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Switch to: References
Citations of:
The Principles of Mathematics
Philosophy 13 (52):481-483 (1938)
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By the theory TT is meant the higher order predicate logic with the following recursively defined types: 1 is the type of individuals and [] is the type of the truth values: [$\tau_l$,..., $\tau_n$] is the type of the predicates with arguments of the types $\tau_l$,..., $\tau_n$. The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals (...) |
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Let serious propositional contingentism (SPC) be the package of views which consists in (i) the thesis that propositions expressed by sentences featuring terms depend, for their existence, on the existence of the referents of those terms, (ii) serious actualism—the view that it is impossible for an object to exemplify a property and not exist—and (iii) contingentism—the view that it is at least possible that some thing might not have been something. SPC is popular and compelling. But what should we say (...) |
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This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) |
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Alan Turing is often portrayed as a materialist in secondary literature. In the present article, I suggest that Turing was instead an idealist, inspired by Cambridge scholars, Arthur Eddington, Ernest Hobson, James Jeans and John McTaggart. I outline Turing’s developing thoughts and his legacy in the USA to date. Specifically, I contrast Turing’s two notions of computability (both from 1936) and distinguish between Turing’s “machine intelligence” in the UK and the more well-known “artificial intelligence” in the USA. According to my (...) |
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Causal slingshots are formal arguments advanced by proponents of an event ontology of token-level causation which, in the end, are intended to show two things: (i) The logical form of statements expressing causal dependencies on token level features a binary predicate ‘‘... causes ...’’ and (ii) that predicate takes events as arguments. Even though formalisms are only revealing with respect to the logical form of natural language statements, if the latter are shown to be adequately captured within a corresponding formalism, (...) |
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RésuméLa théorie des expressions incomplétes dans Principia Mathematica, se fonde sur le principe déja appliqué par Russell dans “On Denoting”, selon lequel il est souhaitable dans certains cas, ?on;établir le statut syntaxique des expressions catégorématiques. Grâce à la théorie intensionnelle ramifyée des types, les expressions incomplétes réféientiellement pourront être logiquement caractérisées par un mode de dérivation principalement basé sur la quantification non‐objectuelle. Ľintroduction des classes cependant, n'est en aucune façon reliée à ce mode intensionnel de dérivation; il en résulte qu'elles (...) |
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The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) |
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The view that the subject matter of epistemology is the concept of knowledge is faced with the problem that all attempts so far to define that concept are subject to counterexamples. As an alternative, this article argues that the subject matter of epistemology is knowledge itself rather than the concept of knowledge. Moreover, knowledge is not merely a state of mind but rather a certain kind of response to the environment that is essential for survival. In this perspective, the article (...) |
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This article introduces the first English translation of one of Tanabe’s early essays on metaphysics. It questions the relation of the universal to the particular in context of logic, phenomenology, Neo-Kantian epistemology, and classical metaphysics. Tanabe provides his reflections on the nature of the concept of universality and its constitutive relation to phenomenal particulars through critical analyses of the issue as it is discussed across various schools of philosophy including: British Empiricism, the Marburg School, the Austrian School, the Kyoto School, (...) |
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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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Dans le deuxième volume, inédit, de La Philosophie de l’Algèbre, Jules Vuillemin fait une lecture inattendue et suggestive de l’œuvre de Richard Dedekind. Nous avons essayé de comprendre, en mobilisant les idées et outils de Vuillemin, les résultats de cette lecture. Ceux-ci nous semblent poser en particulier le problème des rapports entre histoire des sciences et philosophie des sciences. Notre article propose un diptyque pour présenter les questions que nous avons voulu poser au texte de Vuillemin. D’une part, nous analysons (...) |
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Most work on the semantic paradoxes within classical logic has centered around what this essay calls “linguistic” accounts of the paradoxes: they attribute to sentences or utterances of sentences some property that is supposed to explain their paradoxical or nonparadoxical status. “No proposition” views are paradigm examples of linguistic theories, although practically all accounts of the paradoxes subscribe to some kind of linguistic theory. This essay shows that linguistic accounts of the paradoxes endorsing classical logic are subject to a particularly (...) |
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It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, (...) |
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Wittgenstein held that Russell’s multiple relation theory of judgment fails to explain an atomic judgment’s representation of entities as combined. He demonstrated this failure as follows. Under the multiple relation theory, an atomic judgment is a complex whose relating relation is judgment, the universal, and whose terms include the entities the judgment represents as combined. Taking such a complex we may arrive through the substitution of constituents at a complex whose relating relation is again judgment but whose terms do not (...) |
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El objetivo de este artículo es ofrecer una interpretación, en parte novedosa, del pasaje 5.54-5.5423 del Tractatus en el que Wittgenstein examina las proposiciones que representan relaciones intencionales como el juicio. La idea fundamental será que estas proposiciones se consideran como proposiciones que tratan de complejos y, en consecuencia, deberían analizarse de conformidad con el parágrafo 2.0201, lo que me llevará a desarrollar paso a paso el análisis propuesto. Adicionalmente, argumentaré que la teoría de Wittgenstein así entendida excluye la posibilidad (...) |
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This paper compares several models of formalization. It articulates criteria of correct formalization and identifies their problems. All of the discussed criteria are so called “semantic” criteria, which refer to the interpretation of logical formulas. However, as will be shown, different versions of an implicitly applied or explicitly stated criterion of correctness depend on different understandings of “interpretation” in this context. |
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If something changes from being in one state to being in another state, when exactly does it change? And what’s going on at that time? These questions are often discussed under the heading of the ‘moment’ or ‘instant’ of change. In this paper, I will investigate a view on which there is an intrinsically distinguished, atomic time at which something changes, and at that time it is metaphysically indeterminate what is the case. The background metaphysical picture is situationalism, a theory (...) |
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In this article, I show and discuss the relevance of Wittgenstein's arguments as to the spatial structure of sight to recent issues in the philosophy of mind. The first, bearing upon the dimensionality of the manifolds at play in depiction, plays a critical role in Clark's attempt to provide an independent account ofqualiaand of their differentiative properties. The second, pertaining to the properly spatial structure formed by the data of sight, is explicitly appealed to in the debate on the realistic (...) |
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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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This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) |
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In this paper, I motivate a modal account of propositions on the basis of an iterative conception of propositions. As an application, I suggest that the account provides a satisfying solution to the Russell-Myhill paradox. The account is in the spirit of recently developed modal accounts of sets motivated on the basis of the iterative conception of sets. |
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In his 1936 paper,On the Concept of Logical Consequence, Tarski introduced the celebrated definition oflogical consequence: “The sentenceσfollows logicallyfrom the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentenceσ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which (...) |
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Western language philosophy studies meaning from diverse aspects, with a core concern for how meaning is formulated and interpreted. The artificial-language and natural-language schools are two camps in this philosophical undertaking, the former insisting on scientific logic and positivism in meaning verification while the latter emphasizing subjective intention and context in meaning interpretation. Semiotics provides another semantic perspective that tips toward the theory of the natural-language school. This article compares the semantic thought of analytical language philosophers with that of a (...) |
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The claim that scientific explanation is deductive has been attacked on both systematic and historical grounds. This paper briefly defends the claim against the systematic attack. Essential to this defence is a distinction between perfect and imperfect explanation. This distinction is then used to illuminate the differences and similarities between Aristotelian (anthropomorphic) explanations of certain facts and those of classical mechanics. In particular, it is argued that when one attempts to fit classical mechanics into the Aristotelian framework the latter becomes (...) |
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I construe the question Are species sets? as a question about whether species can be conceived of as sets, as the term set is understood by contemporary logicians. The question is distinct from the question Are species classes?: The conception of classes invoked by Hull and others differs from the logician's conception of a set. I argue that species can be conceived of as sets, insofar as one could identify a set with any given species and that identification would satisfy (...) |
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The essay revisits the puzzle of the ‘passage’ of time in relation to EPR-type measurements and asks what philosophical consequences can be drawn from them. Some argue that the lack of invariance of temporal order in the measurement of a space-like related EPR pair, under relativistic motion, casts serious doubts on the ‘reality’ of the lapse of time. Others argue thatcertain features of quantum mechanics establisha tensed theory of time – understood here as Possibilism or the growing block universe. The (...) |
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This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem. |
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RÉSUMÉ: Les philosophes de la période moderne sont souvent présentés comme ayant commis une erreur élémentaire: celle de confondre la force propositionnelle avec le contenu propositionnel. Par l'examen de deux cas saillants, à savoir les philosophes de Port-Royal et John Locke, je montre que l'accusation n'est pas fondée, et que Locke en particulier a les ressources requises pour construire une théorie des attitudes propositionnelles. |
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While Russell’s concerns in developing the theory of descriptions were primarily with his foundation of logic, he was aware of the epistemological uses of both the theory of denoting concepts and the 1905 theory of deWnite descriptions. At the end of “On Denoting” he suggests that the principle of acquaintance is a “result” of the new theory of denoting. In this paper I examine the relation between the theory of descriptions and the principle of acquaintance, and I reject two suggestions, (...) |
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The paper focuses on the inferential role of quantifiers in Frege, Peano and Russell. Two aspects of the early years of mathematical logic are discussed: the gradual perfection of the principles of reasoning with quantifiers, and the presumed conceptual impossibility of posing metatheoretical questions, as embodied in Jean van Heijenoort’s well-known dictum about “logic as calculus and logic as language.”. |
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The paper focuses on the inferential role of quantifiers in Frege, Peano and Russell. Two aspects of the early years of mathematical logic are discussed: the gradual perfection of the principles of reasoning with quantifiers, and the presumed conceptual impossibility of posing metatheoretical questions, as embodied in Jean van Heijenoort’s well-known dictum about “logic as calculus and logic as language.”. |
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In questo paper cercherò di mostrare che la visione tradizionale del mutamento da parte di Russell della sua teoria delle descrizioni definite negli anni che vanno dai Principles of Mathematics del 1903 a «On Denoting» del 1905, visione secondo cui Russell produce una nuova teoria delle descrizioni (anche) per liberarsi dagli impegni ontologici manifesti di stampo meinonghiano ad entità inesistenti connessi alla sua precedente teoria delle descrizioni, non è convincente, perché le due teorie... |
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In this paper I want to claim, first, that despite close similarities, noneism and Crane’s psychological reductionism are different ontological doctrines. For unlike the latter, the former is ontologically committed to objects that are nonentities. Once one splits ontological from existential commitment, this claim, I guess, is rather uncontroversial. Second, however, I want to claim something more controversial; namely, that this ontological interpretation of noneism naturally makes noneism be nonstandardly read as a form of allism, to be however appropriately distinguished (...) |
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This paper surveys the interactions between Russell and Gödel, both personal and intellectual. After a description of Russell’s influence on Gödel, it concludes with a discussion of Russell’s reaction to the incompleteness theorems. |
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ABSTRACT The aim of this paper is to provide a sketch on the way Nāgārjuna deals with the idea of 'relation'. The concept of 'relation' as expressed in the Pāli sources is here theoretically systematized according to three patterns: 1. logical, 2. strictly subordinative existential, 3. non-strictly subordinative existential. After having discussed Nāgārjuna's acceptance and treatment of these three patterns, particular attention is paid to the non-strictly subordinative existential relation. This kind of relation is meant to describe the way the (...) |
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“The Strong Programme” is put forward as a metaphysical theory of sociology by the Edinburgh School (SSK) to study the social causes of knowledge. Barry Barnes and David Bloor are the proponents of the School. They call their programme “the Relativist View of Knowledge” and argue against rationalism in the philosophy of science. Does their relativist account of knowledge present a serious challenge to rationalism, which has dominated 20th century philosophy of science? I attempt to answer this question by criticizing (...) |
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Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) |
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Since the discovery of the paradoxes of Zeno, the problem of infinity was dominated by the meaning of endlessness—a view also adhered to by Herman Dooyeweerd. Since Aristotle, philosophers and mathematicians distinguished between the potential infinite and the actual infinite. The main aim of this article is to highlight the strengths and limitations of Dooyeweerd’s philosophy for an understanding of the foundations of mathematics, including Dooyeweerd’s quasi-substantial view of the natural numbers and his view of the other types of numbers (...) |
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This paper is about a certain view of intentionality, a problem faced by the view, and two ways in which, it has been proposed, the problem might be solved. The view is that every intentional state has an intentional object. The problem is that the putative intentional objects of some intentional states do not, or even cannot, exist. The two strategies to solve the problem and secure the view are those implemented by Tim Crane in his article “Intentional Objects”. In (...) |
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infinite, and offer several arguments in sup port of this thesis. I believe their arguments are unsuccessful and aim to refute six of them in the six sections of the paper. One of my main criticisms concerns their supposition that an infinite series of past events must contain some events separated from the present event by an infinite number of intermediate events, and consequently that from one of these infinitely distant past events the present could never have been reached. I (...) |
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Never shall this be proved, that things that are not are. ParmenidesTo say that something does not exist, or that there is something which is not, is clearly a contradiction in terms; hence “ ” must be true. Moreover, we should certainly expect leave to put any primitive name of our language for the “x” of any matrix “ … x … ”, and to infer the resulting singular statement from “ ”; it is difficult to contemplate any alternative logical (...) |
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The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology which combines a strong foundational requirement with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The (...) |
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In his 1936 paper,On the Concept of Logical Consequence, Tarski introduced the celebrated definition oflogical consequence: “The sentenceσfollows logicallyfrom the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentenceσ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which (...) |
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The purpose of this paper is to assess the prospects for a neo‐logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): ∀P∀Q[Ext(P) = Ext(Q) ≡ [(BAD(P) & BAD(Q)) ∨ ∀x(Px ≡ Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’.1 Background: what (...) |
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When an experimenter uses the experimental method to investigate the effects of the experimenter's expectancy it may be that this research, too, is affected by his expectancy and thus there is an expectancy paradox. To the extent that the experimenter expectancy effect accounts for the variation in the dependent variable and is general, that is to say, universal in psychological research, the expectancy paradox is ineluctable. However, an analysis of the research reviews extant in this area yields the conclusion that (...) |
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Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cognitive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The square is (...) |
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