Peter Milne (2007) poses two challenges to the inferential theorist of meaning. This study responds to both. First, it argues that the method of natural deduction idealizes the essential details of correct informal deductive reasoning. Secondly, it explains how rules of inference in free logic can determine unique senses for the existential quantifier and the identity predicate. The final part of the investigation brings out an underlying order in a basic family of free logics.

This paper is a discussion of Gödel's arguments for a Platonistic conception of mathematical objects. I review the arguments that Gödel offers in different papers, and compare them to unpublished material (from Gödel's Nachlass). My claim is that Gödel's later arguments simply intend to establish that mathematical knowledge cannot be accounted for by a reflexive analysis of our mental acts. In other words, there is at the basis of mathematics some data whose constitution cannot be explained by introspective analysis. This (...) does not mean that mathematics is independent of the human mind, but only that it is independent of our ?conscious acts and decisions?, to use Gödel's own words. Mathematical objects may then have been created by the human mind, but if so, the process of creation cannot be completely analysed and re-enacted. Such a thesis is weaker than some of the statements that Gödel made about his conceptual realism. However, there is evidence that Gödel seriously considered this weak thesis, or a position depending only on this weak thesis. He also criticized Husserl's Phenomenology from this point of view. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

It is well known that in his later years Gödel turned to a systematic reading of phenomenology, whose founder, Edmund Husserl, was highly esteem as a philosopher who sought to elevate philosophy to the standards of a rigorous science. For reasons purportedly related to his earlier attraction to Leibnizian monadology, Gödel was particularly interested in Husserl's transcendental phenomenology and the way it may shape the discussion on the nature of mathematical‐logical objects and the meaning and internal coherence of primitive terms (...) in mathematics. This article, which is less interested in historical facts that are amply presented in other scholars’ accounts, rather tries to point to the mostly indirect influence that the ideas of transcendental phenomenology, including Gödel's reported reference to the notion of (inner) time, had on his philosophy of mathematics, especially with a view to key independent questions of mathematical foundations. A main source for Gödel's philosophical‐epistemological views (of course in addition to his own published material) will be his recorded discussions with H. Wang and S. Toledo, as well as various articles and drafts found in his Nachlass. (shrink)

This paper proposes a possible reconstruction and philosophical-logical clarification of Gödel's idea of causality as the philosophical fundamental concept. The results are based on Gödel's published and non-published texts (including Max Phil notebooks), and are established on the ground of interconnections of Gödel's dispersed remarks on causality, as well as on the ground of his general philosophical views. The paper is logically informal but is connected with already achieved results in the formalization of a causal account of Gödel's onto-theological theory. (...) Gödel's main causal concepts are analysed (will, force, enjoyment, God, time and space, life, form, matter). Special attention is paid to a possible causal account of some of Gödel's logical concepts (assertion, privation, affirmation, negation, whole, part, general, particular, subject, predicate, necessary, possible, implication), as well as of logical antinomies. The problem of mechanical and non-mechanical procedures in the work with and on concepts is addressed in terms of Gödel's causal view. (shrink)

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two (...) decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

This article serves to present a large mathematical perspective and historical basis for the Axiom of Replacement as well as to affirm its importance as a central axiom of modern set theory.

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely-held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single-premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory validating (...) all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker-than-classical K3 logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to--whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non-classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities. (shrink)

This article is an attempt to present Gödel's discussion on concepts, from 1944 to the late 1970s, in particular relation to the thought of Frege and Russell. The discussion takes its point of departure from Gödel's claim in notes on Bernay's review of ?Russell's mathematical logic?. It then retraces the historical background of the notion of intension which both Russell and Gödel use, and offers some grounds for claiming that Gödel consistently considered logic as a free-type theory of concepts, called (...) intensions, considered as the denotations of predicate names. (shrink)

This is an examination, a commentary, of links between some philosophical views ascribed to Gödel and general proof theory. In these views deduction is of central concern not only in predicate logic, but in set theory too, understood from an infinitistic ideal perspective. It is inquired whether this centrality of deduction could also be kept in the intensional logic of concepts whose building Gödel seems to have taken as the main task of logic for the future.

A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, (...) the status of reflection principles as axioms is left open for the Zermelian. (shrink)

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

The paper examines the interrelationship between mathematics and logic, arguing that a central characteristic of each has an essential role within the other. The first part is a reconstruction of and elaboration on Paul Bernays’ argument, that mathematics and logic are based on different directions of abstraction from content, and that mathematics, at its core it is a study of formal structures. The notion of a study of structure is clarified by the examples of Hilbert’s work on the axiomatization of (...) geometry and Hilbert et al.’s formalist proof theory. It is further argued that the structural aspect of logic puts it under the purview of the mathematical, analogously to how the deductive nature of mathematics puts it under the purview of logic. This is then linked, in the second part, to certain aspects of Gödel’s critique of Carnap’s conventionalism, that ‘mere syntax’ cannot capture the full content of mathematics, which is revealed to be closely related to the characteristic of mathematics argued for by Bernays. Finally, this is connected with Gödel’s latter-day views about two kinds of formality, intensional and extensional, and the relationship between them. (shrink)

Nosso objetivo neste artigo é o de lançar luz sobre alguns aspectos das concepções de Gödel acerca da percepção de conceitos. Começamos investigando a natureza e o papel da analogia entre percepção sensível e percepção de conceitos. A seguir, examinamos as conexões entre percepção de conceitos, razão e compreensão, tentando mostrar que a percepção de conceitos é compreensão de conceitos. Por fim, examinamos aqueles aspectos da concepção de Gödel em que a percepção de conceitos de fato se aproxima perigosamente da (...) ideia de um "olho da mente". Contudo, argumentamos que esses aspectos problemáticos não são intrínsecos à noção de percepção de conceitos, mas sim resultam de uma má compreensão da distinção entre conceito e objeto. Our goal in this article is to illuminate some aspects of Gödel's conception of the perception of concepts. We begin by investigating the nature and function of Gödel's analogy between sense perception and perception of concepts. After that, we examine the connections between perception of concepts, reason, and understanding, trying to show that the perception of concepts consists in the understanding of concepts. Finally, we examine those aspects of Gödel's conception where the perception of concepts actually comes dangerously close to the idea of a "mind's eye". However, we argue that these problematic aspects are not intrinsic to the notion of perception of concepts, but result from a misunderstanding of the distinction between concept and object. (shrink)