This book provides an incisive, basic introduction to many-valued logics and to the constructions that are "many-valued" at their origin. Using the matrix method, the author sheds light on the profound problems of many-valuedness criteria and its classical characterizations. The book also includes information concerning the main systems of many-valued logic, related axiomatic constructions, and conceptions inspired by many-valuedness. With its selective bibliography and many useful historical references, this book provides logicians, computer scientists, philosophers, and mathematicians with a valuable survey (...) of the subject. (shrink)
Referential semantics importantly subscribes to the programme of theory of logical calculi. Defined by Wójcicki in [8], it has been subsequently studied in a series of papers of the author, till the full exposition of the framework in [9] and its intuitive characterisation in [10]. The aim of the article is to present several generalizations of referential semantics as compared and related to the matrix semantics for propositional logics. We show, in a uniform way, some own generalizations of referentiality: the (...) first, directed to unrestricted cluster referential semantics, [4], its "discrete" version, a counterpart of algebraic semantics and a many-valued referentiality based on matrices, whose elements are functions from the set of indices to a finite n-element set of values, n≥2, [3]. Next to this we outline pragmatic matrices introduced by Tokarz in [6] as an alternative for cluster referential approach and discuss together all presented versions of referential semantics. (shrink)
The actual introduction of a non-reflexive and non-idempotent q -consequence gave birth to the concept of logical three-valuedness based on the idea of noncomplementary categories of rejection and acceptance. A q -consequence may not have bivalent description, the property claimed by Suszko’s Thesis on logical two-valuedness, ( ST ), of structural logics, i.e. structural consequence operations. Recall that ( ST ) shifts logical values over the set of matrix values and it refers to the division of matrix universe into two (...) subsets of designated and undesignated elements using their characteristic functions as logical valuations, cf. [4] The extension of the idea operates with three-valued function, with the third value ascribed to those elements of the matrix which are neither rejected nor accepted. Accordingly, the logical three-valuedness departs naturally from the division of the matrix universe into three subsets and the ( ST ) counterpart says that any inference based on a structural q -consequence may have a bivalent or a three-valued description. After a short presentation of the three-valued inferential framework, we discuss a solution for further exploration of the idea leading to logical n -valuedness for n > 3. Apparently, the first step in that direction is easy and it consists of a division of the matrix universe into more than three subsets. The next move, i.e. a definition of a matrix consequence-like relation being neither a consequence nor a q -consequence, seems extremely difficult. Therefore, here we consider only finite linear matrices with one-argument functions “labelling” respective matrix subsets. By means of these functions it is possible to represent a q-consequence as a “partial” Tarski’s consequence and, ultimately, to define a logically more-valued consequence-like relation. We believe, that the present partial proposal deserves an attention by itself but also that it may lead to a general approach to logically many-valued inference. (shrink)
The aim of the article is to outline the historical background and the present state of the methodology of deductive systems invented by Alfred Tarski in the thirties. Key notions of Tarski's methodology are presented and discussed through, the recent development of the original concepts and ideas.
In the paper * we discuss a distinctive versatility of the non-Fregean approach to the sentential identity. We present many-valued and referential counterparts of the systems of SCI, the sentential calculus with identity, including Suszko’s logical valuation programme as applied to many-valued logics. The similarity of different constructions: many-valued, referential and mixed, leads us to the conviction of the universality of the non-Fregean paradigm of sentential identity as distinguished from the equivalence, cf. [9].
The paper is a study of properties of quasi-consequence operation which is a key notion of the so-called inferential approach in the theory of sentential calculi established in [5]. The principal motivation behind the quasi-consequence, q-consequence for short, stems from the mathematical practice which treats some auxiliary assumptions as mere hypotheses rather than axioms and their further occurrence in place of conclusions may be justified or not. The main semantic feature of the q-consequence reflecting the idea is that its rules (...) lead from the non-rejected assumptions to the accepted conclusions.First, we focus on the syntactic features of the framework and present the q-consequence as related to the notion of proof. Such a presentation uncovers the reasons for which the adjective inferential is used to characterize the approach and, possibly, the term inference operation replaces q-consequence. It also shows that the inferential approach is a generalisation of the Tarski setting and, therefore, it may potentially absorb several concepts from the theory of sentential calculi, cf. [10]. However, as some concrete applications show, see e.g.[4], the new approach opens perspectives for further exploration. (shrink)
The paper is concerned with the problem of characterization of strengthenings of the so-called Lukasiewicz-like sentential calculi. The calculi under consideration are determined byn-valued Lukasiewicz matrices (n>2,n finite) with superdesignated logical values. In general. Lukasiewicz-like sentential calculi are not implicative in the sense of [7]. Despite of this fact, in our considerations we use matrices analogous toS-algebras of Rasiowa. The main result of the paper says that the degree of maximality of anyn-valued Lukasiewicz-like sentential calculus is finite and equal to (...) the degree of maximality of the correspondingn-valued Lukasiewicz calculus. (shrink)
For decades Ryszard Wójcicki has been a highly influential scholar in the community of logicians and philosophers. Our aim is to outline and comment on some essential issues on logic, methodology of science and semantics as seen from the perspective of distinguished contributions of Wójcicki to these areas of philosophical investigations.
The paper is concerned with the problem of characterization of strengthenings of the so-called Łukasiewicz-like sentential calculi. The calculi under consideration are determined by n-valued Łukasiewicz matrices with superdesignated logical values. In general, Łukasiewicz-like sentential calculi are not implicative in the sense of [7]. Despite of this fact, in our considerations we use matrices analogous to S-algebras of Rasiowa. The main result of the paper says that the degree of maximality of any n-valued Łukasiewicz-like sentential calculus is finite and equal (...) to the degree of maximality of the corresponding n-valued Łukasiewicz calculus. (shrink)
The paper is an extended version of a talk given to the XXXth Conference on the History of Logic devoted to the work of Professor Roman Suszko . Its aim is to present Sentential Calculus with Identity in comparison with other formalizations of propositional identity.
The development of the method of logical matrices at the turn of 19th Century made it possible to define the concept of many-valued logic. Since the first construction of the system of three-valued logic by ukasiewicz in 1918 several matrix based logics have been proposed, cf. [8]. The aim of the present paper is to touch upon some problems related to the topic, which would permit one to get a viewpoint upon the nature of many-valuedness. First, we show that the (...) multiplication of logical values is not a sufficient condition to obtain a non-two-valued logic. Second, we discuss an ingenious solution by R. Suszko [11] explaining through the sentential identity an ontological nature of non-classical logical values. Next, we present a kind of metalogical relation of inference, so-called q-consequence, being three-valued in its spirit. The last chapter will bring a concise description of two ukasiewicz “manyvalued” systems of modalities and an application of the paradigm of q-consequence to these systems. (shrink)
In the paper Weaver's method is adapted to prove interpolation properties of many-valued propositional calculi standard in the sense of Rosser and Turquette. The case of n-valued Lukasiewicz calculi is discussed in connection with the results obtained.
o formalization of intensional functions was made for the purpose of many-valued interpretation of the belief-operators within the scope of the classical logic system. The first aim of the paper is to present and discuss this rather unknown many-valued construction and its properties. The fact that the manyvaluedness of o systems is purely formal - their characteristic matrices are Boolean - calls for further consideration. Departing from intristic similarities of the tables for the epistemic operators to the information functions we (...) show that o structures may be rewritten as special knowledge representation systems. These systems use 0 and 1 as the only values and are called “epistemic”. Their role for the theory of knowledge information systems may be compared to that of the functionally complete matrices in the class of all logical matrices for a given propositional language. (shrink)
