Bulletin of the Section of Logic

Completeness, Categoricity and Imaginary Numbers: The Debate on Husserl

Víctor Aranda — Tue, 30 Jun 2020 00:00:00 +0000

Husserl's two notions of "defi niteness" enabled him to clarify the problem of imaginary numbers. The exact meaning of these notions is a topic of much controversy. A "de finite" axiom system has been interpreted as a syntactically complete theory, and also as a categorical one. I discuss whether and how far these readings manage to capture Husserl's goal of elucidating the problem of imaginary numbers, raising objections to both positions. Then, I suggest an interpretation of "absolute de finiteness" as semantic completeness and argue that this notion does not suffice to explain Husserl's solution to the problem of imaginary numbers.

Computer-supported Analysis of Positive Properties, Ultrafilters and Modal Collapse in Variants of Gödel's Ontological Argument

Christoph Benzmüller, David Fuenmayor — Tue, 30 Jun 2020 00:00:00 +0000

Three variants of Kurt Gödel's ontological argument, proposed by Dana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory reading they are in fact closely related. This has been revealed in the computer-supported formal analysis presented in this article. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.

Cantor on Infinitesimals. Historical and Modern Perspective

Piotr Błaszczyk, Marlena Fila — Tue, 30 Jun 2020 00:00:00 +0000

In his 1887's Mitteilungen zur Lehre von Transfi niten, Cantor seeks to prove inconsistency of infi nitesimals. We provide a detailed analysis of his argument from both historical and mathematical perspective. We show that while his historical analysis are questionable, the mathematical part of the argument is false.

Compounding Objects

Zvonimir Šikić — Tue, 30 Jun 2020 00:00:00 +0000

We prove a characterization theorem for filters, proper filters and ultrafilters which is a kind of converse of Łoś's theorem. It is more natural than the usual intuition of these terms as large sets of coordinates, which is actually unconvincing in the case of ultrafilters. As a bonus, we get a very simple proof of Łoś's theorem.

What Is the Sense in Logic and Philosophy of Language

Urszula Wybraniec-Skardowska — Tue, 30 Jun 2020 00:00:00 +0000

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions.

Formal Methods and Science in Philosophy: Introduction to the Special Issue

Patrick Blackburn, Srećko Kovač, Kordula Świętorzecka — Tue, 30 Jun 2020 00:00:00 +0000

Introduction to the Special Issue.