This volume features essays about and by Paul Benacerraf, whose ideas have circulated in the philosophical community since the early nineteen sixties, shaping key areas in the philosophy of mathematics, the philosophy of language, the philosophy of logic, and epistemology. The book started as a worskhop held in Paris at the Collège de France in May 2012 with the participation of Paul Benacerraf. The introduction addresses the methodological point of the legitimate use of so-called “Princess Margaret Premises” in drawing philosophical (...) conclusions from Gödel’s first incompleteness theorem. The book is then divided into three sections. The first is devoted to an assessment of the improved version of the original dilemma of “Mathematical Truth” due to Hartry Field: the challenge to the platonist is now to explain the reliability of our mathematical beliefs given the very subject matter of mathematics, either pure or applied. The second addresses the issue of the ontological status of numbers: Frege’s logicism, fictionalism, structuralism, and Bourbaki’s theory of structures are called up for an appraisal of Benacerraf’s negative conclusions of “What Numbers Could Not Be.” The third is devoted to supertasks and bears witness to the unique standing of Benacerraf’s first publication: “Tasks, Super-Tasks, and Modern Eleatics” in debates on Zeno’s paradox and associated paradoxes, infinitary mathematics, and constructivism and finitism in the philosophy of mathematics. Two yet unpublished essays by Benacerraf have been included in the volume: an early version of “Mathematical Truth” from 1968 and an essay on “What Numbers Could Not Be” from the mid 1970’s. A complete chronological bibliography of Benacerraf’s work to 2016 is provided.Essays by Jody Azzouni, Paul Benacerraf, Justin Clarke-Doane, Sébastien Gandon, Brice Halimi, Jon Pérez Laraudogoitia, Mary Leng, Antonio Leòn-Sànchez and Ana Leòn-Mejía, Marco Panza, Fabrice Pataut, Philippe de Rouilhan, Andrea Sereni, and Stewart Shapiro. (shrink)

The author of “Parsimony and inference to the best mathematical explanation” argues for platonism by way of an enhanced indispensability argument based on an inference to yet better mathematical optimization explanations in the natural sciences. Since such explanations yield beneficial trade-offs between stronger mathematical existential claims and fewer concrete ontological commitments than those involved in merely good mathematical explanations, one must countenance the mathematical objects that play a theoretical role in them via an application of the relevant mathematical results. The (...) nominalist’s challenge is thus to undermine the platonistic force of such explanations by way of alternative nominalistic ones. The author’s contention is that such nominalistic explanations should provide a paraphrase of the proofs of the mathematical results being applied. There are reasons to doubt that proofs, construed here as formal derivations, actually contribute to the platonistc force to be undermined and, by parity, that nominalized proofs should bear responsability for the corresponding undermining. A discussion of two examples and of associated arguments by Lange, Pincock, Steiner and Tallant, point to a a wealth of worries concerning the construal of this explanatory role. Among those figure the distinction between the weak and strong role of proofs, the distinction between causal or “ordinary” explanations and genuine mathematical ones, and the unifying role of optimization explanations. More generally, the very idea that the explanatory advantages yielded by applied mathematical claims may be construed as gradual or progressive and the associated notion that the feasibility of their nominalistic paraphrases decreases as the generality and force of these claim increases, deserves a closer attention. (shrink)

Many philosophers hold that physical laws have a unique modal status known as nomic necessity which is weaker than metaphysical necessity. This orthodox view has come into question in the past few decades. In particular, the metaphysical view known as essentialism has provided an argument that the laws of nature are necessary in the strongest possible sense. It seems obvious to many that at least some essentialist arguments in favor of the necessity of scientific claims are going to be sound. (...) For example, the view that claims like "water is H2O" are necessary has itself 2 become an orthodox view. However, the question of whether laws, like the law of conservation of energy, or the law of gravity, are necessary is far more contentious. Philosophers divide roughly into two camps, law necessitarians1 who hold that the laws are necessary in the strongest sense and contingency theorists who hold that they are at least in some sense contingent. One argument for the necessitarian position is via an essentialist theory of the transworld identity of properties. In this paper I defend such a theory of the identity of properties and its necessitarian consequences from one major criticism. To focus the paper, I center the discussion on a single critic, E. J. Lowe. In his book, The Four Category Ontology, he offers a criticism of the essentialist argument for necessitarianism via an analogy with other forms of transworld identity and intuitions about the contingency of the physical constants2. I undermine the usefulness of Lowe's analogy by examining the purposes of attributions of properties. I also show that the essentialist's position can allow it to accommodate the intuitions of contingency in a way that fits best with the purpose behind property attributions. (shrink)

In a nutshell, semantic antirealism is the doctrine that if a statement is true, then it must be possible, at least in principle, to determine that it is true. Consider the particular case of self-ascriptions of attitudes such as beliefs, desires and intentions, i.e. statements of the form "I φ [that] p", where φ ranges over propositional attitude verbs and p provides the content of whatever is φd by the self-ascriber. Should we be semantic antirealists about these when the putative (...) bearer of the attitude is the only individual who may retrieve a warrant in favour of his φing that p? We can't provide an answer to the question unless we're clear about (i) whether or not the "at least in principle" clause is too weak, and (ii) what the right construal of positive introspection should look like. 2 Thus two issues: strict finitism on the one hand and the phenomenological nature of introspective warrants on the other. I shall argue that recent views defended by Peacocke and Pryor are found wanting with respect to both. (shrink)

[An Anti-Realist Perspective on Language, Thought, Logic and the History of Analytic Philosophy. .].

Realism is the claim that truth may transcend all possible verification. The familiar Dummettian argument against that modal claim is that there is no way to manifest an understanding of it in actual linguistic practice. The Dummettian anti-realist's provisional conclusion is that the modal claim must be false. ;The attack on truth-conditional semantics and on the principle of bivalence are familiar ingredients of the anti-realist negative programme. I agree that, whether mathematical formulae or ordinary sentences in the past tense are (...) concerned, we cannot manifest a knowledge of the truth-conditions of particular instances of 'It is possible that ' by deciding their truth. There is nevertheless a way to argue in favour of specific instances of the modal claim when s is a sentence in the past tense. The proposed argument does not appeal to the recognitional abilities which, in the standard anti-realist perspective, constitute grasp of meaning. It does not commit one to the endorsement of the principle of bivalence, although the principle implies the modal claim. ;Our understanding of the workings of nature, whether grounded on scientific theories or on common sense, implies the modal claim. Since we are only contingently connected to the evidence pro or con past events and since the existence of evidence is itself a contingent matter, sentences which say that those events occurred could be true whether or not we can, at any point in time, detect that they are true. The possibility that truths about past facts transcend all possible verification is a purely natural possibility, grounded on what we know of the workings of nature. ;This implies that decidable statements in the past tense are not immune to the realism vs. anti-realism debate. Although they do not transcend verifiability, they could transcend whatever our recognitional abilities, no matter how extended, would allow us to establish at any point in time, either in the past, right now, or in an indefinitely extensible future. (shrink)

I contrast two construals of the thesis that truth is independent of verifiability in principle: a modal one and a non modal one. I argue in favor of the modal construal and then, on that basis, that independence holds across the board, i.e., even for statements that are verifiable by us relative to familiar, customary, non-skeptical standards.

May the theory of radical interpretation developed by Donald Davidson on the basis of Quine's arguments for the indeterminacy of translation help fix the meaning of the logical constants? In particular, may the theory exclude ways of conferring meaning on the constants which, although developed within the Davidsonian framework, would lead to unexpected results? Could an interpreter fix the meaning of the constants in a non classical way, although still in accordance with the guiding principles of the interpretative strategy? Or, (...) on the contrary, does the theory incorporate constraints on interpretation which are stronger than those imposed by the possibility of non classical, or deviant logics? I examine the particular case of negation within the framework of the natural deduction system developed by Gentzen and Prawitz, and conclude that Davidson's theory, understood as an empirical theory of truth, leaves the problem of knowing which meaning should be assigned to the constant for negation without a satisfactory solution. Although interpetation has to be carried out according to the principle of charity, disagreements may come up regarding which meaning should be assigned to "not" in a properly regimented natural language. (shrink)

Danielle Macbeth's purpose in Macbeth 2005 is threefold. Her monograph proposes "to provide a logical justification for all aspects of Frege's peculiar notation, to motivate and explain the developments in Frege's views over the course of his intellectual life, and to explicate his most developed, critically reflective conception of his Begriffschrift, his formula language of pure thought". I shall focus here on a few selected aspects of the first and third points and leave on the side the discussion of the (...) historical development of Frege's views on logic, semantics and mathematics. The book's underlying claim is that although Frege's logical language, "can of course be read as a language of quantificational logic, [it] can also be read very differently", so much so, as a matter of fact, that although Frege's logic is usually thought of as a "notational variant of standard quantificational logic", it is nevertheless "unknown to us". This naturally prompts three questions : "What is Frege's logic?", "What is quantificational logic?", and given the way we understand formal languages for first and higher order predicate logic, most notably the notions of argument 2 place, of scope and of binding : "How come Frege's logic says nothing about such notions, or something so different from what we say?". (shrink)

The interview took place in Oxford on 10 September 1992. While working from the tape on the text of the interview, I decided to gather references to books and articles in footnotes so that the reader may have a sense of the flow of the conversation. I then divided the text into sections, according to the topics which were discussed. Some material has been edited from the original transcript.

Professor Prawitz has made four claims in his talk. The first claim is that one should be able to generalize the intuitionistic theory of meaning already available for mathematical discourse to empirical discourse. Since each claim constitutes a step in an argument of a general form in favour of some new kind of antirealistically inclined theory of meaning (with a final pessimistic overtone), I shall go over each claim one by one, check whether the argument which links them in the (...) way described is sound, and assess on that basis the appropriateness of the project (and of the pessimism). (shrink)

There are several ways of conceiving objectivity -- scientific objectivity in particular -- and, accordingly, several ways of defending or attacking particular construals of it. According to one conception sometimes labelled "realism", objectivity in science is a semantic, modal and metaphysical notion: a scientific theory is objective insofar as it tells the truth about the way the world is independently of its epistemic accessibility to us. So, for instance, the Newtonian theory of gravition is objective insofar as it tells the (...) truth about the motion of particles submitted to gravitational forces. What it tells us, the realist contends, holds or is the case whether or not the physical world is accessible to us. The conception is semantic insofar as the notion of truth is involved, modal insofar as the conception of the possibility of a mind-independent natural world is involved and metaphysical insofar as a conception of a certain way the world is is involved. Philosophers as diverse as Berkeley, Kant and Dummett's antirealist will then ask: if this were the case, how could we know anything about it? How could we even form a bona fide conception of its possibility? (shrink)

According to the antirealist view of history, history is something historians construct in the present. Although the warrants they may gather in favour of past events do not form a coherent class, such warrants constitute the assertibility conditions of our statements about the past. They are by nature partial, gradual and defeasible. The antirealist is then faced with two problems. One is to account for a notion of historical significance, either in terms of causal links, broad patterns, or justified historiographic (...) generalizations. A second one is to secure a bound to the holistic construal of historical cognitive content. A coherent antirealist about history must explain why historical claims are in the market for truth, without taking the position of an ideal observer looking at the past from an a-historical fixed point outside time. (shrink)

Danielle Macbeth's purpose in Macbeth 2005 is threefold. Her monograph proposes "to provide a logical justification for all aspects of Frege's peculiar notation, to motivate and explain the developments in Frege's views over the course of his intellectual life, and to explicate his most developed, critically reflective conception of his Begriffschrift, his formula language of pure thought". I shall focus here on a few selected aspects of the first and third points and leave on the side the discussion of the (...) historical development of Frege's views on logic, semantics and mathematics. The book's underlying claim is that although Frege's logical language, "can of course be read as a language of quantificational logic, [it] can also be read very differently", so much so, as a matter of fact, that although Frege's logic is usually thought of as a "notational variant of standard quantificational logic", it is nevertheless "unknown to us". This naturally prompts three questions : "What is Frege's logic?", "What is quantificational logic?", and given the way we understand formal languages for first and higher order predicate logic, most notably the notions of argument 2 place, of scope and of binding : "How come Frege's logic says nothing about such notions, or something so different from what we say?". (shrink)

I offer several reasons for rejecting naturalism as a philosophical viewpoint or program envisaged for two paradigm cases: the case of mathematics and the case of ethics. Semantical, epistemological and metaphysical similarities between the two are investigated and assessed. I then offer a sketch of a different way of understanding the nature of mathematical difficulties and that of ethical puzzles.

As far as logic is concerned, the conclusion of Michael Dummett's manifestability argument is that intuitionistic logic, as first developed by Heyting, satisfies the semantic requirements of antirealism. The argument may be roughly sketched as follows: since we cannot manifest a grasp of possibly justification-transcendent truth conditions, we must countenance conditions which are such that, at least in principle and by the very nature of the case, we are able to recognize that they are satisfied whenever they are. Intuitionistic logic (...) satisfies the semantic requirement that we should either eschew the notion of truth altogether and replace it by provability in principle, or constrain it by provability in principle (Dummett [1973] 1978). (Handout: 1-5). Some philosophers have argued that the traditional antirealist desideratum of decidability in principle is too weak, so that semantic antirealism properly construed must be committed to effective decidability. As such, it either leads to strict finitism (Wright [1982] 1993) [Handout: 6] or to a much stronger kind of logical revisionism than the one considered by intuitionists (whether or not they accept the manifestability argument): substructural logics, and in particular linear logics, rather than intuitionistic logic, satisfy the semantic requirements of strict antirealism (Dubucs and Marion 2004) [Handout: 8-9]. I shall develop two different kinds of replies. The first is concerned with the notion of meaning per se and looks to strict finitism directly, although not on the ground that it would provide a correct way of dealing with Soritic-type paradoxes (the original and primary focus of discussion in Wright [1982] 1993). The second is concerned with the justification of structural and logical rules in a natural deduction system à la Gentzen. It will deal in particular with the criticism of the structural rules of Weakening and Contraction [Handout: 10 and 11]. The first kind of reply, which Dummett has partially taken into consideration, is that if we jettison the effectively vs. in principle distinction, as applied to manifestability-type arguments, we end up with an unsatisfactory explanation of how the meaning of statements covering the practically unsurveyable or pro tempora undecided cases is fixed. The idea is that if we have a method which may be used over some small range, we have determined a way of applying the method everywhere in principle and that this is enough as far as fixing meaning is concerned. Decidability in principle is just what we need with respect to manifestation of grasp of meaning. In this perspective, antirealism shouldn't be strict and manifestability-type arguments need not be applied as far as the strict finitist would want to [Handout: 7]. It follows that there is no reason to think that practical feasibility should be built in assertibility conditions and proofs be construed dynamically as acts in the strict antirealist's acception of that term [Handout: 8]. I shall then look at one radical antirealist principle disqualifying structural rules, namely Token Preservation, arguing against Bonnay and Cozic's criticisms of Dubucs and Marion (Bonnay and Cozic 3 2007) that some conceptual support may be provided for Token Preservation, which doesn't rely on a causal misreading of the turnstile [Handout: 13, 14]. I shall then assess the merits and limits of radical antirealism and the logic of feasible proofs with respect to the original Dummettian argument in favour of semantic antirealism (provided it has indeed revisionist implications for logic), whether the radical antirealist merely stipulates what human feasibility amounts to, or dispenses with structural rules in order to argue in favour of a curb on the epistemic idealizations they unwarrantedly embed. It will be noted here that there is a great difference, conceptually speaking, between the rejection of classical logic via the curbing of the epistemic idealizations embedded in structural rules, and the rejection of classical logic via the criticism of the introduction and elimination rules which fix the meaning of the classical constants. The kind of logical revisionism envisaged by intuitionists from Heyting on is in many respects stronger than the one envisaged by advocates of linear logic, should they ground their arguments on an endorsement of strict antirealism. The crucial case of excluded middle is telling. Because of the splitting of the constants in linear logic (Handout: 12], two distinct laws of excluded middle may be formulated (Handout: 15), but the traditional arguments of Brouwer and Heyting against the classical law yield a stronger revision than the substructural revision with its admission of these two (very) weak versions of the law. (Project: a clearer conception is needed of how the introduction and elimination rules for the logical connectives in the intuitionistic calculus depend on the structural rules which the radical antirealist wishes to reject.). (shrink)

Emotions are part of our culture ; particular emotions like resentment andguilt are part of specific cultural heritages. On the other hand, moral judgementsand imperatives have the appearance of objectivity. There lies - or so it seems -a conflict, even a contradiction. Statements like "Slavery is unjust" may beasserted, agreements may be reached concerning what they claim or express,and they may occur as antecedents in conditionals such as "If slavery is unjust,then it must be abolished". When it is claimed that (...) slavery is unjust or that it iswrong to harm others, it is thereby claimed that it is so objectively. We mean bythis (among other things) that it is so independently of the moral emotionsanyone may feel with respect to slavery or the infliction of pain.It will be argued here that there are grounds to believe that emotions must beevaluated cognitively and that they do have cognitive significance, value andcontent. In this cognitive or quasi-cognitive perspective, the appropriatecharacter of an emotion is akin to the truth of a proposition : just as the truth of aproposition may be evaluated in terms of the justifications we are able toprovide in its favour, the correctness of emotions may be evaluated relatively tocognitive bases and classes of available warrants. Some emotions may be ruledout as incorrect or inapproriate, just as some propositions may be rejected asunjustified or in need of a warrant. The parallel relies on the controversial notionof non-propositional forms of moral justification. The notion may be construedin a variety of ways, the most promising construals being, I believe, thoseprovided by the perceptual model.I shall focus on the following point. Moral judgements and imperatives areexpressed with the usual resources of natural languages. These typically includepredicates like "unjust" and "wrong", and deontic sentential operators like"must" and "should". Moral emotions, on the other hand, may be expressed bybodily movements and facial expressions. Although it may be argued thatemotions do not bear the cognitive marks of objectivity when expressed in thisway, there is no reason to think that a reformed language of ethics, i.e. a purelanguage of emotions, free of all the linguistic or symbolic marks of truth,assertability and objectivity, would thereby be free of normative input, or evenbe inadequate to express normative claims. I shall argue that once nonpropositionalforms of justifications are taken into account, the reformingstrategy of emotivists and neo-emotivists is indeed ineffective. (shrink)

According to semantic antirealism, intuitionistic logic satisfies the requirement that truth should be constrained by provability in principle. Some philosophers have argued that semantic antirealism must be committed to effective provability and that the commitment leads to a stronger kind of logical revisionism exemplified by substructural logics. I shall take into account two different kinds of reply. The first is concerned with meaning per se and grasp or fixing of meaning. It rests on the idea that if we have a (...) method which may be used over some surveyable range, we have determined a way of applying the method everywhere in principle, and that this is enough as far as fixing or grasping meaning is concerned. The second concerns two radical antirealist principles disqualifying structural rules: Token Preservation and Preservation of Local Feasibility. Against criticisms, I shall argue that conceptual support may be provided for both. The main point in this debate is that there is a decisive difference between the rejection of classical logic via the curbing of the epistemic idealizations embedded in structural rules and the rejection of classical logic via the intuitionistic criticism of invalid introduction and elimination rules. It will be explained why the second rejection is stronger than the former. (shrink)

Anti-realists about the past claim that no one has yet manifested a knowledge of the truth of tensed instances of the realist schema '‡,' instances such as '‡. It is true that we cannot decide specific instances of the realist schema and that, consequently, neither our understanding of these instances, nor our knowledge of their truth may be constituted by the recognitional and executive capacities which, according to Michael Dummett's antirealism, constitute grasp of meaning. Although we cannot decide these issues, (...) we can meet Dummett's anti-realist's manifestability challenge by arguing for them from contingency. While no recognitional and decisional skills may constitute our knowledge that their truth-conditions are satisfied, we can, without begging the question, derive that knowledge from our folk and scientific theories of the workings of nature. The evidence we have in favor of the fact that evidential relations between us and past facts are naturally contingent allows us to infer tensed instances of the fundamental realist modal claim. The joint possibility of truth and undecideability pro tempora is a natural possibility and, thereby, a logical and metaphysical possibility. (shrink)

Ontological parsimony requires that if we can dispense with A when best explaining B, or when deducing a nominalistically statable conclusion B from nominalistically statable premises, we must indeed dispense with A. When A is a mathematical theory and it has been established that its conservativeness undermines the platonistic force of mathematical derivations, or that a nonnumerical formulation of some explanans may be obtained so that the platonistic force of the best numerical-based account of the explanandum is also undermined, the (...) parsimony principle has been respected.Since both derivations resorting to conservative mathematics and nonnumerical best explanations also require abstract objects, concepts and principles, ontological parsimony must also be required of nominalistic accounts. One then might of course complain that such accounts turn out to be as metaphysically loaded as their platonistic counterparts. However, it might prove more fruitful to leave this particular worry to one side, to free oneself, as it were, from parsimony thus construed and to look at other important aspects of the defeating or undermining strategies that have been lavished on the disposal of platonism.Two aspects are worthy of our attention: epistemic cost and debunking arguments. Our knowledge that good mathematics is conservative is established at a cost, and so is our knowledge that nominalistic proofs play a theoretical role in best explanations. I will suggest that the knowledge one must acquire to show that nominalistic deductions and explanations do play their respective theoretical role involves some question-begging assumptions regarding the nature of proofs. As for debunking, even if the face value content of either conservative or platonistic mathematical claims didn’t figure in our explanation of why we hold the mathematical beliefs that we do, we could still be justified in holding them so that the distinction between nominalistic deductions and explanations and platonistic ones turns out to be invidious with respect to the relevant propositional attitude, i.e., with respect to belief. (shrink)

Although G¨odel proved the first incompleteness theorem by intuitionistically respectable means, G¨odel’s formula, true although undecidable,seems to offer a counter-example to the general constructivist or anti-realistclaim that truth may not transcend recognizability in principle. It is arguedhere that our understanding of the formula consists in a knowledge of itstruth-conditions, that it is true in a minimal sense and, finally, that it is recognized as such given the consistencyand ω-consistency of P. The philosophical lesson to be drawn from G¨odel’sproof is that (...) our capacities for justification in favour of minimal truth exceedwhat is strictly speaking formally provable in P by means of an algorithm. (shrink)

Ontological parsimony requires that if we can dispense with A when best explaining B, or when deducing a nominalistically statable conclusion B from nominalistically statable premises, we must indeed dispense with A. When A is a mathematical theory and it has been established that its conservativeness undermines the platonistic force of mathematical derivations, or that a non numerical formulation of some explanans may be obtained so that the platonistic force of the best numerical-based account of the explanandum is also undermined, (...) the parsimony principle has been respected. Since derivations resorting to conservative mathematics and proofs involved in non numerical best explanations also require abstract objects, concepts, and principles under the usual reading of “abstract,” one might complain that such accounts turn out to be as metaphysically loaded as their platonistic counterparts. One might then urge that ontological parsimony is also required of these nominalistic accounts. It might, however, prove more fruitful to leave this particular worry to the side, to free oneself, as it were, from parsimony thus construed and to look at other important aspects of the defeating or undermining strategies that have been lavished on the disposal of platonism. Two aspects are worthy of our attention: epistemic cost and debunking claims. Our knowledge that applied mathematics is conservative is established at a cost, and so is our knowledge that nominalistic proofs play a genuine theoretical role in best explanations. I will suggest that the knowledge one must acquire to show that nominalistic deductions and explanations do indeed play their respective theoretical role involves some question-begging assumptions regarding the nature and validity of proofs. As for debunking, even if the face value content of either non numerical claims, or conservative mathematical claims, or platonistic mathematical claims didn’t figure in our causal explanation of why we hold the mathematical beliefs that we do, construed or understood as beliefs about such contents, or as beliefs held in either of these three ways, we could still be justified in holding them, so that the distinction between nominalistic deductions or non numerical explanations on the one hand and platonistic ones on the other turns out to be spurious with respect to the relevant propositional attitude, i.e., with respect to belief. (shrink)

Selon le holisme sémantique, les propriétés sémantiques sont par nature anatomiques, c’est-à-dire intrinsèquement collectives. Selon le holisme de l’interprétation, la signification ou le contenu sont attribués collectivement. La thèse de constitution holistique de la signification peut être raisonnablement défendue à l’aide d’une contrainte de molécularité, qui introduit une hiérarchie dans la complexité logique des phrases, et un ordre ou une articulation dans le langage, en faisant droit à une condition finitaire d’anatomicité.According to semantic holism, semantic properties are anatomic by nature, (...) i.e. intrinsically collective. According to interpretation holism, meaning, or content, is collectively attributed. The thesis that meaning is holistically constituted may be defended provided we take into account a molecularity requirement. The requirement’s rôle is to introduce order or articulation in the langage, via a hierarchy of logical complexity, and by paying due attention to a finitary condition of anatomicity. (shrink)