (4) I show how the two contradictory theses held by Frege are connected with different concerns, compelling Frege to a constant oscillation in terminology. In (5) I summarize two further reasons that prevented Frege from making the distinction between two conceptions of sense clear: (i) the antipsychologism problem and (ii) the overlap of traditions in German literature contemporary to Frege about the concept of value. I conclude with a hint for a reconstruction of the Fregean notion of ‘thought’ which resolves the contradiction between his two theses. (shrink)

The standard inferentialist approaches to modal logic tend to suffer from not being able to uniquely characterize the modal operators, require that introduction and elimination rules be interdefined, or rely on the introduction of possible-world like indexes into the object language itself. In this paper I introduce a hypersequent calculus that is flexible enough to capture many of the standard modal logics and does not suffer from the above problems. It is therefore an ideal candidate to underwrite an inferentialist theory (...) of meaning for modal operators. Here I treat specifically the modal logics K, D, T, S4, B, and S5. I show that the calculi are adequate for each set of models, and show that they meet a large set of criteria that are generally thought necessary for a calculus to underwrite a theory of meaning. (shrink)

Many arguments that show p to be enthymematic (in an argument for q) rely on claims like “if one did not believe that p, one would not have a reason for believing that q.” Such arguments are susceptible to the neg-raising fallacy. We tend to interpret claims like “X does not believe that p” as statements of disbelief (X's belief that not-p) rather than as statements of withholding the belief that p. This article argues that there is a tendency to (...) equivocate in arguments for the enthymematicity of arguments (e.g., Lewis Carroll's paradox, Hume's problem) as well as in arguments for the enthymematicity of action explanations (e.g., arguments for psychologism and for explanatory individualism). The article concludes with a warning, because the equivocation is often helpful in teaching and because neg-raising verbs include philosophically vital verbs: desire, want, intend, think, suppose, imagine, expect, feel, seem, appear. (shrink)

Proof-theoretic semantics is an alternative to model-theoretic semantics. It aims at explaining the meaning of the logical constants in terms of the inference rules that govern their behaviour in proofs. We argue that this must be construed as the task of explaining these meanings relative to a logic, i.e., to a consequence relation. Alas, there is no agreed set of properties that a relation must have in order to qualify as a consequence relation. Moreover, the association of a consequence relation (...) to a logical calculus is not as straightforward as it may seem. We show that these facts are problematic for the proof-theoretic project but the problems can be solved. Our thesis is that the consequence relation relevant for proof-theoretic semantics is the one given by the sequent-to-sequent derivability relation in Gentzen systems. (shrink)

Saul Kripke's thesis that ordinary proper names are rigid designators is supported by widely shared intuitions about the occurrence of names in ordinary modal contexts. By those intuitions names are scopeless with respect to the modal expressions. That is, sentences in a pair like (a) Aristotle might have been fond of dogs, (b) Concerning Aristotle, it is true that he might have been fond of dogs will have the same truth value. The same does not in general hold for definite (...) descriptions. If one, like Kripke, accounts for this difference by means of the intensions of the names and the descriptions, the conclusion is that names do not in general have the same intension as any normal, identifying description. However, this difference can be accounted for alternatively by appeal to the semantics of the modal expressions. On the account we suggest, dubbed 'relational modality', simple singular terms, like proper names, contribute to modal contexts simply by their actual world reference, not by their descriptive content. That account turns out to be fully equivalent with the rigidity account when it comes to truth of modal and non-modal sentence (with respect to the actual world), and hence supports the same basic intuitions. Here we present the relational modality account and compare it with others, in particular Kripke's own. (shrink)

Peter Pagin Is the principle of semantic compositionality compatible with the principle of semantic holism? The question is of interest, since both principles have a lot that speaks for them, and since they do seem to be in conflict. The view that natural languages have compositional structure is almost unavoidable, since linguistic communication by means of new combinations of words would be virtually incomprehensible otherwise. And holism too seems generally plausible, since the meaning of an expression is directly connected with (...) the way that expression interacts with other. (shrink)

This paper is an attack on the Dummett-Prawitz view that the principle of bivalence has a crucial double significance, metaphysical and meaning theoretical. On the one hand it is said that holding bivalence valid is what characterizes a realistic view, i.e. a view in metaphysics, and on the other hand it is said that there are meaning theoretical arguments against its acceptability. I argue that these two aspects are incompatible. If the failure of validity of bivalence depends on properties of (...) linguistic meaning, then there are no metaphysical consequences to be drawn. The case for this view is straightforward as long as we are discussing a language different from our own. But it seems that the distinction between failure because of meaning and failure because of reality cannot be applied to our own language, simply because our own language is just what we use to represent reality. I argue that this impression is illusory. In order to draw a conclusion about reality, meaning must be connected with truth in a non-trivial way, and precisely this cannot be done in the language for which the meaning theory itself is correct. (shrink)

I show that propositional intuitionistic logic is complete with respect to an adaptation of Dummett’s pragmatist justification procedure. In particular, given a pragmatist justification of an argument, I show how to obtain a natural deduction derivation of the conclusion of the argument from, at most, the same assumptions.

The present article aims at showing that it is possible to construct a realist philosophy of mathematics which commits one neither to dream the dreams of Platonism nor to reduce the word ''realism'' to mere noise. It is argued that mathematics is a science of patterns, where patterns are not objects (or properties of objects), but aspects, or aspects of aspects, etc. of objects. (The notion of aspect originates from ideas sketched by Wittgenstein in the Philosophical Investigations.).

According to linguistic conventionalism, necessities are to be explained in terms of the conventionally adopted rules that govern the use of linguistic expressions. A number of influential arguments against this view concerns the ‘Truth-Contrast Thesis’. This is the claim that necessary truths are fundamentally different from contingent ones since they are not made true by ‘the facts’. Instead, they are supposed to be something like ‘true in virtue of meaning’. This thesis is widely held to be a core commitment of (...) the conventionalist position, and the view is frequently rejected on the grounds that this thesis is untenable. I argue that this line of reasoning is mistaken. While the thesis should be rejected, it is not, I argue, entailed by linguistic conventionalism – nor was it invariably accepted by the paradigmatic conventionalists. (shrink)

Alethic pluralists argue truth is a metaphysically robust higher-order property that is multiply realized by a set of diverse and domain-specific subvening alethic properties. The higher-order truth property legitimizes mixed inferences and accounts for a univocal truth predicate. Absent of this higher-order property, pluralists lack an account of the validity of mixed inferences and an adequate semantics for the truth predicate and thereby appear forced to abandon the central tenets of alethic pluralism. I argue the use of many-valued logics to (...) support pluralism fails to address the pluralist’s metaphysical problem regarding mixed inferences and mixed truth functional connectives. The high degree of heterogeneity of the alethic realizers challenges the plausibility of a single higher-order functional property. A functional property with such a heterogeneous base cannot be projectable at a theoretically significant level. The problem with mixed inferences and truth functions is but one symptom of the deeper projectability problem. (shrink)

This essay presents a long, detailed, in many ways critical but also appreciative account, of David Bloor’s recent book The Enigma of the Aerofoil. I take that work as the crowning statement of ideas and principles developed over the past four decades by Bloor and other exponents of the ‘strong programme’ in the sociology of scientific knowledge. It therefore offers both a test-case of that approach and a welcome opportunity to review, clarify and extend some of the arguments brought against (...) it by critical realists. This purpose is particularly well served by Bloor’s having here assembled such a wealth of scientific, historical, political and socio-cultural data concerning one specific, very crucial period of research into theories of aerodynamic lift. His book thus provides an exceptionally useful point of engagement for anyone seeking to adjudicate these issues from a critically angled but engaged and constructive standpoint. (shrink)

This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following, Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also adopts a thought-experimental approach – a (...) variant of Descartes' dream scenario – in order to establish the in-principle possibility that we might be deceived by the apparent self-evidence of basic arithmetical truths or that it might be ‘rational’ to doubt them under some conceivable set of circumstances. Thus Putnam assumes that mathematical realism involves a self-contradictory ‘Platonist’ idea of our somehow having quasi-perceptual epistemic ‘contact’ with truths that in their very nature transcend the utmost reach of human cognitive grasp. On this account, quite simply, ‘nothing works’ in philosophy of mathematics since wecan either cling to that unworkable notion of objective truth or abandon mathematical realism in favour of a verificationist approach that restricts the range of admissible statements to those for which we happen to possess some means of proof or ascertainment. My essay puts the case, conversely, that these hyperbolic doubts are not forced upon us but result from a false understanding of mathematical realism – a curious mixture of idealist and empiricist themes – which effectively skews the debate toward a preordained sceptical conclusion. I then go on to mount a defence of mathematical realism with reference to recent work in this field and also to indicate some problems – as I seethem – with Putnam's thought-experimental approach as well ashis use of anti-realist arguments from Dummett, Kripke, Wittgenstein, and others. (shrink)

In this essay, I offer a critical evaluation of Hilary Putnam's writings on epistemology and philosophy of science, in particular his engagement with interpretative problems in quantum mechanics. I trace the development of his thinking from the late 1960s when he adopted a strong causal-realist position on issues of meaning, reference, and truth, via the "internal realist" approach of his middle-period writings, to the various forms of pragmatist, naturalized, or "commonsense" epistemology proposed in his latest books. My contention is that (...) Putnam's retreat from a full-fledged realist outlook has been prompted in large part by his belief that it cannot possibly be reconciled with the implications of quantum mechanics for our understanding of processes and events in the subatomic domain. However, I suggest, this response should be seen as premature given the range of as-yet unresolved problems with quantum mechanics on the orthodox (Copenhagen) interpretation and also the existence of an alternative account - David Bohm's hidden-variables theory - which perfectly matches the established predictive-observational results while providing a credible realist ontology. I also examine Putnam's case for adopting a nonstandard (three-valued) "quantum logic" in relation to the thinking of other philosophers - Michael Dummett among them - who have espoused a more global or doctrinaire version of anti-realism. I conclude that Putnam's early (causal-realist) position is by no means untenable in light of the various arguments that he now takes as counting decisively against it. (shrink)

Can the world itself _be_ vague, so that rather than vagueness be a deficiency in our mode of describing the world, it is a necessary feature of any true description of it? Gareth Evans famously poses this question in his paper ‘Can There Be Vague Objects’ (Analysis 38(4):208, 1978 ). In his recent paper ‘Indeterminacy and Vagueness: Logic and Metaphysics’, Peter van Inwagen ( 2009 ) elaborates the account of vagueness and, in particular, in the case of sentences, consequent indeterminacy (...) in truth value, to which this conception of ‘worldly’ vagueness is opposed, calling it the ‘sensible’ theory of indeterminacy and rejecting it. In what follows, I defend the sensible theory van Inwagen rejects. I first explain more fully what it involves and, as importantly, what it does not. (shrink)

A lot of good philosophy is done in the armchair, but is nevertheless a posteriori. This paper clarifies and then defends that claim. Among the a posteriori activities done in the armchair are assembling and evaluating commonplaces; formulating theoretical alternatives; and integrating well-known past a posteriori discoveries. The activity that receives the most discussion, however, is the application of theoretical virtues to choose philosophical theories: the paper argues that much of this is properly seen as a posteriori.

A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely proof-theoretical concepts. This kind of (...) analysis is driven by exploiting the Curry–Howard correspondence. In particular, deducibility of identicals is shown to correspond to the computational property of eta expansion, which is essential in the characterization of propositional identity. (shrink)

Richard Heck has recently drawn attention on a new version of the Liar Paradox, one which relies on logical resources that are so weak as to suggest that it may not admit of any “truly satisfying, consistent solution”. I argue that this conclusion is too strong. Heck's Liar reduces to absurdity principles that are already rejected by consistent paracomplete theories of truth, such as Kripke's and Field's. Moreover, the new Liar gives us no reasons to think that (versions of) these (...) principles cannot be consistently retained once the structural rule of contraction is restricted. I suggest that revisionary logicians have independent reasons for restricting such a rule. (shrink)

I argue that the standard anti-realist argument from manifestability to intuitionistic logic is either unsound or invalid. Strong interpretations of the manifestability of understanding are falsified by the existence of blindspots for knowledge. Weaker interpretations are either too weak, or gerrymandered and ad hoc. Either way, they present no threat to classical logic.

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 (2008) 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)

According to logical inferentialists, the meanings of logical expressions are fully determined by the rules for their correct use. Two key proof-theoretic requirements on admissible logical rules, harmony and separability, directly stem from this thesis—requirements, however, that standard single-conclusion and assertion-based formalizations of classical logic provably fail to satisfy :1035–1051, 2011). On the plausible assumption that our logical practice is both single-conclusion and assertion-based, it seemingly follows that classical logic, unlike intuitionistic logic, can’t be accounted for in inferentialist terms. In (...) this paper, I challenge orthodoxy and introduce an assertion-based and single-conclusion formalization of classical propositional logic that is both harmonious and separable. In the framework I propose, classicality emerges as a structural feature of the logic. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s quasi-categoricity (...) theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

The aim of this paper is to reconsider several proposals that have been put forward in order to develop a Proof-Theoretical Semantics, from the by now classical neo-verificationist approach provided by D. Prawitz and M. Dummett in the Seventies, to an alternative, more recent approach mainly due to the work of P. Schroeder-Heister and L. Hallnäs, based on clausal definitions. Some other intermediate proposals are very briefly sketched. Particular attention will be given to the role played by the so-called Fundamental (...) Assumption. We claim that whereas, in the neo-verificationist proposal, the condition expressed by that Assumption is necessary to ensure the completeness of the justification procedure ( from the outside , so to speak), within the definitional framework it is a built-in feature of the proposal. The latter approach, therefore, appears as an alternative solution to the problem which prompted the neo-verificationists to introduce the Fundamental Assumption. (shrink)

I am very happy indeed to contribute to this series of lectures, especially because I owe most of my training in philosophy to Elizabeth Anscombe, whose work has given the series its name. I am deeply indebted to the marvellous generosity of her teaching, to the example she set me of an unrelentingly thorough and serious thinker, to the unobtrusive way she introduced me to Wittgenstein's later philosophy. Through Elizabeth Anscombe I also made the acquaintance of my friend Philippa Foot, (...) whose work in moral philosophy has, over decades and more than anyone else's, influenced my own. I hope it will be possible to recognize in what I am going to say here not indeed the excellence but at least traces of the beneficial influence of both these philosophers. (shrink)