Abstract: This paper deals with several problems concerning notion of existential dependence and ontological notions of existence, necessity and fusion. Following some ideas of Eugenia Ginsberg-Blaustein, the notions are treated in reference to objects, in relation to the concepts of state of affairs and subject of state of affairs. It provides an axiomatic characterization of these concepts within the framework of a multi-modal propositional logic and then presents a semantic analysis of these concepts. The semantics are a slight modification to the standard relational semantics for normal modal propositional logic.
Abstract: This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties.
Abstract: In this paper, two axiomatic theories T− and T′ are constructed, which are dual to Tarski’s theory T+ (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory T+ the primitive notion is the classical consequence function (entailment) Cn+, in the dual theory T− it is replaced by the notion of Słupecki’s rejection consequence Cn− and in the dual theory T′ it is replaced by the notion of the family Incons of inconsistent sets. The author has proved that the theories T+, T−, and T′ are equivalent.
Abstract: During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of an invitation to continue research on these first attempts to axiomatize important concepts and systems in fuzzy set theory. Currently, these attempts are spread over many journals; with this paper they are now collected in a neat overview. In the literature, many axiom systems have been introduced, but as far as we know the axiomatic system of Huntington concerning a Boolean algebra has been the only one where the axioms have been proven independent. Another line of further research could be with respect to the simplification of these systems, in discovering redundancies between the axioms.
Abstract: The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt’s dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal μ-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the μ-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations. Later, Bojanczyk opened a new chapter by introducing an unboundedness quantifier, which allowed for expressing some quantitative properties of programs. Yet another topic, linking the past with the future, is the subject of automata founded in the Fraenkel-Mostowski set theory. The studies on intuitionism found their continuation in the studies of Curry-Howard isomorphism. ukasiewicz’s landmark idea of many-valued logic found its continuation in various approaches to incompleteness and uncertainty.