According to the dialetheist argument from the inconsistency of informal mathematics, the informal version of the Gödelian argument leads us to a true contradiction. On one hand, the dialetheist argues, we can prove that there is a mathematical claim that is neither provable nor refutable in informal mathematics. On the other, the proof of its unprovability is given in informal mathematics and proves that very sentence. We argue that the argument fails, because it relies on the unjustified and unlikely assumption (...) that the informal Gödel sentence is informally provable. (shrink)

In section 1 we argue that the adoption of a tenseless notion of truth entails a realistic view of propositions and provability. This view, in turn, opens the way to the intelligibility of theclassical meaning of the logical constants, and consequently is incompatible with the antirealism of orthodox intuitionism. In section 2 we show how what we call the potential intuitionistic meaning of the logical constants can be defined, on the one hand, by means of the notion of atemporal provability (...) and, on the other, by means of the operator K of epistemic logic. Intuitionistic logic, as reconstructed within this perspective, turns out to be a part of epistemic logic, so that it loses its traditional foundational role, antithetic to that of classical logic. In section 3 we uphold the view that certain consequences of the adoption of atemporal notion of truth, despite their apparent oddity, are quite acceptable from an antirealist point of view. (shrink)

This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.

This paper outlines a framework for the abstract investigation of the concept of canonicity of names and of naming systems. Degrees of canonicity of names and of naming systems are distinguished. The structure of the degrees is investigated, and a notion of relative canonicity is defined. The notions of canonicity are formally expressed within a Carnapian system of second-order modal logic.

I argue that the implementation of theDummettian program of an ``anti-realist'' semanticsrequires quite different conceptions of the technicalmeaning-theoretic terms used than those presupposed byDummett. Starting from obvious incoherences in anattempt to conceive truth conditions as assertibilityconditions, I argue that for anti-realist purposesnon-epistemic semantic notions are more usefully kept apart from epistemic ones rather than beingreduced to them. Embedding an anti-realist theory ofmeaning in Martin-Löf's Intuitionistic Type Theory(ITT) takes care, however, of many notorious problemsthat have arisen in trying to specify suitableintuitionistic (...) notions of semantic value,truth-conditions, and validity, taking into accountthe so-called ``defeasibility of evidence'' forassertions in empirical discourses. (shrink)

Today there appears to be a widespread impression that W. V. Quine's notorious critique of modal logic, based on certain ideas about reference, has been successfully answered. As one writer put it some years ago: “His objections have been dead for a while, even though they have not yet been completely buried.” What is supposed to have killed off the critique? Some would cite the development of a new ‘possible-worlds’ model theory for modal logics in the 1960s; others, the development (...) of new ‘direct’ theories of reference for names in the 1970s.These developments do suggest that Quine's unfriendliness towards any formal logics but the classical and indifference towards theories of reference for any singular terms but variables were unfortunate. (shrink)

This paper briefly traces the evolution of the function concept until its modern set theoretic definition, and then investigates its relationship to the pre-formal notion of variable dependence. I shall argue that the common association of pre-formal dependence with the modern function concept is misconceived, and that two different notions of dependence are actually involved in the classic and the modern viewpoints, namely effective and functional dependence. The former contains the latter, and seems to conform more to our pre-formal conception (...) of dependence. The idea of effective dependence is further investigated in connection with the notions of function content and intensionality. Finally, the relevance of the distinction between the two kinds of dependence to mathematical practice is considered. (shrink)

This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.

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.

This paper is divided in five sections. Section 11.1 sketches the history of the distinction between speech act with negative content and negated speech act, and gives a general dynamic interpretation for negated speech act. “Downdate semantics” for AGM contraction is introduced in Section 11.2. Relying on semantically interpreted contraction, Section 11.3 develops the dynamic semantics for constative and directive speech acts, and their external negations. The expressive completeness for the formal variants of natural language utterances, none of which is (...) a retraction, has been proved in Section 11.4. The last section gives a laconic answer to the question posed in the title of the paper. (shrink)