
Authors:
Robert SOCHACKI
Title:Axiomatic Rejection in the ImplicationalNegational Invariant Sentential Calculi of Lukasiewicz
Pages:16
File:bibtex
Abstract ( + )
In this paper it was proved that any k+1 valued implicationalnegational Lukasiewicz sentential calculus in invariant version is Ldecidable if proper computable set of rejected axioms is given. In the proof the rule of rejection by substitution is not used.

Authors:
Grzegorz MALINOWSKI
Title:That p + q = c(onsequence)
Pages:719
File:bibtex
Abstract ( + )
The famous Tarski's conditions for a mapping on sets of formulas of a language: (ref) reflexivity, (mon) monotonicity, and (cl) closure define the standard notion of consequence. The consequence, viewed as an operation or as a relation, is syntactically described in terms of proof and of rules of inference. In the end, the rules of any structural consequence are schematic and such consequence has a matrix description. Accordingly, it closely reflects Tarski's intuitions expressed in his seminal work.
Our aim is to raise a discussion on two revisions of the set (ref), (mon), and (cl), originating in matrix semantics and resulting in generalizations of the concept of consequence. One generalization changes the notion of proof; the other the form of rule of inference. The first generalization affects on semantics, which becomes essentially manyvalued. The second dually extends the line of approach.

Authors:
Tarek Sayed AHMED
Title:A Nonfinitizability Result in Algebraic Logic
Pages:2127
File:bibtex
Abstract ( + )
We show that if we expand the language of CA_{ω} by finitely many substitutions corresponding to bijective maps, then no quantifier free set of formulas containing only finitely many variables axiomatize RCA_{ω}.

Authors:
Tarek Sayed AHMED
Title:Neat Embedding is not Sufficient for Complete Representability
Pages:2935
File:bibtex
Abstract ( + )
We have already characterized the class of countable completey representable relation and cylindric algebras via special neat embeddings. In this note we provide a counterexample showing that the condition of countability cannot be omitted.

Authors:
Zofia KOSTRZYCKA
Title:On the Existence of a Continuum of Logics in NEXT(KTB+L^{2}p>L^{3}p)
Pages:3743
File:bibtex
Abstract ( + )
In this paper we consider formulas in one variable in the normal logic T2 = KTB+ L^{2}p>L^{3}p. Next, we use the formulas to define a continuum of logics over T2.

Authors:
Jose M. MENDEZ, Gemma ROBLES and Francisco SALTO
Title:The Basic Constructive Logic for NegationConsistency Defined with a Propositional Falsity Constant
Pages:4557
File:bibtex
Abstract ( + )
The logic B_{K+} is Routley and Meyer's basic positive logic B_{+} plus the K axiom. The logic B_{Kc4} is a negation extension of B_{K+} in which consistency can be understood as in the standard sense, i.e. as the absence of any contradiction. The logic B_{Kc4} is a weak logic, but we prove that a definitionally equivalent logic formulated with a falsity constant can be defined.

Authors:
Szymon FRANKOWSKI
Title:Pure Strict Implication Logics
Pages:5965
File:bibtex
Abstract ( + )
Most authors considering the logic of strict implication usually take into account languages which involve the others connectives, i.e. conjunction, disjunction, negation, and sometimes modal operators. We find infinite axiomatizations of two logics of strict implication, K and T, defined on pure implicational language. In this note we provide a simple proof of the fact that neither of them is finitely axiomatizable, and point out some their inconsistency properties.

Authors:
Andrew SCHUMANN
Title:NonArchimedean Valued Predicate Logic
Pages:6778
File:bibtex
Abstract ( + )
In this paper I propose the nonArchimedean multiplevalidity. Further, I build an infiniteorder predicate logical language in that predicates of various order are considered as fuzzy relations. Such a language can have nonArchimedean valued semantics. For instance, infiniteorder predicates can have an interpretation in the set [0,1] of hyperreal (hyperrational) numbers. Notice that there exists an effectively axiomatizable part of nonArchimedean valued predicate logic, namely the class of higherorder formulas such that all their predicate quantifiers are universal (or existential).

Authors:
Lidia BADURA and Marek ZAIONC
Title:Parametrizability by Regular Expressions for Equations on Words
Pages:7993
File:bibtex
Abstract ( + )
By well known Makanin's result it is decidable whether or not an equation on free monoid has solution. In case of infinitely many solutions for constantfree equations the structure of the set of solutions can be described by the proces of parametrization introduced by Khmielewski. It is proved by Khmielewski that solutions of any up to three unknowns constantfree equations are parameterizable but some equations with four unknowns are not. We propose another way of parametrization of solutions based on regular expressions in which it is possible to express nested substitutions like for example x > (x*y)* not expressible by standard Khmielewski's parametrization. For example it is proved that the class of quadratic equations is parameterizable by regular expression description while it is not parameterizable by the standard one.