What is quantum logic?

The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic.     

Cardinalities of proper ideals in some lattices of strengthenings of the intuitionistic propositional logic

We prove that each proper ideal in the lattice of axiomatic, resp. standard strengthenings of the intuitionistic propositional logic is of cardinality 2^?0. But, each proper ideal in the lattice of structural strengthenings of the intuitionistic propositional logic is of cardinality 2^2?0. As a corollary we have that each of these three lattices has no atoms.     

Second order propositional operators over cantor space

We consider propositional operators defined by propositional quantification in intuitionistic logic. More specifically, we investigate the propositional operators of the formA* :p q(p A(q)) whereA(q) is one of the following formulae: (¬¬q q) V ¬¬q, (¬¬q q) (¬¬q V ¬q), ((¬¬q q) (¬¬q V ¬q)) ((¬¬q q) V ¬¬q). The equivalence ofA*(p) to ¬¬p is proved over the standard topological interpretation of intuitionistic second order propositional logic over Cantor space.We relate topological interpretations of second order intuitionistic propositional logic over Cantor space with the interpretation of propositional quantifiers (as the strongest and weakest interpolant in Heyting calculus) suggested by A. Pitts. One of the merits of Pitts' interpretation is shown to be valid for the interpretation over Cantor space.Presented byJan Zygmunt     

On P-compatible hybrid identities and hyperidentities

P-compatible identities are built up from terms with a special structure. We investigate a variety defined by a set ofP-compatible hybrid identities and answer the question whether a variety defined by a set ofP-compatible hyperidentities can be solid.Presented byJan Zygmunt;     

A note on the system of propositional calculus with primitive rule of extensionality

Summary The present paper deals with a systemS of propositional calculus, conjunction, equivalence and falsum being its primitive terms. The only primitive rule inS is the rule of extensionality defined by the scheme:      

Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus

We present a general framework for proof systems for relational theories. We discuss principles of the construction of deduction rules and correspondences reflecting relationships between semantics of relational logics and the rules of the respective proof systems. We illustrate the methods developed in the paper with examples relevant for the Lambek calculus and some of its extensions.