Timothy Smiley’s wonderful paper ‘Rejection’ (1996) is still perhaps not as well known or well understood as it should be. This note first gives a quick presentation of themes from that paper, though done in our own way, and then considers a putative line of objection – recently advanced by Julien Murzi and Ole Hjortland (2009) – to one of Smiley’s key claims. Along the way, we consider the prospects for an intuitionistic approach to some of the issues discussed in (...) Smiley’s paper. (shrink)

Deflationists about truth hold that the function of the truth predicate is to enable us to make certain assertions we could not otherwise make. Pragmatists claim that the utility of negation lies in its role in registering incompatibility. The pragmatist insight about negation has been successfully incorporated into bilateral theories of content, which take the meaning of negation to be inferentially explained in terms of the speech act of rejection. We implement the deflationist insight in a bilateral theory by taking (...) the meaning of the truth predicate to be explained by its inferential relation to assertion. We combine this account of the meaning of the truth predicate with a new diagnosis of the liar paradox: its derivation requires the truth rules to preserve evidence, but these rules only preserve commitment. The result is a novel inferential deflationist theory of truth, which solves the liar paradox in a principled manner. We end by showing that our theory and simple extensions thereof have the resources to axiomatize the internal logic of several supervaluational hierarchies, including Cantini’s. This solves open problems of Halbach (2011) and Horsten (2011). (shrink)

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and (...) the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. (shrink)

We introduce the rejection game, designed to formalize the interaction between interlocutors in a Stalnakerian conversation: a speaker who asserts something and a listener who may accept or reject. The rejection game is similar to other signalling games known to the literature in economics and biology. We point out similarities and differences, and propose an application in linguistics. We uncover basic conditions under which the Gricean maxim of quality emerges from incentives among the players, providing evidence for a functionalist understanding (...) of the Gricean program. (shrink)

We present epistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati and Schlöder 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic S5 modulo an appropriate translation. The logical framework developed provides (...) the basis for a novel, proof-theoretic approach to the study of epistemic modality. To demonstrate the fruitfulness of the approach, we show how the framework allows us to reconcile classical logic with the contradictoriness of so-called Yalcin sentences and to distinguish between various inference patterns on the basis of the epistemic properties they preserve. (shrink)

We discuss a well-known puzzle about the lexicalization of logical operators in natural language, in particular connectives and quantifiers. Of the many logically possible operators, only few appear in the lexicon of natural languages: the connectives in English, for example, are conjunction _and_, disjunction _or_, and negated disjunction _nor_; the lexical quantifiers are _all, some_ and _no_. The logically possible nand (negated conjunction) and Nall (negated universal) are not expressed by lexical entries in English, nor in any natural language. Moreover, (...) the lexicalized operators are all upward or downward monotone, an observation known as the Monotonicity Universal. We propose a logical explanation of lexical gaps and of the Monotonicity Universal, based on the dynamic behaviour of connectives and quantifiers. We define update potentials for logical operators as procedures to modify the context, under the assumption that an update by \( \phi \) depends on the logical form of \( \phi \) and on the speech act performed: assertion or rejection. We conjecture that the adequacy of update potentials determines the limits of lexicalizability for logical operators in natural language. Finally, we show that on this framework the Monotonicity Universal follows from the logical properties of the updates that correspond to each operator. (shrink)

ABSTRACTLinguistic evidence supports the claim that certain, weak rejections are less specific than assertions. On the basis of this evidence, it has been argued that rejected sentences cannot be premisses and conclusions in inferences. We give examples of inferences with weakly rejected sentences as premisses and conclusions. We then propose a logic of weak rejection which accounts for the relevant phenomena and is motivated by principles of coherence in dialogue. We give a semantics for which this logic is sound and (...) complete, show that it axiomatizes the modal logic KD45 and prove that it still derives classical logic on its asserted fragment. Finally, we defend previous logics of strong rejection as being about the linguistically preferred interpretations of weak rejections. (shrink)

Deflationists about truth hold that the function of the truth predicate is to enable us to make certain assertions we could not otherwise make. Pragmatists claim that the utility of negation lies in its role in registering incompatibility. The pragmatist insight about negation has been successfully incorporated into bilateral theories of content, which take the meaning of negation to be inferentially explained in terms of the speech act of rejection. We implement the deflationist insight in a bilateral theory by taking (...) the meaning of the truth predicate to be explained by its inferential relation to assertion. We combine this account of the meaning of the truth predicate with a new diagnosis of the Liar Paradox: its derivation requires the truth rules to preserve evidence, but these rules only preserve commitment. The result is a novel inferential deflationist theory of truth, which solves the Liar Paradox in a principled manner. We end by showing that our theory and simple extensions thereof have the resources to axiomatize the internal logic of several supervaluational hierarchies, including Cantini’s. This solves open problems of Halbach (2011) and Horsten (2011). (shrink)

We present an inferentialist account of the epistemic modal operator might. Our starting point is the bilateralist programme. A bilateralist explains the operator not in terms of the speech act of rejection ; we explain the operator might in terms of weak assertion, a speech act whose existence we argue for on the basis of linguistic evidence. We show that our account of might provides a solution to certain well-known puzzles about the semantics of modal vocabulary whilst retaining classical logic. (...) This demonstrates that an inferentialist approach to meaning can be successfully extended beyond the core logical constants. (shrink)

We develop a novel solution to the negation version of the Frege-Geach problem by taking up recent insights from the bilateral programme in logic. Bilateralists derive the meaning of negation from a primitive *B-type* inconsistency involving the attitudes of assent and dissent. Some may demand an explanation of this inconsistency in simpler terms, but we argue that bilateralism’s assumptions are no less explanatory than those of *A-type* semantics that only require a single primitive attitude, but must stipulate inconsistency elsewhere. Based (...) on these insights, we develop a version of B-type expressivism called *inferential expressivism*. This is a novel semantic framework that characterises meanings by inferential roles that define which *attitudes* one can *infer* from the use of terms. We apply this framework to normative vocabulary, thereby solving the Frege-Geach problem generally and comprehensively. Our account moreover includes a semantics for epistemic modals, thereby also explaining normative terms under epistemic modals. (shrink)

Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, supervaluationists (...) can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic _NT_. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented. (shrink)

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, (...) I outline a different approach, which circumvents these problems by dispensing with the priority or dependence relation altogether. Along the way, I show how this approach enables the mathematical structuralist to defuse an objection recently raised against her view. (shrink)

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink)

The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA (...) is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation. (shrink)

We provide a framework for understanding agnosticism. The framework accounts for the varieties of agnosticism while vindicating the unity of the phenomenon. This combination of unity and plurality is achieved by taking the varieties of agnosticism to be represented by several agnostic stances, all of which share a common core provided by what we call the minimal agnostic attitude. We illustrate the fruitfulness of the framework by showing how it can be applied to several philosophical debates. In particular, several philosophical (...) positions can be aptly conceived of as instances of agnosticism whilst retaining their differences and distinguishing features. (shrink)

Maddy gave a semi-formal account of restrictiveness by defining a formal notion based on a class of interpretations and explaining how to handle false positives and false negatives. Recently, Hamkins pointed out some structural issues with Maddy's definition. We look at Maddy's formal definitions from the point of view of an abstract interpretation relation. We consider various candidates for this interpretation relation, including one that is close to Maddy's original notion, but fixes the issues raised by Hamkins. Our work brings (...) to light additional structural issues that we also discuss. (shrink)

The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.

Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...) and should be restricted as little as possible. The view might even have been held by Ernst Zermelo (1908), who,according to Penelope Maddy (1988), subscribed to a ‘one step back from disaster’ rule of thumb: if a natural principle leads to contra-diction, the principle should be weakened just enough to block the contradiction. We prove a generalization of McGee’s Theorem, anduse it to show that the situation for set theory is the same as that for truth: there are multiple incompatible sets of instances of Naïve Comprehension, none of which, given minimal assumptions, is recursively axiomatizable. This shows that the view adumbrated by Goldstein, Quine and perhaps Zermelo is untenable. (shrink)

Some bilateralists have suggested that some of our negative answers to yes-or-no questions are cases of rejection. Mark Textor (2011. Is ‘no’ a force-indicator? No! Analysis 71: 448–56) has recently argued that this suggestion falls prey to a version of the Frege-Geach problem. This note reviews Textor's objection and shows why it fails. We conclude with some brief remarks concerning where we think that future attacks on bilateralism should be directed.

Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist (...) want to attribute to it. I consider two strategies to deal with the problem --- one of which is outlined by McGee himself (2000) --- and argue that both of them fail. I end with some remarks on the prospects for mathematical realism in the light of my discussion. (shrink)

At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.

Anti-realists typically contend that truth is epistemically constrained. Truth, they say, cannot outstrip our capacity to know. Some anti-realists are also willing to make a further claim: if truth is epistemically constrained, classical logic is to be given up in favour of intuitionistic logic. Here we shall be concerned with one argument in support of this thesis - Crispin Wright's Basic Revisionary Argument, first presented in his Truth and Objectivity. We argue that the reasoning involved in the argument, if correct, (...) validates a parallel argument that leads to conclusions that are unacceptable to classicists and intuitionists alike. (shrink)

Leon Horsten has recently claimed that the class of mathematical truths coincides with the class of theorems of ZFC. I argue that the naturalistic character of Horsten’s proposal undermines his contention that this claim constitutes an analogue of a thesis that Daniel Isaacson has advanced for PA. I argue, moreover, that Horsten’s defence of his claim against an obvious objection makes use of a distinction which is not available to him given his naturalistic approach. I suggest a way out of (...) the objection which is in line with the naturalistic spirit of Horsten’s proposal but which further weakens the analogy with Isaacson’s Thesis. I conclude by evaluating the prospects for providing an analogue of Isaacson’s Thesis for ZFC. (shrink)

Starting from the case of insurance claims, I investigate the dynamics of acceptance, rejection and denial. I show that disagreement can be more varied than one might think. I illustrate this by looking at the Warren/Sanders controversy in the 2020 democratic primaries and at religious agnosticism.

A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. (...) We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic. (shrink)

According to the received view, formalism – interpreted as the thesis that mathematical truth does not outrun the consequences of our maximal mathematical theory – has been refuted by Goedel's theorem. In support of this claim, proponents of the received view usually invoke an informal argument for the truth of the Goedel sentence, an argument which is supposed to reconstruct our reasoning in seeing its truth. Against this, Field has argued in a series of papers that the principles involved in (...) this argument – when applied to our maximal mathematical theory – are unsound. This paper defends the received view by showing that there is a way of seeing the truth of the Goedel sentence which is immune to Field's strategy. (shrink)

Several philosophers have argued that the logic of set theory should be intuitionistic on the grounds that the open-endedness of the set concept demands the adoption of a nonclassical semantics. This paper examines to what extent adopting such a semantics has revisionary consequences for the logic of our set-theoretic reasoning. It is shown that in the context of the axioms of standard set theory, an intuitionistic semantics sanctions a classical logic. A Kripke semantics in the context of a weaker axiomatization (...) is then considered. It is argued that this semantics vindicates an intuitionistic logic only insofar as certain constraints are put on its interpretation. Wider morals are drawn about the restrictions that this places on the shape of arguments for an intuitionistic revision of the logic of set theory. (shrink)

Data involving epistemic modals suggest that some classically valid argument forms, such as _reductio_, are invalid in natural language reasoning as they lead to modal collapses. We adduce further data showing that the classical argument forms governing the existential quantifier are similarly defective, as they lead to a _de re–de dicto_ collapse. We observe a similar problem for disjunction. But if the classical argument forms for negation, disjunction and existential quantification are invalid, what are the correct forms that govern the (...) use of these items? Our diagnosis is that epistemic modals interfere with hypothetical reasoning. We present a modal first-order logic and model theory that characterizes hypothetical reasoning with epistemic modals in a principled manner. One upshot is a sound and complete natural deduction system for reasoning with epistemic modals in first-order logic. (shrink)

The long-standing dispute between absolutists and relativists traditionally focuses on whether there are absolute truths, absolute epistemic norms, and absolute.

This is a review of 'The Facts in Logical Space' by Jason Turner.

Recenze knihy:Luca Incurvati, Conceptions of Set and the Foundations of Mathematics. Cambridge University Press, 2020, 238 s.

Conceptions of Set and the Foundations of Mathematics By IncurvatiLucaCambridge University Press, 2020. xvi + 238 pp.

This is an introductory handbook for some of the main themes around the notion of predication in Aristotle. It does not aim at being exhaustive, but only sketches some important lines about the subject; it contains an introductory essay, besides the translation (into Portuguese) and commentary of basic texts (such as Posterior Analytics I-22, Categories 1-5, Interpretation 1-6 etc.).

This book examines phenomenal conservatism, one of the most influential and promising internalist conceptions of non-inferential justification debated in current epistemology and philosophy of mind. It also explores the significance of the findings of this examination for the general debate on epistemic justification. According to phenomenal conservatism, non-inferential justification rests on seemings or appearances, conceived of as experiences provided with propositional content. Phenomenal conservatism states that if it appears to S that P, in the absence of defeaters, S thereby has (...) some justification for believing that P. This view provides the basis for foundationalism and many ordinary epistemic practices. This book sheds new light on phenomenal conservatism by assessing objections to it and examining epistemological merits and advantages attributed to it. In a nutshell, phenomenal conservatism is actually compatible with Bayesian reasoning, and it is unaffected by bootstrapping problems and challenges that appeal to the cognitive penetrability of perception. Nevertheless, appearance-based justification proves unstable or elusive and its antisceptical bite is more limited than expected. These difficulties could be surmounted if phenomenal conservatism were integrated with a theory of inferential justification. The book appeals to scholars and postgraduates in the field of epistemology and philosophy of mind who are interested in the rational roles of appearances. (shrink)

In this paper we focus on transmission and failure of transmission of warrant. We identify three individually necessary and jointly sufficient conditions for transmission of warrant, and we show that their satisfaction grounds a number of interesting epistemic phenomena that have not been sufficiently appreciated in the literature. We then scrutinise Wright’s analysis of transmission failure and improve on extant readings of it. Nonetheless, we present a Bayesian counterexample that shows that Wright’s analysis is partially incoherent with our analysis of (...) warrant transmission and prima facie defective. We conclude exploring three alternative lines of reply: developing a more satisfactory account of transmission failure, which we outline; dismissing the Bayesian counterexample by rejecting some of its assumptions; reinterpreting Wright’s analysis to make it immune to the counterexample. (shrink)

I discuss a short string of five sentences in Metaphysics V.5, 1015b6-9 relating demonstration to necessity. My proposal is that Aristotle focuses his attention on the demonstration as a demonstration. Other interpretations reduce the necessity in question to the modality of the component sentences of the demonstrations (the conclusion and the premises). My view does not deny that the modality of the component sentences is important, but takes seriously the idea that a demonstration itself should be understood as necessary—as not (...) capable of being otherwise. A demonstration cannot be different from what it is in the sense that [i] its components cannot be different from what they are, [ii] its components must be related to each other exactly in the way they are related. Demonstrations aim at the fully appropriate explanation of a given explanandum—and each demonstration is individuated by the explanandum it takes. Thus, the basic idea is that, for the target explanandum that individuates a given demonstration, the premises delivering the fully appropriate explanation cannot be replaces with different ones. I show how this proposal, which explains Aristotle’s language in 1015b6-9 accurately, does not make demonstrations ‘melt down into conditional necessity’, first, because the modality of the component sentences is still importantly involved, second, because the explanatory relation expressed in a demonstration is a necessary fact in the real world, so that the demonstration itself is also necessary (in the way I have explained) inasmuch as it captures that fact. (shrink)

Mindreading (or folk psychology, Theory of Mind, mentalizing) is the capacity to represent and reason about others’ mental states. The Simulation Theory (ST) is one of the main approaches to mindreading. ST draws on the common-sense idea that we represent and reason about others’ mental states by putting ourselves in their shoes. More precisely, we typically arrive at representing others’ mental states by simulating their mental states in our own mind. This entry offers a detailed analysis of ST, considers theoretical (...) arguments and empirical data in favour of and against it, discusses its philosophical implications, and illustrates some alternatives to it. (shrink)

The assumption that coercion is largely responsible for our legal systems’ efficacy is a common one. I argue that this assumption is false. But I do so indirectly, by objecting to a thesis I call “(Compliance)”, which holds that most citizens comply with most legal mandates most of the time at least partly in virtue of being motivated by legal systems’ threats of sanctions and other unwelcome consequences. The relationship between (Compliance) and the efficacy of legal systems is explained in (...) section 1. There I also show that (Compliance) must be rejected for it relies on unsubstantiated empirical assumptions. In section 2, I claim that an alternative, and more refined, formulation of (Compliance) also lacks adequate support. I conclude with a few general remarks about the centrality of coercion in our thought and talk about legal systems. (shrink)

Translation of Aristotle's Metaphysics IV and VI, with notes. The translation is preliminary and intended as a provisional teaching tool to be also used in seminars and discussions with peers in order to reach a more elaborated version.

This chapter has two main sections. In the first section I briefly sketch Immanuel Kant’s moral theory as laid out in his Groundwork of the Metaphysics of Morals (1785). I explain Kant’s claim that morality must be grounded on what he calls a categorical imperative and examine his three formulations of this categorical imperative. In the second section I explain the distinction between “deontological” and “teleological” ethical theories. Kantian ethics is often presented as the paradigm example of a deontological ethical (...) theory, but I question whether Kant’s ethics should be understood as purely deontological. (shrink)

Transmission of justification across inference is a valuable and indeed ubiquitous epistemic phenomenon in everyday life and science. It is thanks to the phenomenon of epistemic transmission that inferential reasoning is a means for substantiating predictions of future events and, more generally, for expanding the sphere of our justified beliefs or reinforcing the justification of beliefs that we already entertain. However, transmission of justification is not without exceptions. As a few epistemologists have come to realise, more or less trivial forms (...) of circularity can prevent justification from transmitting from p to q even if one has justification for p and one is aware of the inferential link from p to q. In interesting cases this happens because one can acquire justification for p only if one has independent justification for q. In this case the justification for q cannot depend on the justification for p and the inferential link from p to q, as genuine transmission would require. The phenomenon of transmission failure seems to shed light on philosophical puzzles, such as Moore's proof of a material world and McKinsey's paradox, and it plays a central role in various philosophical debates. For this reason it is being granted continued and increasing attention. (shrink)

In this paper, I present an interpretation of Wittgenstein's remarks on the experience of wonder at the existence of the world. According to this interpretation, Wittgenstein's feeling of wonder stems from perceiving the existence of the world as an absolute miracle, that is, as a fact that is in principle beyond explanation. Based on this analysis, I will suggest that Wittgenstein's experience is akin to what has been described by other authors such as Coleridge, Pessoa, Heidegger, Scheler, Sartre, and Hadot, (...) among many others. Through a comparison between Wittgenstein and Coleridge on the experience of existence, I shall highlight some core features of this experience, chief among them the use of the notion of nothingness in clarifying what is understood in the experience, the role of intuition, and the presence of a specific pathos. As a whole, this paper aims to provide a contribution to the project of a phenomenology of the experience of existence and intends to create a bridge for a dialogue between Wittgenstein and the phenomenological tradition on the enigma of existence. (shrink)