
Authors:
Mohamed KHALED and Tarek Sayed AHMED
Title:Vaught's Theorem holds for L_{2} but fails for L_{n} when n > 2
Pages:107122
File:bibtex
Abstract ( + )
Vaught's theorem says that if T is a countable atomic first order theory, then T has an atomic model. Let L_{n} denote the finite variable fragment of first order logic with n variables. We show that a strong form of Vaught's theorem holds for L_{2} while it fails for L_{n} when n<2. An analogous result is proved for the finite variable fragments without equality.

Authors:
Andrzej WIŚNIEWSKI and Jerzy POGONOWSKI
Title:A Note On Diagonalization
Pages:123131
File:bibtex
Abstract ( + )
We present a diagonal method of constructing a denumerable family of infinite recursive subsets of a recursive set all of which are different from elements of an effectively given family of infinite r.e. subsets of this set. The construction in question leads to some incompleteness results, e.g. in problem solving systems.

Authors:
Andrew SCHUMANN
Title:Towards Theory of MassiveParallel Proofs. Cellular Automata Approach
Pages:133145
File:bibtex
Abstract ( + )
In the paper I sketch a theory of massively parallel proofs using cellular automata presentation of deduction. In this presentation inference rules play the role of cellularautomatic local transition functions. In this approach we completely avoid axioms as necessary notion of deduction theory and therefore we can use cyclic proofs without additional problems. As a result, a theory of massiveparallel proofs within unconventional computing is proposed for the first time.

Authors:
Gábor SÁGI
Title:A Short Proof for the Completeness of Paramodulacion
Pages:147152
File:bibtex
Abstract ( + )
In this note we provide a short and elementary proof for the well known fact that first order resulution extended with paramodulation is a sound and refutation complete calculus.

Authors:
Ewa GRACZYŃSKA
Title:Dependence Spaces
Pages:153160
File:bibtex
Abstract ( + )
This is a continuation of my lecture presented on 77th Workshop on General Algebra, 24th Conference for Young Algebraists in Potsdam (Germany) on 21st March 2009. The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finitedimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Steinitz.

Authors:
M. SPINKS and R. VEROFF
Title:Slaney's Logic F^{**} is Constructive Logic with Strong Negation
Pages:161174
File:bibtex
Abstract ( + )
In [19] Slaney et al. introduced a little known deductive system F^{**} in connection with the problem of the indeterminacy of future contingents. The main result of this paper shows that, up to definitional equivalence, F^{**} has a familiar description: it is precisely Nelson's constructive logic with strong negation [25].

Authors:
George WEAVER, Irena PENEV
Title:Simple Expansions of Classes Satisfying FraenkelCarnap Properties
Pages:175186
File:bibtex
Abstract ( + )
In the 1920's Fraenkel and Carnap raised the question of whether or not every finitely axiomatizable semantically complete theory formulated in the theory of types is categorical. The question remains open. Recent papers have provided partial answers to this and a related question for theories formulated in secondorder languages. These papers singled out subclasses of interpretations and showed that the secondorder theories of members of these classes are categorical, if the theories are finitely axiomatizable. This paper continues the search for partial answers. It focuses on simple expansions of classes previously studied.

Authors:
A.V. FIGALLO, C. GALLARDO and A. ZILIANI
Title:Weak Implication on Generalized Łukasiewicz Algebras of Order n
Pages:187198
File:bibtex
Abstract ( + )
J. Vaz De Carvalho and T. Almada in
A generalization of the Łukasiewicz algebras, Studia Logica 69 (2001), 329338 introduced the variety
,
m≥1,
n≥ 2, of
mgeneralized Łukasiewicz algebras of order
n as a generalization of Łukasiewicz algebras of order
n and a particular case of Ockham algebras. In this note, bearing in mind the important role that weak implication played in the study of Łukasiewicz algebras of order
n, we introduce an implication operation on
mgeneralized Łukasiewicz algebras of order
n. As this operation generalizes the one indicated above, we will call it with the same name. The deductive systems associated with this implication enable us to establish an isomorphism between the congruence lattice of an
mgeneralized Łukasiewicz algebras of order
n A and the lattice of all the deductive systems of
A. This result turns out to be quite useful for characterizing the principal congruences on these algebras simplier than the one described in the above mentioned paper.

Authors:
Jacob VOSMAER
Title:A New Version of an Old Modal Incompleteness Theorem
Pages:199204
File:bibtex
Abstract ( + )
Thomason [5] showed that a certain modal logic L⊂ S4 is incomplete with respect to Kripke semantics. Later Gerson [3] showed that L is also incomplete with respect to neighborhood semantics. In this paper we show that L is in fact incomplete with respect to any class of complete Boolean algebras with operators, i.e.~that it is completely incomplete.

Authors:
Norihiro KAMIDE
Title:An EmbeddingBased Completeness Proof for Nelson's Paraconsistent Logic
Pages:205214
File:bibtex
Abstract ( + )
It is known that a syntactical embedding theorem of Nelson's paraconsistent logic N4 into the positive intuitionistic logic LJ is useful to show the cutelimination and decidability theorems for N4. In this paper, a semantical embedding theorem of N4 into LJ is shown. An alternative proof of the Kripkecompleteness theorem for N4 is obtained by combining both the syntactical and semantical embedding theorems. Thus, the completeness, cutelimination and decidability theorems can uniformly be obtained from these embedding theorems. A singleconsequence Kripke semantics for N4 is also addressed based on a modification of the semantical embedding theorem.