Krystian Jobczyk

In 1969, Lindström proved his celebrated theorem characterizing first-order logic and established criteria for the first-order definability of formal theories for discrete structures. Barwise, Shelah, Väänänen, and others extended Lindström’s characterizability program to classes of infinitary logic systems, including a recent paper by Džamonja and Väänänen on Karp’s chain logic, which satisfies interpolation, undefinability of well-order, and is maximal in the class of logic systems with these properties. From the perspective of the article analysis, research on chain logic delivers a promising idea of chain models and a new definition of satisfiability, particularly for quantifier expressions. Our paper gives a framework for Lindström-type characterizability of predicate logic systems interpreted semantically in models with objects based on measures (analytic structures). In particular, Hájek’s logic of integrals (HLI) is redefined as an abstract logic with a new type of Hájek’s satisfiability and constitutes a maximal logic in the class of logic systems for describing analytic structures with Lebesgue integrals and satisfying compactness, elementary chain condition, and weak negation.

In “Models and Reality”, H. Putnam formulated his model-theoretic argument against “metaphysical realism”. The article proposes a meta-reconstruction of Putnam’s model-theoretic argument in the light of two mutually compatible interpretations of it–elaborated by Manuel Garcia-Carpintero and Igor van Douven. A critical reflection on these interpretations and their adequacy for Putnam’s argument allows us to expose new theses coherent with Putnam’s reasoning and indicate new paths to improve this argument for our reconstruction task. In particular, we show that Putnam’s position may be coherent with van Douven’s versions of Global Descriptivism under some conditions, but Putnam cannot reject realism as quickly as Carpintero suggests. We show that Suszko’s canonic axiomatic system and Sneed’s concept of theory may provide valuable support for Putnam’s argument. Finally, we critically evaluate Carpintero’s theses about the genesis of unintended interpretations of our languages, adopting the machinery of the Upward Skolem–Loewenheim Theorem and Knight’s Theorem.