Bulletin of the Section of Logic 38/34 (2009)

Authors:Jan WOLENSKI
Title:The Principle of Bivalence and Suszko Thesis
Pages:99110File:bibtexAbstract ( + )According to Suszko, every logic with structural consequence operation is bivalent, that is, evaluates sentences as either true or false. Truth and falsehoods are logical valuations. On the other hand, the Suszko thesis (ST) admits algebraic valuations as related to semantic correlates of sentences. Due to Frege's axiom (materially equivalent sentences are identical), standard propositional calculus identifies logical and algebraic valuations. SCI (sentential calculus with identity) considers identity and equivalence as different connectives. Hence, we can ascribe separate semantic correlates to different true sentences. Thus, SCI is logically bivalent and algebraically multivalent. This paper argues that although ST can be defended, its justification is pragmatic, but not purely logical. 
Authors:Grzegorz MALINOWSKI
Title:Concerning Intuitions on Logical ManyValuedness
Pages:111121File:bibtexAbstract ( + )The widely acclaimed Suszko's Thesis on logical twovaluedness of structural logics raised a question on whether it is possible to define a relation of inference that would not have description in terms of zeroone valuations. A positive answer to this question was possible in the result of a division of the logic algebra of sentence referents into three subsets of elements: rejected, accepted, and neutral. The division affords possibilities to define a natural inference using the rules of inference which from nonrejected premises lead to the accepted conclusions. Accordingly, Tarski consequence is weakened to a qconsequence, the most cases of which are logically threevalued, since they can only be described by means of threeelement valuations.
Our aim is to discuss logically threevalued qconsequence matrixbased operations that have representation in terms of twovalued consequence operations. The possibility of such representation sheds new light on logical threevaluedness and calls for further exploration of nonorthodox inference operations possible  it was already used, e.g. for the construction of nvalued matrix inferences, for n>3. We also believe, that such an approach is a good start to the investigation of potentially rich spectrum of possible nonclassical inferences, founded on linear and nonlinear qmatrices.
The present paper is a version of the lecture given to the 2nd Conference on Nonclassical Logic. Theory and aplication, Lodz, 1719 September 2009. The talk was a tribute to Roman Suszko, whose ideas influenced my way of thinking on inferential manyvaluedness. 
Authors:Kordula SWIETORZECKA
Title:Between the Logic of Parmenides and the Logic of Liar
Pages:123133File:bibtexAbstract ( + )An epistemological realist usually places his opinions somewhere between two  in his opinion equally improbable – concepts of being: Parmenides’ theory of the impossibility of any change and variabilism by Heraclitus. Obviously, if we reject these extreme visions, we are still left with the wide range of opinions that may be assumed on the issue of the existence of various changes in the reality. In the presented text we shall focus on some specific concept of changes that – in opposition to Parmenides – does not generate any contradiction, what is more, it allows to talk consistently about some objects usually considered as paradoxical  about the socalled Eubulidenian sentences. The inspiration to take up this subject I owe to the late Professor Jerzy Perzanowski, to whom I dedicate the following considerations. 
Authors:Jerzy PLUTA
Title:Formalization of Leibniz's Notion of Existence and of God with Suszko's NonFregean Logic
Pages:135150File:bibtexAbstract ( + )The present study contains an attempt to formalize the essential part of Leibniz's metaphysics, and namely: of the notion of existence and of the notion of God. To this end we use the nonFregean logic created by Roman Suszko under the direct inspiration of Ludwig Wittgenstein's Tractatus LogicoPhilosphicus. In particular, we use the nonFregean sentential calculus with identity (SCI), which constitutes the core of Wlanguages, created by Suszko in order to formalise Wittgenstein's ontology of situations. In the present work we use the SCI with quantifiers to build Leibniz's Language (LL), that allows us to formulate the formal definition of existence (in the sense of Leibniz) for the universe of situations and next  of the Leibniz's notion of God. On the ground of these definitions we prove several ontological theorems, including  the theistic thesis. 
Authors:Andrzej INDRZEJCZAK
Title:Suszko's Contribution to the Theory of Nonaxiomatic Proof Systems
Pages:151161File:bibtexAbstract ( + )The paper is an exposition of the early works of Suszko devoted to the construction of nonaxiomatic proof systems. He provided a formalization of firstorder logic with identity and definite descriptions of quite an original character, based on the application of proper inference rules. A rule is proper, in Suszko's sense, iff schemata of premisses and conclusion are schemata of contingent formulae. According to Suszko, neither of nonaxiomatic systems proposed so far (e.g. Jaskowski's and Gentzen's work on natural deduction and sequent calculi) satisfies this requirement. The system proposed formalizes, in a proper way the firstorder logic but without identity; for the latter one axiomatic rule is needed. Suszko's system may be seen as an early attempt to provide a kind of sequent calculus which is in a sense opposite to Gentzen's approach. In the latter only one primitive sequent is postulated and all logical constants are characterised by means of rules of transformation of sequents. In Suszko's system the only rules transforming sequents are structural whereas all constants are characterised by primitive sequents representing (proper) inference rules. 
Authors:Andrzej PIETRUSZCZAK
Title:Simplified Kripke Style Semantics for Modal Logics K45, KB4 and KD45
Pages:163171File:bibtexAbstract ( + )In this paper we show that logics K45, KB4 (= KB5) and KD45 are determined by some classes of simplified Kripke frames without binary accessibility relations between possible worlds. These frames are ordered pairs of sets (W, A), where W is a nonempty set of worlds and A ⊆ W (a set of common alternatives to all worlds in W). From a frame (W, A) we can construct models of the form (W, A, V), where V is a standard valuation which to formulae and words assigns truthvales with respect to the set A. For K45 we use the class of all simplified frames; for KB4 we have the case that A = Ø or A = W; and for KD45 we use frames with A ≠ Ø. Moreover, to each one of these logics we also assign a suitable class of finite euclidean relational frames which satisfy conditions for normal extensions of K5 presented by Nagle in [1981]. 
Authors:Marek NASIENIEWSKI and Andrzej PIETRUSZCZAK
Title:Semantics for Regular Logics Connected with Jaskowski's D2
Pages:173187File:bibtexAbstract ( + )Jaskowski's discussive logic D_{2} was defined with the help of the modal logic S5. The formulation of D_{2} does not change, if instead of S5 we use some weaker logics. In the present paper we give semantics for the weakest regular modal logic defining D_{2}, we also recall semantics results for some other known logics defining D_{2}. Thanks to these results one can partially describe the lattice of logics defining D_{2}. 
Authors:Krystyna MRUCZEKNASIENIEWSKA and Marek NASIENIEWSKI
Title:Beziau's Logics Obtained by Means of QuasiRegular Logics
Pages:189203File:bibtexAbstract ( + )The paper extends results from [MruczekNasieniewska, Nasieniewski, BSL, 2005], [Marcos, L&A, 2005] and [MruczekNasieniewska, Nasieniewski, BSL, 2008] for the case of quasiregular modal logics. The original idea comes from [Béziau, LLP, 2006] where the logic Z is formulated with the help of the logic S5. The problem arises to use another modal logic instead of S5. In the present paper we give a framework for considering logics expressed by means of quasiregular modal logics and present an example of such a logic. The present paper is a written version of results presented during Conference “Nonclassical logics. Theory and applications” Lodz 2009, 17–19.09.2009 and basically relies on results from [MruczekNasieniewska, Nasieniewski, BSL, 2008]. 
Authors:Tomasz JARMUZEK
Title:Master Argument vs. Seafight Tomorrow
Pages:205214File:bibtexAbstract ( + )This paper deals with the Aristotelian problem of Seafight tomorrow, but it is analyzed from a Diodorean point of view. Precisely, we examine whether the lost argument of Diodor Cronus, called Master Argument, could be conducted within a frame of future open time. Since we do not know the argument itself, we follow the known reconstructions of Master Argument. Each of them requires special logical assumptions, but also semantic constraints concerning a frame of time. The last detail interests us the most. Showing shortly, step by step, all constraints, we come to the conclusion that the strictly treated reconstructions of Master Argument must prefer a linear model of future. 
Authors:Adam NIEWIADOMSKI
Title:On Type2 Fuzzy Logic and Linguistic Summarization of Databases
Pages:215227File:bibtexAbstract ( + )Type2 fuzzy logic is a generalization of fuzzy logic by Zadeh . The keypoint of this generalization is that membership degrees in type2 fuzzy sets (T2FSs) are traditional fuzzy sets in [0,1], while in traditional fuzzy sets (type1 fuzzy sets, T1FSs), membership degrees are real numbers in [0,1]. This provides more versatile means for modelling imprecise information, e.g. natural language expressions. Indent Computational methods based on T2FSs are broadly applied in many domains related to computer science. In this approach, we present the calculus of linguistically quantified statements in which linguistic expressions are represented by T2FSs. This provides techniques to define and generate linguistic summaries of databases in the sense of Yager but based on T2FSs. Linguistic summaries are natural language sentences generated by dedicated software from large databases (e.g. a few milion of records). Original elements are introduced by the author in a generalization of the calculus of linguistically quantified statements from type1 to type2, and definitions of cardinalities of T2FSs which cover real cardinalities of T1FSs. Hence, a point of departure to formalize and to apply type2 linguistic summaries of databases, their generation, their degrees of truth as measures of their informativeness, is presented. 
Authors:Szymon FRANKOWSKI
Title:Bisimulations and pmorphisms
Pages:229235File:bibtexAbstract ( + )One of the fundamental notions of model theory for modal logics is a bisimulation. Indeed it not only preserves logical value of formulas but it plays the important role for the modal definable classes of models). The main scope of the paper is to present several types of bisimulations that are proper for the different types of model structures  ordinary Kripke models, their manyvalued counterparts, or those based on topological spaces. Moreover, the important part of our work is to show connections between bisimulation and the socalled pmorphism. However, we will not limit ourselves to the well known statement that every pmorphism is a bisimulation. Some reduction of the notion of bisimulation to the notion of pmorphism will be shown.