
Authors:
Kazimiera DYRDA and Irena JANICKAZUK
Title:Axiomatization of the logics determined by finite relational systems of natural numbers with identity
Pages:152155
File:bibtex
Abstract ( + )
In this paper we show that the logics determined by relational systems N_{n} = < N_{n} ; = >, where N_{n} = {1, 2,..., n} are axiomatizable.

Authors:
Vladimir SOTIROV
Title:Arithmetizations of the syllogistic a la Leibniz
Pages:156163
File:bibtex
Abstract ( + )
Lukasiewicz had every reason to suppose that Leibniz's winged Calculemus! had been connected with the Aristotelian syllogistic. Indeed, after Louis Couturat's pioneer efforts in commenting and publishing Leibniz's logical opuscula. The basic idea of the arithmetization of syllogistic was to establish a correspondence between terms and suitable integers (the characteristic numbers of notions), so that the logical truth of a proposition would turn into an arithmetical truth of unsuccessful. The second one used a translation of terms into pairs of coprime numbers and was successful, as Slupecki proved. This second translation obviously was more soffisticated but the bigger trouble was in the paper we justify the viability of the earlier (and less complicated) Leibniz idea. Moreover, we propose two translations into arithmetic which are appropriate for syllogistic with all Boolean term operations.

Authors:
Vladimir V.RYBAKOV, M. TERZILER and C. GENCER
Title:Description of selfadmissible quasicharacterizing inference rules
Pages:164171
File:bibtex
Abstract ( + )
We study quasicharacterizing inference rules which were introduced into consideration by A.Citkin (1977). The main result of our paper consists of a complete description of all quasicharacteristic rules which are selfadmissible. It turns out that a rule is selfadmissible iff the frame of the algebra generating this rule is not rigid. We also prove that selfadmissible rules are alwaysadmissible in canonical, in a sense, logics S4 or IPC regarding the type of algebra generating rules. Since the limitation on volume of the papers submitted to this journal to 8 pages, we are not a position to submit complete proofs of our results. We intend to submit the complete version of our paper to the journal Studia Logica.

Authors:
Vladimir V.RYBAKOV and T. ONER
Title:The structure of rigid frames of restricted depth
Pages:172181
File:bibtex
Abstract ( + )
In this paper we describe in precise the structure of all rigid modal rooted frames of depth 2. The interest to rigid frames came from the paper Description of selfadmissible quasicharacterizing inference rules (V.V.RYBAKOV, M.TERZILER and C.GENCER) where it has been showed that any givenquasicharacterizing inference rule is selfadmissible if and only if the frame of the algebra generated that rule is not rigid. The study of admissibility of inference rules was, in particular, motivated by H.Friedman problem of determining the admissibility forinference rules in intuitionistic propositional logic IPC. We note here that the introduction of quasicharacteristic rules by Citkin, as a generalization of Jankov's characterizing formulas, brought important results and a description of all quasicharacterizing rules admissible in IPC. As we noted above, it is proved that quasicharacterizing rule is selfadmissible iff the frame of the algebra originated that rule is not rigid. Therefore in this paper we restrict our attention to describe the structure of all rigid frames of restricted depth. Since in whole the structure can be very complicated we will deal with only frames of depth 2. By a considerably simple reasoning we will be in a position to give very transparent description for all rigid frames of depth 2. And hence we are in a position to specify in precise selfadmissible quasicharacteristic rules of algebras of depth 2.

Authors:
Eiko ISODA and Tatsuya SHIMURA
Title:Kripke incompleteness of some predicate extensions of modal subframe logics without finite embedding property
Pages:182189
File:bibtex
Abstract ( + )
Let QL be the least predicate extension of a normal extension L of S4 and BF be the Barcan formula. Ghilardi showed that it is rare that QL is complete with respect to Kripke semantics. On the other hand, if L is a subframe logic with the finite embedding property, we can show the completeness of QL+BF by the method of canonical models. It is natural to ask whether QL+BF is complete if L is a subframe logic without finite embedding property. Cresswell described a proof due to Fine of the incompleteness of QS4M = QS4M + (LMp implies MLp) + BF and asked whether QS4.3.1 + BF is complete or not. In this note, we solve the problem negatively by proving the following theorem since QS4.3.1 + BF is in this interval.

Authors:
Gabor SAGI
Title:A model Theoretic characterization of complexity of theories
Pages:190195
File:bibtex
Abstract ( + )
Our goal is to characterize the (recursion theoretic) complexity of the axiom systems of a first order theory in terms of the models of that theory.

Authors:
Zbigniew STACHNIAK
Title:Reasoning with partial situations
Pages:196206
File:bibtex
Abstract ( + )
The semantic framework of Z.Stachniak's Nonmonotonic Theories and their Axiomatic Varieties for modeling of propositional nonmonotonic reasoning is extended to other forms of nonmonotonic reasoning including reasoning about actions and change.