
Authors:
Yehuda SCHWARTZ, George TOURLAKIS
Title:A Proof Theoretic Tool for FirstOrder Modal Logic
Pages:93110
File:bibtex
Abstract ( + )
The authors of[11] initiated the(syntactic) proof theoretic study of the firstorder modal logics M^{3} and BM that were originally introduced in[5,14,15]. The principal tool in[11] was the "Gentzenisation'' of said logics, in the paradigm of[8,9,16], and cut elimination. In this paper we introduce a new technique based on formula to formula mappings that we call "formulators'', which preserve provability. As applications we present both positive and negative metatheoretical results for three logics:the aforementioned M^{3}; another logic,ML^{3},that we introduce here as a firstorder counterpart of the logic GL(of [2,6,8,9,16]);and for the quantified version of GL,QGL,obtained from classical first order logic by adding the GL axiomschemata without requiring  as we do for M^{3} and ML^{3}  that □A is always closed. QGL is a logic that does not allow cut elimination([1])  that is, its natural Gentzen equivalent does not  and this demonstrates the significance of the formulator tool;not only allows simple proofs, bypassing two highly non trivial steps for the logic under study  namely, Gentzenisation and cut elimination  but it also finds application to predicative modal logics,such as QGL, where cut elimination is provably impossible. Among the positive results we prove that the weak reflection property and the conservation property hold for all of M^{3}, ML^{3} and QGL. We also establish the negative results that neither A→□A nor □A→A are provable in any of the three logics.Our positive and negative result for QGL are,as far as we know,new in the literature.

Authors:
Karel CHVALOVSKY
Title:Note on DCompleteness and Prelinearity
Pages:111118
File:bibtex
Abstract ( + )
We prove that any set of axioms containing B, B', and the prelinearity axiom is Dcomplete, i.e. the very same formulae are provable with the rule of condensed detachment as are with modus ponens and substitution.

Authors:
Veronica QUIROGA
Title:An Alternative Definition of FStructures for the logic C_{1}
Pages:119134
File:bibtex
Abstract ( + )
In this work we give a characterization of the Fstructures defined by M.M. Fidel, which are the basis of the algebraicrelational semantics for the paraconsistent logic C_{1}. The new formulation is simpler and it allows a deeper study of such structures, as we will show in some applications in this article.

Authors:
Rafal GRUSZCZYNSKI
Title:Mereological Fusion as an Upper Bound
Pages:135150
File:bibtex
Abstract ( + )
Among characterizations of mereological set that can be found in the literature, the following two are the most important: these of mereological sum and mereological fusion. As it was proven in [PietruszczakM] the two relations coincide in the class of separative partial orders. In this paper we show that the distinctive element which is sufficient for fusion to be sum is that fusion is an upper bound of its constituents. This motivates investigation of structures in which we axiomatically demand that fusion is an upper bound. Moreover, we compare fusion relation with the standard relation of supremum and prove that in very weak structures interconnections between fusions and suprema coincide with those between suprema and mereological sums. For this reason the paper may be considered as the sequel to [GruszczynskiPietruszczakMSSPOSETS]

Authors:
Tomasz POLACIK
Title:Bisimulation Reducts of FirstOrder Kripke Models
Pages:151160
File:bibtex
Abstract ( + )
We consider elementary submodels of a given intuitionistic Kripke model K meant as models that share the same theory with K and result in restricting the frame of K and/or replacing some of its worlds with their elementary substructures. We introduce the notion of bisimulation reduct of the Kripke model wich allows us to construct elementary submodels of given Kripke models in the sense of the definition above.

Authors:
Lidia TYPANSKA
Title:A Note on Postcomplete Extensions of a Logic of Values of A. Ivin
Pages:161168
File:bibtex
Abstract ( + )
In this note we prove that the logic introduced by A. Ivin has infinitely many Postcomplete extensions.

Authors:
Janusz CIUCIURA
Title:NonAdjunctive Discursive Logic
Pages:169182
File:bibtex
Abstract ( + )
The discursive logic is sometimes treated as an example of the socalled nonadjunctive logic, which is correct but only as long as the definition of the discursive conjunction is deleted. In [1], we indicated a possible source of this confusion.
A nonadjunctive logic is not closed under the rule P, Q /P∧ Q. From a practical viewpoint, it may be justly called an obstacle to be overcome at any rate.
Jaskowski's nonadjunctive discursive logic clearly shows the importance of a philosophical explanation for a formal approach to natural language. He points to the fact that people taking part in the discussion may disagree while being (self) consistent. They may use some vague words or terms either purposefully or unintentionally. In such a case, ``from the fact that a thesis P and a thesis Q have been advanced in a discourse it does not follow that the thesis P ∧ Q has been advanced because it may happen that P and Q have been advanced by different persons" ([7], p. 49.)
The aim of this paper is to present a semantical study of the nonadjunctive discursive logic. We, additionally, show that the logic has a weak adjunctive property (in a sense given below).

Authors:
Slawomir KOST
Title:Countable Frames for Bimodal Logics S5 ⊗ S5 and Grz.3⊗ Grz
Pages:183198
File:bibtex
Abstract ( + )
In this paper we consider bimodal logics S5⊗ S5 and Grz.3⊗ Grz.3. We construct and describe two single countable frames which characterize systems S5⊗S5 and Grz.3⊗Grz3, respectively.