According to Frege’s principle the denotation of a sentence coincides with its truthvalue. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense. A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A ded...

Classical logic is embedded into constructive logic, through a definition of the classical connectives and quantifiers in terms of the constructive ones. The history of the notion of constructivity started with a dispute on the deduction rules that one should or should not use to prove a theorem. Depending on the rules accepted by the ones and the others, the proposition P ∨ ¬P , for instance, ...

We consider the sabotage modal logic SML which was suggested by van Benthem. SML is the modal logic equipped with a ‘transition-deleting’ modality and hence a modal logic over changing models. It was shown that the problem of uniform model checking for this logic is PSPACE-complete. In this paper we show that, on the other hand, the formula complexity and the program complexity are linear, resp...

First-order modal logic is clearly more than classical first-order logic, because modalities can be used to phrase more advanced necessity and possibility properties. At the same time, first-order modal logic and non-modal first-order logic are not entirely different either. Modalities quantify over all ( ) or some (♦) worlds. Quantifiers also quantify over all (∀) or some (∃) objects (yet not ...

Queries over temporal databases involve references to time. We study diierences between two approaches of including such references into a rst-order query language (e.g., relational calculus): explicit (using typed variables and quantiiers) vs. implicit (using a nite set of modal connectives). We also show that though the latter approach|a rst-order query language with implicit references to ti...

This is an overview a study of interpretability logic in Zagreb for the last twenty years: a brief history and some planes for further research. The idea of treating a provability predicate as a modal operator goes back to Gödel. The same idea was taken up later by Kripke and Montague, but only in the mid–seventies was the correct choice of axioms, based on Löb’s theorem, seriously considered b...

Abstract Natural language does not express all connectives definable in classical logic as simple lexical items. Coordination English is expressed by conjunction and , disjunction or negated nor . Other languages pattern similarly. Non-lexicalized are typically compositionally: English, combining negation ( both ). This surprising: if $$\wedge $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math...

It is proved that all the isomorphisms in the Cartesian category freely generated by a set of objects (i.e., a graph without arrows) can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free Cartesian category is an isomorphism. Introduction. We believe that a logi...

This paper establishes undecidability of satisfiability for multi-modal logic equipped with the hybrid binder ↓ , with respect to frame classes over which the same language with only one modality is decidable. This is in contrast to the usual behaviour of many modal and hybrid logics, whose uni-modal and multi-modal versions do not differ in terms of decidability and, quite often, complexity. T...