The identity concept developed in the Homotopy Type theory supports an analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities. In the context of this analysis we consider the traditional distinction between the extension and the intension of concepts as it appears in HoTT, discuss an ontological significance of this distinction and, finally, provide a homotopical reconstruction of a basic kinematic scheme, which is used in the Classical Mechanics, and discuss its relevance in the (...) Quantum Mechanics. (shrink)

In his Tractatus Logico-Philosophicus Ludwig Wittgenstein (1889–1951) presents the concept of order in terms of a notational iteration that is completely logical but not part of logic. Logic for him is not the foundation of mathematical concepts but rather a purely formal way of reflecting the world that at the minimum adds absolutely no content. Order for him is not based on the concepts of logic but is instead revealed through an ideal notational series. He states that logic is “transcendental”. (...) As such it requires an ideal that his philosophical method eventually forces him to reject. I argue that Wittgenstein’s philosophy is more dialectical than transcendental. (shrink)

Following Neale, I call the notion that there can be no such thing as a structured referring expression ‘structure skepticism’. The specific aim of this paper is to defuse some putative counterexamples to structure skepticism. The general aim is to bolster the case in favor of the thesis that lack of structure—in a sense to be made precise—is essential to reference.

The paper first formalizes the ramified type theory as (informally) described in the Principia Mathematica [32]. This formalization is close to the ideas of the Principia, but also meets contemporary requirements on formality and accuracy, and therefore is a new supply to the known literature on the Principia (like [25], [19], [6] and [7]).As an alternative, notions from the ramified type theory are expressed in a lambda calculus style. This situates the type system of Russell and Whitehead in a modern (...) setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see [3]. (shrink)

The aim of this paper is to give a detailed reconstruction of Frege's solution to his puzzle about the cognitive function of truth, which is this: On the one hand, the concept of truth seems to play an essential role in acquiring knowledge because the transition from the mere hypothetical assumption that p to the acknowledgement of its truth is a crucial step in acquiring the knowledge that p, while, on the other hand, this concept seems to be completely redundant (...) because the sense of the word 'true' does not make any essential contribution to the senses of the sentences in which it occurs. (shrink)

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, (...) and making an estimate of future prospects. The symptom was the contradiction that had crept into naïve set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege. (shrink)

Michael Dummett has shown that the fragment ‘17 Kernsätze zur Logik’ is evidence that Frege knew Lotze's Logik Dummett’s dating of this fragment prior to 1879, however, must be rejected.The present paper shows that there are other articles of Frege’s which bear clear traces of Lotze's LogikFirst of all, the expressions Vorstellungsverlauf from ‘Über die wissenschaftliche Berechtigung einer Begriffsschrift’, and veranlassenden Ursachen, from ‘Logik’, certainly are borrowed from Lotze.Second, there are links between ‘Booles rechnende Logik und die Begriffsschrift’ and Lotze's (...) Logik. Furthermore, it is shown that Frege’s ‘Kernsätze’, the ‘Dialog mit Pünjer über Existenz’, and his ‘Logik’ are intimately connected.All of this indicates that these texts were written in roughly the same period, namely the early 1880s.Conclusive evidence for this is that the terms Vorstellung and Vorstellungsverbindung are used indiscriminately in both a psychological and a logical sense in the ‘Begriffsschrift’, a fact which contradicts the ‘Kernsätze’. (shrink)

Mathematics in the 19th century underwent explosive development. The discussion among philosophers was motivated by new results which conflicted with Kant’s very influential conception. One of the most important areas which lead to the meeting of philosophy and mathematics was a rigorous account of the infinitesimal calculus. In his article the author shows that considerations of this type found in Bolzano and Frege are important not only for the overcoming of the Kantian conception, but as for what we today call (...) the linguistic turn. (shrink)

Logic and philosophy of science share a long history, though contacts have gone through ups and downs. This paper is a brief survey of some major themes in logical studies of empirical theories, including links to computer science and current studies of rational agency. The survey has no new results: we just try to make some things into common knowledge.

We reconstruct Frege’s treatment of certain deducibility problems posed by Boole. It turns out that in his formalization and solution of Boole’s problems Frege anticipates the idea of propositional resolution.

I examine the difficulties that several philosophers of language are liable to encounter in their attempts to provide an account of the connection between truth and assertion. I then attempt to provide an account of this connection. The analysis is concerned chiefly with difficulties which consist in elucidating the conceptual connection between truth and assertion in a way that respects certain linguistic intuitions while at the same time rendering the concept of truth amenable to a semantic interpretation. The proposed view (...) suggests one way in which we might go about meeting the theoretical demands implicit in addressing this concern, among others, demonstrating the extent to which a theory of truth should be regarded as belonging to the province of epistemology. Insofar as semantical considerations figure into such a theory, a more systematic investigation of the interface between epistemology and natural language semantics is recommended. The solution to many problems at this interface, I argue, lay in an analysis of judgment. (shrink)

We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set (...) theory, which undoubtedly point towards the mathematics of twentieth and twenty-first centuries. (shrink)

A discussion of the connections and differences between completeness and representation theorems in logic, with examples drawn from classical and modal logic, the logic of friendliness, and nonmonotonic reasoning.

We construct an extension P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of P;rfs preceded by theassertion sign constituteelementary assertive formulas of P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of P is endowed with atruth value defined classically, (...) and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic. (shrink)

Jarmo Pulkkinen, The threat of logical mathematism. A study on the critique of mathematical logic in Germany at the turn of the 20th century. Frankfurt a.M:Peter Lang, 1994. Scandinavian University Studies in the Humanities and Social Sciences; 7). 186 pp. 24 DM. ISBN 3-631-47409-1.

I first show that most authors who developed Plural Quantification Logic (PQL) argued it could capture various features of natural language better than can other logic systems. I then show that it fails to do so: it radically departs from natural language in two of its essential features; namely, in distinguishing plural from singular quantification and in its use of an relation. Next, I sketch a different approach that is more adequate than PQL for capturing plural aspects of natural language (...) semantics and logic. I conclude with a criticism of the claim that PQL should replace natural language for specific philosophical or scientific purposes. (shrink)

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so that (...) nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

Las interpretaciones tradicionales de Frege tienden a explicar su antipsicologismo como indicando que ninguna consideración subjetivista debe interferir en la elaboración de un lenguaje lógico; dichas interpretaciones, sin embargo, tienen problemas para explicar el énfasis dado por el mismo Frege a conceptos como el de fuerza asertiva. Este artículo pretende solucionar esta extrañeza investigando, a partir de su concepto de ciencia, algunas de las ideas epistemológicas de Frege. Esto nos llevará a concluir que una concepción de la verdad como fenómeno (...) esencialmente público puede resolver este y otros problemas interpretativos. (shrink)

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent logicism is compatible with intuitionism.

Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics.

In British and American anthropological literature, the ethnology of Central and Eastern European countries has shared in the plight of descriptions of this part of the world: it was seen as exotic, foreign, remote, a backwater, focused on sideline problems and situated on the periphery of this field of science. This state of affairs has been the case since at least the beginning of the Cold War as the descriptions of the national characters of Eastern Bloc communities, drafted by American (...) scholars. The ethnography of these countries was automatically aligned with the same categories of description which were applied to the characteristics of these communities. This was, in part, an effect of entirely different research traditions in the subject matter, objectives and methods of these remote parts of the science. The differences of ’cultural wholes’ did have a decisive impact on the image of ethnology. Like all these communities, Polish ethnology was perceived as a homogeneous whole, which it never was. (shrink)

Cognitive sciences are grouped together according to their substantial disciplines such as neurobiology, psychology, linguistics and many others. From more than ten years a search has been undertaken for the best fitting cognitive research method, that does not influence the scientific output. More recently attention has been turned to systems which might operate through their dynamic aspect, called the dynamic approach. Their great advantage is to see the old linguistic axioms such as connectivity in a new way.

way on the information available in the contexts in which they are used, it’s not surprising that there is a minor but growing industry of work in semantics and the philosophy of language concerned with the precise nature of the context-dependency of epistemically modalized sentences. Take, for instance, an epistemic might-claim like..