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Is there anything like an experiment in mathematics? And if this is the case, what would distinguish a mathematical experiment from a mathematical thought experiment? In the present paper, a framework for the practice of mathematics will be put forward, which will consider mathematics as an experimenting activity and as a proving activity. The relationship between these two activities will be explored and more importantly a distinction between thought-experiments, real experiments, quasi experiments and proofs in pure mathematics will be provided. (...) |
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In this paper we present a solution to Saul Kripke’s Puzzle About Belief Meaning and use, Dordrecht, 1979) based on Kaplan’s metaphysical picture of words. Although it is widely accepted that providing such a solution was one of the main incentives for the development of Kaplan’s theory, it was never presented by Kaplan in a systematic manner and was regarded by many as unsatisfactory. We agree with these critiques, and develop an extension of Kaplan’s theory by introducing the notion of (...) |
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There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...) |
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Among philosophers, Fregeanism and sententialism are widely considered two of the leading theories of the semantics of attitude reports. Among linguists, these approaches have received little recent sustained discussion. This paper aims to bridge this divide. I present a new formal implementation of Fregeanism and sententialism, with the goal of showing that these theories can be developed in sufficient detail and concreteness to be serious competitors to the theories which are more popular among semanticists. I develop a modern treatment of (...) |
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It is often assumed that two linguistic agents can come to understand one another in part because they use the same words. That is, many philosophical theories of communication posit an intersubjective same-word relation. However, giving an account of this relation is complicated by what I call “The Variation Problem”—a problem resulting from the fact that the same word can be pronounced differently. In this paper, I first argue that previous models of the same-word relation, including Kaplanian and Chomskyan models, (...) |
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This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in (...) |
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Temporal ontology is concerned with the ontological status of the past, the present and the future, with presentism and eternalism as main contenders since the second half of the last century. In recent years several philosophers have argued that the presentism/eternalism dispute is not substantial. They have embraced, one may say, deflationism. Denying or downplaying the meaningfulness of tenseless language and wielding the so-called triviality objection have been their main argumentative tools. Other philosophers have opposed this trend, thereby holding fast (...) |
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Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) |
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During the past few decades, a radical shift has occurred in how philosophers conceive of the relation between science and philosophy. A great number of analytic philosophers have adopted what is commonly called a ‘naturalistic’ approach, arguing that their inquiries ought to be in some sense continuous with science. Where early analytic philosophers often relied on a sharp distinction between science and philosophy—the former an empirical discipline concerned with fact, the latter an a priori discipline concerned with meaning—philosophers today largely (...) |
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The strategy of this paper is twofold: First, we carry out a systematic investigation of the question of what specific kind of meaning quotation marks contribute to the overall meaning of an utterance. We consider the following kinds of meaning: literal meaning (§ 2.1), conventional implicature (§ 2.2), presupposition (§ 2.3), and conversational implicature (§ 2.4). We present arguments in favor of a pragmatic analysis of quotation marks, claiming that the notion of conversational implicature seems to be the most promising (...) |
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Building on recent work by Delia Graff Fara and Ora Matushansky on appellative constructions like ‘Mirka called Roger handsome’, I argue that if Millianism about proper names is true, then the quantifier ‘something’ in ‘Mirka called Roger something’ is best understood as a kind of substitutional quantifier. Any adequate semantics for such quantifiers must explain both the logical behavior of ‘Mirka called Roger something’ and the acceptability of ‘so’-anaphora in ‘Mirka called Roger something, and everyone so called is handsome’. Millianism (...) |
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Jim Mackenzie; Authority, Journal of Philosophy of Education, Volume 22, Issue 1, 30 May 2006, Pages 57–65, https://doi.org/10.1111/j.1467-9752.1988.tb00177.x. |
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O presente artigo busca debater a posição de Charles Sanders Peirce e dos primeiros estudantes peirceanos de Lógica (Christine Ladd e O. H. Mitchell nos Studies in Logic, 1883) dentro do debate inspirador da visão da linguagem dentro da Filosofia Analítica, conhecido como “Lingua Universalis contra Calculus Ratiocinator”, cujos primórdios podem ser traçados desde a filosofia de Gottfried Leibniz. Para isso, comparamos esse campo do pensamento peirceano com o debate crucial entre a conceitografia de Gottlob Frege (Begriffsschrift, 1879) e a (...) |
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Some philosophers, notably Professors Quine and Geach, have stressed the analogies they see between pronouns of the vernacular and the bound variables of quantification theory. Geach, indeed, once maintained that ‘for a philosophical theory of reference, then, it is all one whether we consider bound variables or pronouns of the vernacular'. This slightly overstates Geach's positition since he recognizes that some pronouns of ordinary language do function differently from bound variables; he calls such pronouns ‘pronouns of laziness'. Geach's characterisation of (...) |
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The mereological predicate ‘is part of’ can be used to define the predicate ‘is identical with’. I argue that this entails that mereological theories can be ideologically simpler than nihilistic theories that do not use the notion of parthood—contrary to what has been argued by Ted Sider. Moreover, if one accepts an extensional mereology, there are good philosophical reasons apart from ideological simplicity to give a mereological definition of identity. |
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Chapin reviewed this 1972 ZEITSCHRIFT paper that proves the completeness theorem for the logic of variable-binding-term operators created by Corcoran and his student John Herring in the 1971 LOGIQUE ET ANALYSE paper in which the theorem was conjectured. This leveraging proof extends completeness of ordinary first-order logic to the extension with vbtos. Newton da Costa independently proved the same theorem about the same time using a Henkin-type proof. This 1972 paper builds on the 1971 “Notes on a Semantic Analysis of (...) |
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In a series of recent works, Kit Fine, 605–631, 2003, 2007) has sketched a novel solution to Frege’s puzzle. Radically departing from previous solutions, Fine argues that Frege’s puzzle forces us to reject compositionality. In this paper we first provide an explicit formalization of the relational semantics for first-order logic suggested, but only briefly sketched, by Fine. We then show why the relational semantics alone is technically inadequate, forcing Fine to enrich the syntax with a coordination schema. Given this enrichment, (...) |
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In this paper, I examine Quine's views on the epistemology of logic. According to Quine's influential holistic account, logic is central in the “web of belief” that comprises our overall theory of the world. Because of this, revisions to logic would have devastating systematic consequences, and this explains why we are loath to make such revisions. In section1, I clarify this idea and thereby show that Quine actually takes the web of belief to have asymmetrical internal structure. This raises two (...) |
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Kit Fine has reawakened a puzzle about variables with a long history in analytic philosophy, labeling it “the antinomy of the variable”. Fine suggests that the antinomy demands a reconceptualization of the role of variables in mathematics, natural language semantics, and first-order logic. The difficulty arises because: (i) the variables ‘x’ and ‘y’ cannot be synonymous, since they make different contributions when they jointly occur within a sentence, but (ii) there is a strong temptation to say that distinct variables ‘x’ (...) |
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Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals (...) |
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Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”—making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an (...) |
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Recent scholarship indicates that Quine’s “Truth by Convention” does not present the radical critiques of analytic truth found fifteen years later in “Two Dogmas of Empiricism.” This prompts a historical question: what caused Quine’s radicalization? I argue that two crucial components of Quine’s development can be traced to the academic year 1940–1941, when he, Russell, Carnap, Tarski, Hempel, and Goodman were all at Harvard together. First, during those meetings, Quine recognizes that Carnap has abandoned the extensional, syntactic approach to philosophical (...) |
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We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set-theoretic (...) |
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Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it is (...) |
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This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) |
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The title of this paper may seem to involve a contradiction: my purpose is to show that it does not.Individuals fall into two categories; those which depend for their existence upon the existence of other individuals, and those which do not. In the second category are found such things as shoes, ships, cabbages, kings, and discrete bits of sealing wax. These may be calledindividual substances, and the way in which the existence of a cabbage depends upon water and earth, or (...) |
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Some philosophers, notably Professors Quine and Geach, have stressed the analogies they see between pronouns of the vernacular and the bound variables of quantification theory. Geach, indeed, once maintained that ‘for a philosophical theory of reference, then, it is all one whether we consider bound variables or pronouns of the vernacular'. This slightly overstates Geach's positition since he recognizes that some pronouns of ordinary language do function differently from bound variables; he calls such pronouns ‘pronouns of laziness'. Geach's characterisation of (...) |
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The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom (...) |
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Philosophical dialetheism, whose main exponent is Graham Priest, claims that some contradictions hold, are true, and it is rational to accept and assert them. Such a position is naturally portrayed as a challenge to the Law of Non-Contradiction (LNC). But all the classic formulations of the LNC are, in a sense, not questioned by a typical dialetheist, since she is (cheerfully) required to accept them by her own theory. The goal of this paper is to develop a formulation of the (...) |
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LetCbe an axiom system formalized within the first order functional calculus, and letC′ be related toCas the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. Ilse Novak [5] and Mostowski [8] have shown that, ifCis consistent, thenC′ is consistent. Mostowski has also proved the stronger result that any theorem ofC′ which can be formalized inCis a theorem ofC.The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction inCfrom a contradiction inC′. We could, (...) |
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Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much more, (...) |
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For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) |
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Sections 1 through 4 define, in the usual inductive style, various classes of object including one which is called the “combinatory terms of polymorphic type”. Section 5 defines a reduction relation on these terms. Section 6 shows that the weak normalizability of the combinatory terms of polymorphic type entails the weak normalizability of the lambda terms of polymorphic type. The entailment is not vacuous, because the combinatory terms of polymorphic type are indeed weakly normalizable, as is proven in Sect. 7 (...) |
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Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie... |
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Beginning from some passages by Vācaspati Miśra and Bhāskararāya Makhin discussing the relationship between a crown and the gold of which it is made, this paper investigates the complex underlying connections among difference, non-difference, coreferentiality, and qualification qua relations. Methodologically, philological care is paired with formal logical analysis on the basis of ‘Navya-Nyāya Formal Language’ premises and an axiomatic set theory-based approach. This study is intended as the first step of a broader investigation dedicated to analysing causation and transformation in (...) |
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In this paper, we show how to construct for a given consistent theory U a $$\varSigma ^0_1$$ Σ 1 0 -predicate that both satisfies the Löb Conditions and the Kreisel Condition—even if U is unsound. We do this in such a way that U itself can verify satisfaction of an internal version of the Kreisel Condition. |
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According to Hume’s principle, a sentence of the form ⌜The number of Fs = the number of Gs⌝ is true if and only if the Fs are bijectively correlatable to the Gs. Neo-Fregeans maintain that this principle provides an implicit definition of the notion of cardinal number that vindicates a platonist construal of such numerical equations. Based on a clarification of the explanatory status of Hume’s principle, I will provide an argument in favour of a nominalist construal of numerical equations. (...) |
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In his “Wissenschaftslogik: The Role of Logic in the Philosophy of Science,” Michael Friedman argues that Carnap’s philosophy of science “is fundamentally anti-metaphysical—he aims to use the tools of mathematical logic to dissolve rather [than] solve traditional philosophical problems—and it is precisely this point that is missed by his logically-minded contemporaries such as Hempel and Quine”. In this paper, I take issue with this claim, arguing that Quine, too, is a part of this anti-metaphysical tradition. I begin in section I (...) |
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Montague and Kaplan began a revolution in semantics, which promised to explain how a univocal expression could make distinct truth-conditional contributions in its various occurrences. The idea was to treat context as a parameter at which a sentence is semantically evaluated. But the revolution has stalled. One salient problem comes from recurring demonstratives: "He is tall and he is not tall". For the sentence to be true at a context, each occurrence of the demonstrative must make a different truth-conditional contribution. (...) |
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This paper systematically outlines Wittgenstein's ab-notation. The purpose of this notation is to provide a proof procedure in which ordinary logical formulas are converted into ideal symbols that identify the logical properties of the initial formulas. The general ideas underlying this procedure are in opposition to a traditional conception of axiomatic proof and are related to Peirce's iconic logic. Based on Wittgenstein's scanty remarks concerning his ab-notation, which almost all apply to propositional logic, this paper explains how to extend his (...) |
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