
Authors:
Tadao ISHII
Title:Propositional calculus with identity
Pages:96104
File:bibtex
Abstract ( + )
We will introduce the classical propositional logic with identity (PCI in short), and show that our logic PCI corresponds exactly to the modal logic K. Next, we will introduce Kripketype semantics for PCI and prove the completeness theorem, by using the completeness result for K. Then, we will introduce several extensions of PCI, each of which corresponds to modal logics KT,KB,K4,KD,K5,KT4,KT5, respectively and prove their completeness. In the last section, we will discuss relations of our system PCI to Corsi's weak logic F with strict implication and Angell's system AC of analytic containment.

Authors:
Ewa GRACZYNSKA
Title:G. Birkhoff's theorems for Msolid varieties
Pages:105116
File:bibtex
Abstract ( + )
First G. Birkhoff's Theorem 1935, asserts that a set Σ of identities of a given type τ : I > N can be represented in the form Σ = θ (K), i.e. Σ is an equational theory if and only if Σ is closed under rules (1)(5) of derivation (called Birkhoff's rules of derivation). Second Birkhoff's Theorem asserts that a nonempty class of algebras of a given type τ : I > N is a variety (i.e. is closed under operation of passing to subalgebras, homomorphic images and also arbitrary direct products) if and only if it is an equational theory. Our aim is to present proofs for similar theorems but for the concept of Msolid variety.

Authors:
Paulo A.S. VELOSO
Title:On eight independent equational axiomatisations for fork algebras
Pages:117129
File:bibtex
Abstract ( + )
A fork algebra is a relational algebra enriched with a binary operation, called fork, where the projections can be defined by terms. Fork algebras can be axiomatised by equations giving properties of the fork operation and the projections. We examine the interplay between these equations and obtain eight independent equational axiomatisations for fork algebras with orientation ranging from fork to projections.

Authors:
Yvon GAUTHIER
Title:A polynomial translation of Goedel's functional interpretation
Pages:130137
File:bibtex
Abstract ( + )
I expand in this note a remark about Goedel's consistency proof for arithmetic (the Dialectica interpretation). Goedel introduced functionals (recursive functions of higher types) over all finite types as abstract objects beyond the (concrete) natural numbers. The Dialectica interpretation has been extended by Spector, Howard and Kreisel and others in the intuitionistic spirit of barinduction and barrecursion of finite type. Although Goedel was animated by intuitionistic motives, his proof for Heyting arithmetic can be translated for Peano arithmetic where its constructive content can be carried over. I propose a different approach to the consistency problem. Arithmetic here is not Peano arithmetic, but Fermat (or FermatKronecker) arithmetic with Fermat's infinite descent replacing Peano's induction and Kronecker's indeterminates instead of functional variables. The ``general arithmetic'' of polynomials (or forms, in Kronecker's terminology) is built upon ``effinite'' (infinitely proceeding, in Brouwerian terminology) sequences. Finite sequences are sets and the Cauchy (convolution) product for polynomials is used as a mapping from sequences to sequences in N while the degree of a polynomial replaces the type of a formula.

Authors:
Wojciech BUSZKOWSKI and Ewa ORLOWSKA
Title:Relational logics for formalization of database dependencies
Pages:138143
File:bibtex
Abstract ( + )
Dependencies between information items play an important role in knowledge representation. In this paper we present relational logical systems for representation of and reasoning about dependencies of attributes in information systems and we discuss relationships between these formalisms.

Authors:
Adam OBTULOWICZ
Title:Remark on visual presentation of deductions in Jaskowski's method of suppositions
Pages:144147
File:bibtex
Abstract ( + )
In E.Orlowska's On the Jaskowski's Method of Suppositions some formalization of Jaskowski's method of suppositions is presented and discussed. In the present paper we propose a geometrization of Jaskowski's method which means a uniform geometrical presentation of deductions formed according to the rules of the method. The proposed geometrical presentation is some precise mathematical description of the visual presentations of deductions given by Jaskowski in On the rules of suppositions in formal logic.