An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these (...) results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics. (shrink)

The purpose of this paper is to provide a new system of logic for existence and essence, in which the traditional distinctions between essential and accidental properties, abstract and concrete objects, and actually existent and possibly existent objects are described and related in a suitable way. In order to accomplish this task, a primitive relation of essential identity between different objects is introduced and connected to a first order existence property and a first order abstractness property. The basic idea is (...) that possibly existent objects are completely determinate and that essentially identical objects are just different individuations of the same individual essence. Accordingly, essential properties are defined as properties that are invariant with respect to this kind of identity, while abstract objects are determined by being characterized by essential properties only. Once such ideas are implemented, a number of classical intuitions about objects, their essence, and their way of existence can be consistently interpreted. (shrink)

In recent years numerous attempts have been made by analytic philosophers to _naturalize _various different domains of philosophical inquiry. All of these attempts have had the common goal of rendering these areas of philosophy amenable to empirical methods, with the intention of securing for them the supposedly objective status and broad intellectual appeal currently associated with such approaches. This volume brings together internationally recognised analytic philosophers, including Alvin Plantinga, Peter van Inwagen and Robert Audi, to question the project of naturalism. (...) The articles investigate what it means to _naturalize_ a domain of philosophical inquiry and look at how this applies to the various sub-disciplines of philosophy including epistemology, metaphysics and the philosophy of the mind. The issue of whether naturalism is desirable is raised and the contributors take seriously the possibility that excellent analytic philosophy can be undertaken without naturalization. Controversial and thought-provoking, _Analytic Philosophy Without Naturalism_ examines interesting and contentious methodological issues in analytic philosophy and explores the connections between philosophy and science. (shrink)

In recent years numerous attempts have been made by analytic philosophers to naturalize various different domains of philosophical inquiry. All of these attempts have had the common goal of rendering these areas of philosophy amenable to empirical methods, with the intention of securing for them the supposedly objective status and broad intellectual appeal currently associated with such approaches. This volume brings together internationally recognised analytic philosophers, including Alvin Plantinga, Peter van Inwagen and Robert Audi, to question the project of naturalism. (...) The articles investigate what it means to naturalize a domain of philosophical inquiry and look at how this applies to the various sub-disciplines of philosophy including epistemology, metaphysics and the philosophy of the mind. The issue of whether naturalism is desirable is raised and the contributors take seriously the possibility that excellent analytic philosophy can be undertaken without naturalization. Controversial and thought-provoking, Analytic Philosophy Without Naturalism examines interesting and contentious methodological issues in analytic philosophy and explores the connections between philosophy and science. (shrink)

This note clarifies the significance of the proof of Gödel’s first disjunct obtained through the formalisation of Penrose’s second argument within the DTK system. It analyses two formulations of the first disjunct – one general and the other restricted – and dwells on the demonstration of the restricted version, showing that it yields the following result: if by F we denote the set of propositions derivable from any formalism and by K the set of mathematical propositions humanly knowable, then, given (...) certain conditions, F necessarily differs from K. Thus it is possible that K surpasses F but also, on the contrary, that F surpasses K. In the latter case, however, the consistency of F is humanly undecidable. (shrink)

The aim of this essay is twofold. First, it outlines the concept of ontological frame. Secondly, two models are distinguished on this structure. The first one is connected to Kant’s concept of possible object and the second one relates to Leibniz’s. Leibniz maintains that the source of possibility is the mere logical consistency of the notions involved, so that possibility coincides with analytical possibility. Kant, instead, argues that consistency is only a necessary component of possibility. According to Kant, something is (...) possible if there is a cause capable of bringing it into existence; to this end consistency alone is not sufficient. Thus, while the Leibnizian notion of consistency is at the root of the concept of analytical possibility, the Kantian notion of possibility is the source of real possibility. This difference plays an important role in the discussion of Gödel’s ontological proof, which can be formally interpreted on the ontological frame of the pure perfections. While this proof, under some emendation condition, is conclusive in the context of Leibniz’s ontological model, it is not so within the Kantian one. This issue will be the subject of the second part of the present essay. (shrink)

Cet ouvrage offre une introduction accessible à la théorie de la démonstration : il donne les détails des preuves et comporte de nombreux exemples et exercices pour faciliter la compréhension des lecteurs. Il est également conçu pour servir d’aide à la lecture des articles fondateurs de Gerhard Gentzen. L’ouvrage introduit également aux trois principaux formalismes en usage : l’approche axiomatique des preuves, la déduction naturelle et le calcul des séquents. Il donne une démonstration claire et détaillée des résultats fondamentaux du (...) domaine : traduction de l’arithmétique classique vers l’arithmétique intuitionniste, élimination des coupures, théorème de normalisation et conduit ensuite pas à pas le lecteur vers l’exposé de la célèbre preuve de cohérence de Gentzen pour l’arithmétique de Peano du premier ordre. Il comble ainsi une importante lacune éditoriale en présentant à la fois la théorie structurelle et la théorie ordinale de la démonstration. (shrink)

In this essay, we will try to provide a formal analysis of practical inference, attentive to the various phases in which it is articulated, and being so capable of explaining both the logical conclusiveness of the inference and the probabilistic nature of its conclusion. An innovative purpose of this article is to show how the final deliberation leading to action—the ultimate practical judgment—takes place according to a logic consistent with the use of Bayes’ theorem. This is why we refer to (...) Bayesian practical inference in the title of this essay. (shrink)

This article aims to examine Koellner’s reconstruction of Penrose’s second argument – a reconstruction that uses the DTK system to deal with Gödel’s disjunction issues. Koellner states that Penrose’s argument is unsound, because it contains two illegitimate steps. He contends that the formulas to which the T-intro and K-intro rules apply are both indeterminate. However, we intend to show that we can correctly interpret the formulas on the set of arithmetic formulas, and that, as a consequence, the two steps become (...) legitimate. Nevertheless, the argument remains partially inconclusive. More precisely, the argument does not reach a result that shows there is no formalism capable of deriving all the true arithmetic propositions known to man. Instead, it shows that, if such formalism exists, there is at least one true non-arithmetic proposition known to the human mind that we cannot derive from the formalism in question. Finally, we reflect on the idealised character of the DTK system. These reflections highlight the limits of human knowledge, and, at the same time, its irreducibility to computation. (shrink)

Published in honor of Sergio Galvan, this collection concentrates on the application of logical and mathematical methods for the study of central issues in formal philosophy. The volume is subdivided into four sections, dedicated to logic and philosophy of logic, philosophy of mathematics, philosophy of science, metaphysics and philosophy of religion. The contributions adress, from a logical point of view, some of the main topics in these areas. The first two sections include formal treatments of: truth and paradoxes; definitions by (...) abstraction; the status of abstract objects, such as mathematical objects and universal concepts; and the structure of explicit knowledge. The last two sections include papers on classical problems in philosophy of science, such as the status of subjective probability, the notion of verisimilitude, the notion of approximation, and the theory of mind and mental causation, and specific issues in metaphysics and philosophy of religion, such as the ontology of species, actions, and intelligible worlds, and the logic of religious belonging. (shrink)

In this article I defend a form of classical possibilism with an actualist foundation. As a matter of fact, I believe that this position is more in keeping with the classical metaphysical tradition. According to this form of possibilism, I construe possible objects as possible non-existing objects of an existing producing power. Consequently, they are nothing vis-à -vis the modality of their own actual being, although they do exist with regard to the modality of the producing power’s being. The actualist (...) requirement prescribed by the Frege-Quinean criterion of the quantification domain is thus fulfilled; indeed, really possible objects are not actual objects, but their possibility is actual. (shrink)

The paper studies two formal schemes related to -completeness.LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where (x) is a formula with the only free variable x): (a) (for anyn) ( T (n)), (b) T x Pr T (–(x)–) and (c) T x(x) (the notational conventions are those of Smoryski [3]). The aim of this paper is to examine the (...) meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) (c) and (a) (b), where ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) (c) is equivalent to ¬Cons T and 2. the schema corresponding to (a) (b) is not consistent with 1-CON T. The former result follows from a simple adaptation of the -incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma. (shrink)

One 1120 pages volume, with 4000 entries covering - Western philosophy: authors, schools, concepts and terminology; - religions, cultural anthropology, eastern philosophies; - Psychology and psychoanalysis; - linguistics and semiotics; - sociology and political theory.

The 'Enciclopedia Filosofica' is an encyclopaedia of philosophical topics promoted by the Centre for Philosophical Studies of Gallarate and published, in its third and last edition in 2006, by the Bompiani publishing house in Milan. The first edition of the 'Enciclopedia Filosofica' was promoted by the Centre for Philosophical Studies of Gallarate in the 1950s, seeing the light in 1957-58. A second edition, published by the Sansoni publishing house in Florence, was published in 1968-69 and reprinted in 1979. The third (...) and last edition, published by Bompiani in 2006, is intended to cover the whole of philosophical knowledge and related disciplines with over ten thousand entries edited by about a thousand academics, both Italian and non-Italian, and distributed in twelve volumes for a total of 12,496 pages. (shrink)