The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms (...) of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms. (shrink)

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of (...) accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND. (shrink)

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S...

Tarski, Carnap and Quine spent the academic year 1940?1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap's Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking (...) place. In my article, I present a survey of these notes with special emphasis on Tarski's rejection of the analytic/synthetic distinction, the passage from typed languages to first-order languages, Tarski's finitism/nominalism, and the construction of a finitist language for mathematics and science. (shrink)

Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

The present paper is the result of a long struggle to understand how the notion of compositionality can be used to motivate the structure of a sentence. While everyone seems to have intuitions about which proposals are compositional and which ones are not, these intuitions generally have no formal basis. What is needed to make such arguments work is a proper understanding of what meanings are and how they can be manipulated. In particular, we need a definition of meaning that (...) bans all mentioning of syntactic structure; it is not the task of semantics to state in which way things are put together in syntax. The present paper presents such a theory of meaning. This, in tandem with some minimal assumptions on the syntactic process (that there can be no deletion) yield surprisingly deep insights into natural language. First, it rehabilitates a lot of linguistic work as necessary on semantic grounds and defends it against potential claims of redundancy. For example, θ-roles and linking are an integral part of semantics, and not syntax. To assume the latter is to put the cart before the horse. Second, as a particular example we shall show that Dutch is not strongly context free even if weakly context free. To our knowledge, this is the first formal proof of this fact. (shrink)

Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what (...) is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo—Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect. (shrink)

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...) analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom). (shrink)

The paper is about 'absolute logic': an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. In our system (...) the set-builder notation is an expression-building pattern. In our system we can easily express second-order, third order and any-order conditions. The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved. We discuss on the completeness of our deductive systems. We also discuss how our system relates to the most know types of paradoxes, from the discussion no specific vulnerability to paradoxes comes out. The paper provides both the theoretical material and a fully documented example of deduction. (shrink)

Logicism finds a prominent place in textbooks as one of the main alternatives in the foundations of mathematics, even though it lost much of its attraction from about 1950. Of course the neologicist trend has revitalized the movement on the basis of Hume’s Principle and Frege’s Theorem, but even so neologicism restricts itself to arithmetic and does not aim to account for all of mathematics. The present contribution does not focus on the classical logicism of Frege and Dedekind, nor on (...) the Russell-Carnap period, and also not on recent neologicism; its aim is to call attention to some forms of heritage from logicism that normally go quite unnoticed. In the 1920s, 1930s and 1940s, the logicist thesis became a stimulus for some deep innovations in the field of mathematical logic. One can argue, in particular, that two key ideas linked with formal semantics had their origins in the conception of logic associated with the logicist trend – the expansion of metamathematics brought about by Tarski, opening the way to model theory, and the insistence on the “full” set-theoretic semantics as “standard” for second-order logic. The paper proposes an analysis of those inheritances and argues that that logical theory ought to avoid some of their implications. (shrink)

Arnold Cusmariu ABSTRACT: The Cogito formulation in Discourse on Method attributes properties to one conceptual category that belong to another. Correcting the error ends up defeating Descartes’ response to skepticism. His own creation, the Evil Genius, is to blame. Download PDF.

Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity--including such revolutionary areas as black hole physics, relativistic computers, new cosmology--are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc 2. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.