This paper argues against Oaksford and Chater's claim that logicist cognitive science is not possible. It suggests that there arguments against logicist cognitive science are too closely tied to the account of Pylyshyn and of Fodor, and that the correct way of thinking about logicist cognitive science is in a mental models framework.

According to the received view, Russell rediscovered about 1900 the logical definition of cardinal number given by Frege in 1884. In the same way, we are told, he stated and developed independently the idea of logicism, using the principle of abstraction as the philosophical ground. Furthermore, the role commonly ascribed in this to Peano was only to invent an appropriate notation to be used as mere instrument. In this paper I hold that the study of Russell's unpublished manuscripts and Peano's (...) (and disciples) writings (as part of a larger investigation only pointed out here) shows, on the contrary, that Russell obtained the method to transform definitions by abstraction into nominal definitions and the general logicist idea from Peano (and his school). The only original insight from Russell, the principle of abstraction, partially derived from Moore's early philosophy, was finally abandoned because it was not of practical use. (shrink)

The paper focuses on the lectures on the philosophy of mathematics delivered by Wittgenstein in Cambridge in 1939. Only a relatively small number of lectures are discussed, the emphasis falling on understanding Wittgenstein’s views on the most important element of the logicist legacy of Frege and Russell, the definition of number in terms of classes—and, more specifically, by employing the notion of one-to-one correspondence. Since it is clear that Wittgenstein was not satisfied with this definition, the aim of the essay (...) is to propose a reading of the lectures able to clarify why that was the case. This reading shows that his better known views on language and mind expressed in Philosophical Investigations illuminate his conception of mathematics. (shrink)

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 (...) and why?2 Framework3 GOOD candidates, indefinite extensibility4 The framework of (RV) alone, or almost alone5 The axioms6 Brief closing. (shrink)

The paper argues that Frege’s primary foundational purpose concerning arithmetic was neither that of making natural numbers logical objects, nor that of making arithmetic a part of logic, but rather that of assigning to it an appropriate place in the architectonics of mathematics and knowledge, by immersing it in a theory of numbers of concepts and making truths about natural numbers, and/or knowledge of them transparent to reason without the medium of senses and intuition.

The question, “Was Aristotle the first logicist?”, may appear anachronistic and elicit skepticism since the doctrine of logicism as a fully-fledged idea emerged only in the nineteenth century in the context of the debates surrounding the foundation of mathematics. Indeed, Bertrand Russell credits Gottlob Frege with being the first in “logicising” mathematics (Russell 1919, p. 7), where the thesis espouses that mathematical concepts and propositions are ultimately reducible to or derivable from a number of fundamental logical concepts and principles. However, (...) anachronistic appearances aside, in a fresh reexamination of some of the specific Aristotelian texts in Metaphysics and Prior Analytics, and especially focusing on Aristotle’s particular remarks on the status and significance of the principle of non-contradiction, one may textually argue for a nascent and burgeoning form of logicism in Aristotle, albeit within a much larger metaphysical context than mathematics. (shrink)

L'AUTEUR PROCEDE EN TROIS ETAPES A UNE LECTURE HISTORIQUE DE LA PHILOSOPHIE MATHEMATIQUE DE RUSSELL. *LA PREMIERE PHASE - PRINCIPLES OF MATHEMATICS, 1903 - OPERE LA CONSTRUCTION DE LA LOGIQUE FORMELLE ET LA REDUCTION DES MATHEMATIQUES A CETTE NOUVELLE LOGIQUE. RUSSELL PREND POUR GUIDE LA GRAMMAIRE PHILOSOPHIQUE POUR ELABORER SA LOGIQUE, DEVELOPPE UNE CONCEPTION REFERENTIELLE DE LA SIGNIFICATION ET ADOPTE UNE PHILOSOPHIE REALISTE (RELATIONS EXTERNES, ATOMISME LOGIQUE...) *LA SECONDE PHASE -"ON DENOTING", 1905- MONTRE COMMENT LA NOUVELLE LOGIQUE, DEVENUE AUTONOME, PRODUIT, (...) EN RETOUR, UNE ANALYSE DU LANGAGE NATUREL. LA QUANTIFICATION AUTORISE UNE DEFINITION CONTEXTUELLE DES DESCRIPTIONS DEFINIES QUI DISPENSE DE TOUT ENGAGEMENT ONTOLOGIQUE. *LA TROISIEME PHASE -PRINCIPIA MATHEMATICA, 1910-13- ACHEVE LA REALISATION DU PROJET LOGICISTE. LA THEORIE RAMIFIEE DES TYPES EVITE LES PARADOXES ET L'USAGE SYSTEMATIQUE DES DEFINITIONS CONTEXTUELLES REDUIT LES CLASSES, RELATIONS, NOMBRES... (RASOIR D'OCCAM). FORT DE CE SUCCES, RUSSELL REND COMPTE DE NOTRE CONNAISSANCE DES ELEMENTS IRREDUCTIBLES : DONNEES SENSIBLES, FAITS ET UNIVERSSAUX NON DEFINISSABLES. IL VA MEME JUSQU'A ETENDRE LA REDUCTION LOGICISTE A NOTRE CONNAISSANCE, COMMUNE ET SCIENTIFIQUE, DU MONDE EXTERIEUR. MAIS LA GIGANTESQUE CONSTRUCTION DES PRINCIPIA ECHOUE PARCE QUE RUSSELL ET WHITEHEAD DOIVENT, IN FINE, ADMETTRE TROIS ASSOMPTIONS NON LOGIQUES : LES AXIOMES MULTIPLICATIF, D'INFINI ET DE REDUCTIBILITE. EN 1925, RUSSELL ET RAMSEY SE DISPENSERONT DE L'AXIOME DE REDUCTIBILITE. (shrink)

Sections “Introduction: Hume’s Principle, Basic Law V and Cardinal Arithmetic” and “The Julius Caesar Problem in Grundlagen—A Brief Characterization” are peparatory. In Section “Analyticity”, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle, bearing in mind that with its analytic or non-analytic status the intended logical foundation of cardinal arithmetic stands or falls. Section “Thought Identity and Hume’s Principle” is concerned with the two criteria of thought identity that Frege states in 1906 and (...) their application to Hume’s Principle. In Section “The Nature ofion: A Critical Assessment of Grundlagen, §64”, I scrutinize Frege’s characterization of abstraction in Grundlagen, §64 and criticize in this context the currently widespread use of the terms “recarving” and “reconceptualization”. Section “Frege’s Proof of Hume’s Principle” is devoted to the formal details of Frege’s proof of Hume’s Principle. I begin by considering his proof sketch in Grundlagen and subsequently reconstruct in modern notation essential parts of the formal proof in Grundgesetze. In Section “Equinumerosity and Coextensiveness: Hume’s Principle and Basic Law V Again”, I discuss the criteria of identity embodied in Hume’s Principle and in Basic Law V, equinumerosity and coextensiveness. In Section “Julius Caesar and Cardinal Numbers—A Brief Comparison Between Grundlagen and Grundgesetze ”, I comment on the Julius Caesar problem arising from Hume’s Principle in Grundlagen and analyze the reasons for its absence in this form in Grundgesetze. I conclude with reflections on the introduction of the cardinals and the reals by abstraction in the context of Frege’s logicism. (shrink)

Although the notion of logical object plays a key role in Frege's foundational project, it has hardly been analyzed in depth so far. I argue that Marco Ruffino's attempt to fill this gap by establishing a close link between Frege's treatment of expressions of the form ‘the concept F’ and the privileged status Frege assigns to extensions of concepts as logical objects is bound to fail. I argue, in particular, that Frege's principal motive for introducing extensions into his logical theory (...) is not to be able to make in-direct statements about concepts, but rather to define all numbers as logical objects of a fundamental kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Contrary to what Ruffino claims, reducibility to extensions cannot be the ‘ultimate criterion’ for Frege of what is to be regarded as a logical object. (shrink)

Part of Frege's concern about whether number words are properties of objects was that if they could be construed as such it would lend support to the view that truths of arithmetic were empirical truths. Such concern is ill-founded. Even if number words do apply to objects as predicates, this does not entail that numerical truths would be empirical, any more than the fact that ‘bachelor’ and ‘unmarried’ are predicates of objects entails that their relationship is an empirical one. The (...) account of number words as predicates given here, consequently, should not be taken as supportive of any one particular theory concerning arithmetic truth rather than another.The view of number words that emerges above from the criticism of Frege's arguments is that sentences containing number words can and do say something about objects in a collective and syncategorematic way. That is, they say something about classes of objects, when it is clear what is to count as an object. Whether this view commits us to accepting certain ontological entities, classes of individuals, above and beyond individuals themselves, I am not prepared to say, and at any rate this question falls beyond the intended scope of this essay. To those who would eschew abstract entities, however, it may be noted that the ontological problems that arise are not peculiar to this view; they seem no more or less severe than those that arise in connection with other collective predicate statements, as, for example, “Blue whales are becoming extinct.”The view presented here also does not pretend to be a fatal criticism of the logicist program. Taking our cue from the above noted fact that syncategorematic predicates share in a common ‘partial’ abstract sense, it seems reasonable to say that it is just the set of such partial senses of number words, and the relationships that hold between such senses, that the logicist and the pure number theorist investigate. (shrink)

It has been known for a few years that no more than Π 1 1 ∆ 1 3 comprehension is needed for the proof of “Frege’s Theorem.” One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no (...) part of my view that, say, Delta-3-1 comprehension axioms are not logical truths. What I suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called “ancestral logic” is developed, and it is then extended to yield what I call “Arché logic.”. (shrink)

The logicist position in the philosophy of mathematics, in short, assumes that mathematics can be reduced to a sufficiently comprehensive formal logic and therefore mathematics is a part of logic.

Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it (...) isn’t. In fact, 2FA is not conservative over n-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic. (shrink)

The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...) established by the neo-logicist. (shrink)

This note presents an analysis of Symbolic Knowledge from Leibniz to Husserl, a collection of works from some members of The Southern Cone Group for the Philosophy of Formal Sciences. The volume delineates an outlook of the philosophical treatments presented by Leibniz, Kant, Frege, and the Booleans, as well as by Husserl, of some questions related to the conceptual singularities of symbolic knowledge –whose standard we find in the arts of algebra and arithmetic. The book’s unity of themes and (at (...) least in part) style is examined with the aim of showing the articulation of its parts. (shrink)

Frege’s logicist program is a program of scientific unification of arithmetic and logic via the reduction of arithmetic to logic. Logic on this view is the prior science, indeed, the most fundamental of all sciences. The coherence of this picture has been questioned, based on the claim that the Basic Laws of logic are not justifiable as judgements. That Frege’s conception of logic suffers from this fatal flaw is incorrect, and in this paper I explore why. The discussion has three (...) primary parts. The first explores Frege’s view of logic, distinguishing pure from applied logic. The second delves into Frege’s conception of science as applications of logic, and how arithmetic is understood as such an application. The third concerns Frege’s account of judgement, and how it carefully distinguishes a judgement from judging, the cognitive act of recognizing truths. The final section turns to how on Frege’s account of judgement, the Basic Laws of logic are in fact justified as judgements, and accordingly can serve as the axioms of the science of arithmetic. The paper concludes with reflections on Frege’s understanding of scientific unification and reduction, and its relation to his logical realism. (shrink)

Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of (...) a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume's Principle and its connections to the root of the contradiction in Frege's system. (shrink)

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...) potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. (shrink)

Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is extremely (...) implausible because science treats the logical structure of quantities as subject to experimentally and theoretically motivated refinements and revisions. (shrink)

Gottlob Frege abandoned his logicist program after Bertrand Russell had discovered that some assumptions of Frege’s system lead to contradiction (so called Russell’s paradox). Nevertheless, he proposed a new attempt for the foundations of mathematics in two last years of his life. According to this new program, the whole of mathematics is based on the geometrical source of knowledge. By the geometrical source of cognition Frege meant intuition which is the source of an infinite number of objects in arithmetic. In (...) this article, I describe this final attempt of Frege to provide the foundations of mathematics. Furthermore, I compare Frege’s views of intuition from The Foundations of Arithmetic (and his later views) with the Kantian conception of pure intuition as the source of geometrical axioms. In the conclusion of the essay, I examine some implications for the debate between Hans Sluga and Michael Dummett concerning the realistic and idealistic interpretations of Frege’s philosophy. (shrink)

Ludwig Wittgenstein studied with Russell in Cambridge from 1911 to 1913, and wrote the _Tractatus_ while on active service in the Austrian army during the First World War. Large parts of the book are devoted to explaining and correcting errors in the conception of logic to be found in _Principia_. Wittgenstein did not, as is sometimes suggested, reject the idea of a hierarchy of types, but he did reject the notion that mathematics (and in particular arithmetic) could be based, as (...) in _Principia_, on classes. For this reason although the account of arithmetic given in the _Tractatus_ is in a sense logicist, it is very different from Russell's. (shrink)

This chapter states and explains all the formal rules of inference that are involved in the Constructive Logicist account of the natural numbers. Natural-deduction rules of introduction and elimination govern the primitives 0, _s_, and #, as well as various pasigraphs (such as _Nx,_ for ‘_x_ is a natural number’) that are inferentially definable in terms of the primitives. We set out important inferences about 1‒1 mappings, and define ancestrals of one-place functions by means of special introduction and elimination rules. (...) The ancestral of the successor function affords a definition of _Nx_. All the reasoning involving these notions, by means of these rules, will be both constructive and relevant. (shrink)

Is Plato a logicist? A comparative study between The Republic and The Sofist shows at one side how Plato mantains the trascendence of Good over the rest of the Ideas conserving it in this manner not just like a form but as that which mades possible the Ideas and the inteligibility of all that is. On the other hand one notice in The Sofist that being which Plato is talking about is surely the being (and not being) of propositions. Being (...) is for Plato like an Idea, but this doesn't imply logicism because in his thought there's a place for a Foundation, a Principle -the Good- which trascends all the forms: which mades possible also "the seen" and seing itself. (shrink)

Le programme logiciste, préconisé par J.-C. Gardin, propose de restituer l'architecture de nos constructions scientifiques sous forme de schématisations. Ces schématisations sont des arborescences òu sont énoncées les principales composantes de nos constructions, à savoir les bases de faits, les conclusions et les propositions intermédiaires reliant les premières aux secondes. Lorsqu'elles sont jouées sur multimédia et mises en scène sur 4 écrans selon le format SCD , elles permettent d'envisager de nouvelles pratiques éditoriales qui sont une réponse puissante à la (...) crise des publications en sciences humaines et sociales.The logicist programme, advocated by J. -C. Gardin, proposes to model the architecture of our constructs in the form of schematisations. These schematisations are arborescences composed of the main components of our constructs, i. e. a data base, conclusions and intermediary propositions linking the first group to the second. When they are played using multimedia equipment and produced according to the SCD format, they enable us to consider new editorial practices that are a powerful answer to the crisis of publications in human and social sciences. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about (...) their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic. (shrink)

Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)

It is difficult to describe this book without praising it. Collected here in one volume are some thirty-six high quality translations into English of the most important foreign-language works in mathematical logic, as well as articles and letters by Whitehead, Russell, Norbert Weiner and Post. The contents of the volume are arranged in chronological order, beginning with Frege's Begriffsschrift—translated in its entirety—and concluding with Gödel's famous "On Formally Undecidable Propositions" and Herbrand's "On the Consistency of Arithmetic". The translation of the (...) Gödel article is a small masterpiece, and is especially welcome in view of some of the rather poor efforts which have been made to render this important work into English. But this is more than a paste-pot job of anthologizing. Van Heijenoort and others have written prefaces to each of the included selections, in which are provided background historical and bibliographical information, comments on the intellectual currents which led to the production of the article, and, in many cases, a digest of the important theorems and concepts in the article itself. Each selection is a classic in its own right. In reading through the selections in this volume, one is struck by the extent to which purely mathematical problems and issues have controlled recent logical inquiry and have determined the development of various philosophies of mathematics. In the early stages, Frege and Peano were concerned with the logical reconstruction of arithmetic, and this led to the work of Russell and the logicist philosophy. Already in the development of the new logic there were certain problems which were resolved with Russell's theory of types—a device which many mathematicians felt to be rather artificial. The use of set theory, beginning with Cantor and reaching its maturity in Zermelo's 1904 proof of the well-ordering theorem, gave rise to a new set of difficulties concerning the axioms of choice, the continuum problem, and infinite sets. The two traditions are brought together in a common nexus of foundational problems by Weiner in 1914 and by Skolem's form of the Löwenheim theorem, given in 1920. Hereafter, the way is open for the rise of mathematical inquiry into the foundations of number systems, set theory and logic by Post, Skolem, von Neumann, Hilbert, Bernays, Gödel and many others. The work was far from completed by 1931, but a new and important tradition in mathematical inquiry had been established, and had come into its own. This book is, in effect, the record of an important chapter in the history of thought. No serious student of logic or foundations of mathematics will want to be without it.—H. P. K. (shrink)