Studies on Church's calculus

An extended version of this abstract will appear in Reports on Mathematical Logic. 1. In Church's calculus we establish classes of equivalent formulas built from only one propositional variable p in order to obtain the theorem on existence exactly one Lindenbaum's extension for Church's system. Moreover, we construct a class of finitely axiomatizable systems between Church's and Grzegorczyk's systems and we consider the problem of structural completeness of Church's calculus reduced to formulas formed from the variable p (Fragment tekstu)

Countable frames for bimodal logics S5oS5 and Grz.3oGrz.3.

In this paper we consider bimodal logics S5®S5 and Grz.3®Grz.3. We construct and describe two single countable frames which characterize systems S5 ® S5 and Grz.3 ® Grz.3, respectively

Tarski and Lesniewski on Languages with Meaning versus Languages without Use: A 60th Birthday Provocation for Jan Wolenski

Constructivism, Judgement and Meanin

On logics with coimplication

This paper investigates (modal) extensions of Heyting–Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gödel-embedding of intuitionistic logic into S4 , it is shown that all (modal) extensions of Heyting–Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok–Esakia-Theorem is proved for this embedding

Analysing the familiar : reasoning about space and time in the everyday world

The development of suitable explicit representations of knowledge that can be manipulated by general purpose inference mechanisms has always been central to Artificial Intelligence (AI). However, there has been a distinct lack of rigorous formalisms in the literature that can be used to model domain knowledge associated with the everyday physical world. If AI is to succeed in building automata that can function reasonably well in unstructured physical domains, the development and utility of such formalisms must be secured. This thesis describes a first order axiomatic theory that can be used to encode much topological and metrical information that arises in our everyday dealings with the physical world. The formalism is notable for the minimal assumptions required in order to lift up a very general framework that can cover the representation of much intuitive spatial and temporal knowledge. The basic ontology assumes regions that can be either spatial or temporal and over which a set of relations and functions are defined. The resulting partitioning of these abstract spaces, allow complex relationships between objects and the description of processes to be formally represented. This also provides a useful foundation to control the proliferation of inference commonly associated with mechanised logics. Empirical information extracted from the domain is added and mapped to these basic structures showing how further control of inference can be secured. The representational power of the formalism and computational tractability of the general methodology proposed is substantiated using two non-trivial domain problems - modelling phagocytosis and exocytosis of uni-cellular organisms, and modelling processes arising during the cycle of operations of a force pump