Fabrice Pataut

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" (p. vii). 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" (p. vii), 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" (p. 1). 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 (for a predicate symbol), of scope and of binding (of variables - either individual or predicate of first or higher order): "How come Frege's logic says nothing about such notions, or something so different from what we say?"

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 (Field), 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 (Rizza), 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.

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 (Field), 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 (Rizza), 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.

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?"

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.

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.

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.

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