Paul Égré

Belnap (1970, 1973) proposed to formalize the restriction of first-order quantifiers ∀ and ∃ by means of a single sentential connective: a trivalent conditional that takes the semantic value ‘void’ when the antecedent is false. Thus, ‘every A is B’ is represented as ∀x(Ax → Bx) and ‘some A is B’ is represented as ∃x(Ax → Bx)—a notable unification compared to the standard representation in first-order logic that uses distinct connectives. This paper implements Belnap’s program, henceforth called Conditional Reduction, in full generality and applies it to all conservative ⟨1, 1⟩ quantifiers. By combining Conditional Reductions with the Conservativity Theorem, due to Keenan and Stavi, any conservative type ⟨1, 1⟩ quantifier turns out to be equivalent to Boolean combinations of three basic type ⟨1⟩ quantifiers (‘everything’, ‘at least λ many things’ and the possessives) scoping over a trivalent conditional.

Belnap (1970, 1973) proposed to formalize the restriction of first-order quantifiers ∀ and ∃ by means of a single sentential connective: a trivalent conditional that takes the semantic value ‘void’ when the antecedent is false. Thus, ‘every A is B’ is represented as ∀x(Ax → Bx) and ‘some A is B’ is represented as ∃x(Ax → Bx)—a notable unification compared to the standard representation in first-order logic that uses distinct connectives. This paper implements Belnap’s program, henceforth called Conditional Reduction, in full generality and applies it to all conservative ⟨1, 1⟩ quantifiers. By combining Conditional Reductions with the Conservativity Theorem, due to Keenan and Stavi, any conservative type ⟨1, 1⟩ quantifier turns out to be equivalent to Boolean combinations of three basic type ⟨1⟩ quantifiers (‘everything’, ‘at least λ many things’ and the possessives) scoping over a trivalent conditional.

This chapter gives some elements of the history of trivalent logics and presents the key technical notions. We stress that Boole and Frege were aware of reasons to go beyond bivalence, in ways that influenced Łukasiewicz in particular. Then we put particular emphasis on the 1930s as a pivotal moment in the application of trivalence to a range of interconnected phenomena, such as probability and hypothetical reasoning, quantum indeterminacy, computability theory, and the semantic paradoxes. The chapter goes on to present basic elements regarding the connection of trivalent logics to other logics, including intuitionistic logic, modal logics, fuzzy and many-valued logics, and finally to classical and non-classical logics more broadly. Along the way, we present basic systems of mixed consequence, and provide glimpses into the theory of ternary computation.

Résumé — Toute vérité est-elle connaissable en principe ? Une réponse négative à cette question suit d’un argument logique dû à F. Fitch, voisin du paradoxe de Moore, et connu sous le nom de paradoxe de la connaissabilité. Le paradoxe de Fitch constitue un obstacle à la conception antiréaliste de la vérité et, plus généralement, semble-t-il, à l’idéal positiviste d’après lequel toute vérité devrait nous être accessible en principe. Dans cet article, j’examine différentes tentatives pour préserver le principe selon lequel toute vérité est connaissable, chacune inspirée d’une forme propre d’antiréalisme . Ces différentes approches sont comparées à une conception réaliste du positivisme, évoquée récemment par Burgess, qui postulerait seulement que toute vérité nécessaire est connaissable. Selon cette conception, certaines vérités contingentes sont effectivement inconnaissables, mais l’accessibilité de principe des vérités nécessaires demeure suffisante pour garantir la confiance positiviste dans la science.— Are all truths knowable ? A negative answer to this question follows from a logical argument related to the Moore Paradox and due to F. Fitch, also known as Fitch’s Knowability Paradox. Fitch’s paradox is widely considered to be an obstacle to the antirealist conception of truth, but even for a realist, it appears to threaten the positivist faith in the accessibility of all truths to our minds. In this paper, I first review different strategies to circumvent the paradox, each of them inspired by a different form of antirealism . I then compare these approaches to a realist version of positivism, discussed recently by Burgess, which would only postulate that all necessary truths are knowable. On that view, some contingent truths are indeed unknowable, but the idea that all necessary truths are within our reach remains sufficient to maintain the positivist faith in scientific knowledge

According to Aristotle, virtue is the mean between deficiency and excess. While Aristotle’s concept of “mean” has been the subject of some debate, his geometric and arithmetic comparisons suggest a natural interpretation according to which the virtuous mean lies on a straight path connecting the extremes. We critically discuss this interpretation, first by focusing on the mathematical elaboration of the concept of mean offered by Aristotle, and then by investigating how contemporary English speakers conceptualize the relation between some of the moral character traits discussed in the Nicomachean Ethics. Our main finding, based on studies of five Aristotelian triads, is that for most of them, alleged extreme traits tend to be roughly as opposed to each other as they are to the alleged mean. We use these findings to defend a more nuanced interpretation of the doctrine of the mean, situating the relevant character traits in a higher-dimensional space. This higher-dimensional interpretation remains compatible with the Aristotelian idea that reaching the mean involves finding the right balance between several dimensions. However, a philosophical consequence we draw is that simply moving away from deficiency toward excess may not always bring one closer to virtue. Going from one extreme to the other will sometimes lead one further astray and is no substitute for aiming at virtue proper.

This chapter is an enquiry into the goals and methods of linguistics, with the aim of understanding both the specifics of the discipline and the relationships between linguists’ take on the methodology of their field and general principles of philosophy of science. The first part highlights linguistics as an inquiry into language, as opposed to languages. The second part describes the shift from structural linguistics to generative grammar as a paradigm shift, involving major changes in both what is studied and how it is studied. The next section focuses on the empirical import of contemporary linguistics, discussing standards of explanation and prediction, as well as confirmation and refutation of linguistic hypotheses. The last part introduces linguistic universals, what they are and how they may be identified and explained, thus making explicit the connection with the goal of understanding not only the variety of languages but the faculty of language.

Trivalent theories of indicative conditionals are thought to be at odds with trivalent theories of presupposition, and in particular, with an adequate analysis of the presuppositions of conditionals. We address this challenge based on (i) a semantics for conditionals and modals that combines trivalence with context updates, and (ii) a principle for felicitous assertion that we call Avoid Void: do not assert a sentence that takes the value void everywhere in the context. This account highlights the semantic differences between conditionals and other presupposition carriers, and yields the intuitively correct presuppositions of conditionals. Finally, we develop a trivalent logic C# ("C-sharp"), which implements Avoid Void at the level of reasoning. This yields a Strawsonian notion of entailment, on which false antecedents as well as contradictory premises do not allow one to infer arbitrary information.

This paper provides a new framework for formalizing conditional obligations in natural language: it pairs a unary deontic operator with trivalent semantics for the indicative conditional and Kratzer's assumption that the antecedents of conditionals restrict the scope of modals in the consequent. Combining these three ideas, we obtain a fully compositional theory of "if" and "ought'" that validates plausible principles for deontic reasoning. Moreover, it resolves classical challenges such as the "if A then ought A" problem, the paradox of the miners and the modeling of contrary-to-duty obligations (e.g., Chisholm's quartet). All in all, our proposal is attractive from a logical point of view and squares well with general theories of natural language reasoning.

This paper introduces a systems engineering framework for understanding the fundamental principles of counting and the nature of natural numbers, arguing that traditional axiomatic approaches overlook the essential functional and computational components. We define counting as a Wittgensteinian "counting game" in which an agent, the Counter, must employ robust perception and classification capabilities within a given environment. Central to this approach is the claim that counting is a stateful computational activity that requires memory, leading to the definition of a natural number as the unique state of a counting system stored in that memory. We examine the role of symbolic representation, identifying numerals as efficient encoding schemes for these numerical states, and argue that this analysis contributes to improving symbolic features in Artificial Intelligence (AI) systems. Furthermore, our model contributes to a long-standing philosophical debate, suggesting that ordinals come first rather than cardinals due to practical memory storage limitations.

One important aspect of conceptual engineering concerns the evaluation of concepts for usefulness, which lets us decide whether to keep those concepts or to revise them. Among the concepts that pose a particular challenge for that project are empty concepts. These are concepts like ‘unicorn’, or ‘phlogiston’, that do not refer to any actual objects. We use such concepts quite often—but how could they be useful when they do not refer to anything, and how could they come about in the first place? In earlier work, we proposed an account of concept utility that treated concepts as groupings of objects, implying that empty concepts would have zero utility, and leaving no explanation for how they could arise. While we believe they indeed have no utility in natural contexts once known to be empty, here we extend our theory in two ways to account for empty concepts. We show how they can first arise—as hypothetical groupings of features rather than objects; and we show how they can have a measurable utility—in scientific theorizing, and in fiction.

Editorial: Objects and Sound Perception Content Type Journal Article Pages 5-17 DOI 10.1007/s13164-009-0006-3 Authors Nicolas J. Bullot, École des Hautes Études en Sciences Sociales Centre de Recherches sur les Arts et le Langage (CRAL/CNRS) 96 Bd Raspail 75006 Paris France Paul Égré, Institut Jean-Nicod (ENS/EHESS/CNRS) Département d’Etudes Cognitives de l’ENS 29 rue d’Ulm 75005 Paris France Journal Review of Philosophy and Psychology Online ISSN 1878-5166 Print ISSN 1878-5158 Journal Volume Volume 1 Journal Issue Volume 1, Number 1.

The principle of tolerance characteristic of vague predicates is sometimes presented as a soft rule, namely as a default which we can use in ordinary reasoning, but which requires care in order to avoid paradoxes. We focus on two ways in which the tolerance principle can be modeled in that spirit, using special consequence relations. The first approach relates tolerant reasoning to nontransitive reasoning; the second relates tolerant reasoning to nonmonotonic reasoning. We compare the two approaches and examine three specific consequence relations in relation to those, which we call:strict-to-tolerantentailment,pragmatic-to-tolerantentailment, andpragmatic-to-pragmaticentailment. The first two are nontransitive, whereas the latter two are nonmonotonic.

This paper explores the relations between two logical approaches to vagueness: on the one hand the fuzzy approach defended by Smith (Vagueness and degrees of truth. Oxford University Press, Oxford, 2008), and on the other the strict-tolerant approach defended by Cobreros et al. (“Tolerant, Classical, Strict”, J Philos Logic 41(2):347–385, 2012). Although the former approach uses continuum many values and the latter implicitly four, we show that both approaches can be subsumed under a common three-valued framework. In particular, we defend the claim that Smith’s continuum many values are not needed to solve what Smith calls ‘the jolt problem’, and we show that they are not needed for his account of logical consequence either. Not only are three values enough to satisfy Smith’s central desiderata, but they also allow us to internalize Smith’s closeness principle in the form of a tolerance principle at the object-language. The reduction, we argue, matters for the justification of many-valuedness in an adequate theory of vague language.

In a previous paper (see ‘Tolerant, Classical, Strict’, henceforth TCS) we investigated a semantic framework to deal with the idea that vague predicates are tolerant, namely that small changes do not affect the applicability of a vague predicate even if large changes do. Our approach there rests on two main ideas. First, given a classical extension of a predicate, we can define a strict and a tolerant extension depending on an indifference relation associated to that predicate. Second, we can use these notions of satisfaction to define mixed consequence relations that capture non-transitive tolerant reasoning. Although we gave some empirical motivation for the use of strict and tolerant extensions, making use of them commits us to the view that sentences of the form ‘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p {\vee} {\neg} p}$$\end{document}’ and ‘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p {\wedge} {\neg} p}$$\end{document}’ are not automatically valid or unsatisfiable, respectively. Some philosophers might take this commitment as a negative outcome of our previous proposal. We think, however, that the general ideas underlying our previous approach to vagueness can be implemented in a variety of ways. This paper explores the possibility of defining mixed notions of consequence in the more classical super/sub-valuationist setting and examines to what extent any of these notions captures non-transitive tolerant reasoning.

We present an investigation of the ways in which speakers' subjective perspectives are likely to affect the meaning of gradable adjectives like tall or heavy. We present the results of a study showing that people tend to use themselves as a yardstick when ascribing these adjectives to human figures of varied measurements: subjects' height and weight requirements for applying tall and heavy are found to be positively correlated with their personal measurements. We draw more general lessons regarding the definition of subjectivity and the ways in which a standard of comparison and a significant deviation from that standard are specified.

Why is ordinary language vague? We argue that in contexts in which a cooperative speaker is not perfectly informed about the world, the use of vague expressions can offer an optimal tradeoff between truthfulness (Gricean Quality) and informativeness (Gricean Quantity). Focusing on expressions of approximation such as “around”, which are semantically vague, we show that they allow the speaker to convey indirect probabilistic information, in a way that can give the listener a more accurate representation of the information available to the speaker than any more precise expression would (intervals of the form “between”). That is, vague sentences can be _more informative_ than their precise counterparts. We give a probabilistic treatment of the interpretation of “around”, and offer a model for the interpretation and use of “around”-statements within the Rational Speech Act (RSA) framework. In our account the shape of the speaker’s distribution matters in ways not predicted by the Lexical Uncertainty model standardly used in the RSA framework for vague predicates. We use our approach to draw further lessons concerning the semantic flexibility of vague expressions and their irreducibility to more precise meanings.

This paper extends a trivalent semantics for indicative conditionals to a language including the modal operators "might" and "must". Specifically, we combine Cooper's (1968) truth-functional, trivalent analysis of the conditional connective with Kratzer's (1986, 2012) idea that if-clauses restrict modal operators. By hard-wiring both trivalence and the restriction operation into the truth conditions of conditional-modal expressions, we obtain an attractive theory that yields plausible predictions for the interaction of conditionals and modals, explains the intuitive appeal of the Restrictor View and avoids the typical problems of Kratzer-style accounts, especially regarding the probability of conditional-modal expressions.

We say that a sentence A is a permissive consequence of a set X of premises whenever, if all the premises of X hold up to some standard, then A holds to some weaker standard. In this paper, we focus on a three-valued version of this notion, which we call strict-to-tolerant consequence, and discuss its fruitfulness toward a unified treatment of the paradoxes of vagueness and self-referential truth. For vagueness, st-consequence supports the principle of tolerance; for truth, it supports the requisite of transparency. Permissive consequence is non-transitive, however, but this feature is argued to be an essential component to the understanding of paradoxical reasoning in cases involving vagueness or self-reference.

The set of $$\textsf{ST}$$ ST -valid inferences is neither the intersection, nor the union of the sets of $$\textsf{K}_3$$ K 3 -valid and $$\textsf{LP}$$ LP -valid inferences, but despite the proximity to both systems, an extensional characterization of $$\textsf{ST}$$ ST in terms of a natural set-theoretic operation on the sets of $$\textsf{K}_3$$ K 3 -valid and $$\textsf{LP}$$ LP -valid inferences is still wanting. In this paper, we show that it is their relational product. Similarly, we prove that the set of $$\textsf{TS}$$ TS -valid inferences can be identified using a dual notion, namely as the relational sum of the sets of $$\textsf{LP}$$ LP -valid and $$\textsf{K}_3$$ K 3 -valid inferences. We discuss links between these results and the interpolation property of classical logic. We also use those results to revisit the duality between $$\textsf{ST}$$ ST and $$\textsf{TS}$$ TS. We present a notion of duality on which $$\textsf{ST}$$ ST and $$\textsf{TS}$$ TS are dual in exactly the same sense in which $$\textsf{LP}$$ LP and $$\textsf{K}_3$$ K 3 are dual to each other.

This paper develops a trivalent semantics for the truth conditions and the probability of the natural language indicative conditional. Our framework rests on trivalent truth conditions first proposed by Cooper (1968) and Belnap (1973) and it yields two logics of conditional reasoning: (i) a logic C of certainty-preserving inference; and (ii) a logic U for uncertain reasoning that preserves the probability of the premises. We show systematic correspondences between trivalent and probabilistic representations of inferences in either framework, and we use the distinction between the two systems to cast light on the validity of inferences such as Modus Ponens, Or-To-If, and Conditional Excluded Middle. Specifically, the conditional behaves monotonically in C, but non-monotonically in U; Modus Ponens is valid in C, but valid in U only for non-nested conditionals. The result is a unified account of the semantics and epistemology of indicative conditionals that can be fruitfully applied to analyzing the validity of conditional inferences.

We present the results of four experiments concerning the evaluation people make of sentences involving “many”, showing that two sentences of the form “many As are Bs” vs. “many As are Cs” need not be equivalent when evaluated relative to a background in which B and C have the same cardinality and proportion to A, but in which B and C are predicates with opposite semantic and affective values. The data provide evidence that subjects lower the standard relevant to ascribe “many” for the more negative predicate, and that judgments involving “many” are sensitive to moral considerations, namely to expectations involving a representation of the desirability as opposed to the mere probability of an outcome. We relate the results to similar semantic asymmetries discussed in the psychological literature, in particular to the Knobe effect and to framing effects.

This paper examines an hypothesis put forward by Pettit and Knobe 2009 to account for the Knobe effect. According to Pettit and Knobe, one should look at the semantics of the adjective “intentional” on a par with that of other gradable adjectives such as “warm”, “rich” or “expensive”. What Pettit and Knobe’s analogy suggests is that the Knobe effect might be an instance of a much broader phenomenon which concerns the context-dependence of normative standards relevant for the application of gradable expressions. I adduce further evidence in favor of this view and go on to examine the predictions one obtains when assuming that “intentional” involves a two-dimensional scale, which implies evaluating how much an action or outcome is desired on the one hand, and how much it can be foreseen as a consequence of one’s actions on the other.

Eubulide de Milet est connu comme l’auteur d’un ensemble considérable d’arguments logiques déconcertants, encore appelés sophismes ou paradoxes. Parmi ceux-ci figurent l’antinomie du Menteur et le paradoxe du Sorite. Peut-on donner une solution unifiée de ces paradoxes? La réponse examinée dans cet essai est positive et vise à faire connaître la théorie stricte-tolérante, développée à l’origine pour résoudre le Sorite avant d’être transposée au Menteur. Selon cette théorie, la conclusion d’un argument suit logiquement des prémisses quand la vérité stricte des prémisses implique la vérité tolérante de la conclusion. Si les notions de vérité stricte et tolérante coïncident dans un cadre bivalent, elles se recouvrent seulement partiellement dans un cadre trivalent. Nous expliquons en quel sens la théorie stricte-tolérante confère aux notions mêmes de vérité et de conséquence logique un caractère d’équivocité. Or l’identification d’une équivocité préside à la solution d’autres paradoxes, comme dans l’argument du Cornu également dû à Eubulide.