Informal and Absolute Proofs: Some Remarks from a Gödelian Perspective

Topoi - After a brief discussion of Kreisel’s notion of informal rigour and Myhill’s notion of absolute proof, Gödel’s analysis of the subject is presented. It is shown how...     

Decomposition Proof Systems for Gödel-Dummett Logics

The main goal of the paper is to suggest some analytic proof systems for LC and its finite-valued counterparts which are suitable for proof-search. This goal is achieved through following the general Rasiowa-Sikorski methodology for constructing analytic proof systems for semantically-defined logics. All the systems presented here are terminating, contraction-free, and based on invertible rules, which have a local character and at most two premises.     

Some Notes on Boolos’ Semantics: Genesis,Ontological Quests and Model-Theoretic Equivalence to Standard Semantics

The main aim of this work is to evaluate whether Boolos’ semantics for second-order languages is model-theoretically equivalent to standard model-theoretic semantics. Such an equivalence result is, actually, directly proved in the “Appendix”. I argue that Boolos’ intent in developing such a semantics is not to avoid set-theoretic notions in favor of pluralities. It is, rather, to prevent that predicates, in the sense of functions, refer to classes of classes. Boolos’ formal semantics differs from a semantics of pluralities for Boolos’ plural reading of second-order quantifiers, for the notion of plurality is much more general, not only of that set, but also of class. In fact, by showing that a plurality is equivalent to sub-sets of a power set, the notion of plurality comes to suffer a loss of generality. Despite of this equivalence result, I maintain that Boolos’ formal semantics does not committ (directly) second-order languages (theories) to second-order entities (and to set theory), contrary to standard semantics. Further, such an equivalence result provides a rationale for many criticisms to Boolos’ formal semantics, in particular those by Resnik and Parsons against its alleged ontological innocence and on its Platonistic presupposition. The key set-theoretic notion involved in the equivalence proof is that of many-valued function. But, first, I will provide a clarification of the philosophical context and theoretical grounds of the genesis of Boolos’ formal semantics.     

Gödel on Concepts

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.     

Standard Gödel Modal Logics

We prove strong completeness of the □-version and the ?-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.     

Gödelizing the Yablo Sequence

We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser’s provability predicate.     

Normative Objectivity Without Ontological Commitments?

Several non-naturalist philosophers look for ways to maintain the objectivity of morals without making any (robust) ontological commitments. Recently Derek Parfit proposed an account of non-ontologically existing irreducible moral properties. My first aim in this paper is to outline that such an account is doomed to fail. My second aim in this paper is to argue that irreducible moral properties can be integrated with adaptions into an ontological framework such as E.J. Lowe’s four-category ontology. If it can be shown that irreducible moral properties have a proper place in such an ontology, then there is no need to distinguish between an ontological and non-ontological mode of existence, which, in turn helps to eschew the obscurities that this distinction brings in its wake.     

Emotion Education without Ontological Commitment?

Emotion education is enjoying new-found popularity. This paper explores the ‘cosy consensus’ that seems to have developed in education circles, according to which approaches to emotion education are immune from metaethical considerations such as contrasting rationalist and sentimentalist views about the moral ontology of emotions. I spell out five common assumptions of recent approaches to emotion education and explore their potential compatibility with four paradigmatic moral ontologies. I argue that three of these ontologies fail to harmonise with the common assumptions. Either those three must therefore be rejected or, if we want to retain one or more of them (for instance, Jesse Prinz’s recent rebranding of hard sentimentalism that I explore in detail), we need to revise our assumptions about the practice of emotion education in ways that are both radical and, I argue, ultimately unacceptable.     

Gödel’s Natural Deduction

This is a companion to a paper by the authors entitled “Gödel on deduction”, which examined the links between some philosophical views ascribed to Gödel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Gödel presented with the help of Gentzen’s sequents, which amounts to Ja?kowski’s natural deduction system of 1934, and which may be found in Gödel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Gödel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Gödel’s Notre Dame course, and is followed by comments concerning Gödel’s natural deduction system.     

Skolem's Discovery of Gödel-Dummett Logic

Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.     

On interpreting Gödel's Second Theorem

Summary In this paper I have considered various attempts to attribute significance to G2.²⁵ Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are literally false. Two others (BCR and Resnik's Interpretation), I have argued, are groundless.I would like to thank Dale Gottlieb, Stephen Barker, Tim McCarthy, Philip Kitcher, Michael Resnik and Richmond Thomason for extensive and helpful discussion of this work.