Grzegorz Malinowski

Formal Methods in the Methodology of Empirical Sciences

A.- Some problems of formal methodology.- Approximate truth and truthlikeness.- A multiple sentential logic for empirical theories.- An axiomatic foundation for the logic of inductive generalization.- A two-dimensional continuum of a priori probability distributions on constituents.- Inductive logic and theoretical concepts.- A pragmatic approach to the formalization of empirical theories.- Uncertainty, probability and empirical knowledge.- The concept of empirical data.- Interpretation of theoretical terms: In defence of an empiricist dogma.- Definability problems in the methodology of science.- Laws, identities and reduction.- On logical analysis of methods.- Axiomatization in expected utility theory.- A logical model for game-like situations and the transformation of game-like situations.- Indeterminate probabilities.- Theoretical laws.- Causality, ontology and subsumptive explanation.- On the introduction of intensions into set theory.- Types of information and their role in the methodology of science.- Classification and ranking models in the discrete data analysis.- What have physicists learned from experience about inductive inference?.- B (Papers presented by title).- Verisimilitude: Popper, Miller and Hattiangadi.- On a general scheme of causal analysis.- Logic of quantum mechanics.- On possibilities and limits of the application of inductive methods.- Correspondence principle and the idealization.- Pragmatic meaning and truth.- Semantic complementarity in quantitative empirical sciences.- Marxs concept of law of science.- The impossibility theorem for universal theory of prediction.- Scientific knowledge-formation.- The methodology of behavioral theory construction: Nomological-deductive and axiomatic aspects of formalized theory.- Intertheory relations on the formal and semantical level.

Inferential Many-Valuedness

The problem of interpretation of logical values in addition to truth and falsity is still among the most controversial questions of contemporary logic. In connection with this the investigations of logical formalizations bore several descriptions of many-valued constructions in terms of zero-one valuations, cf. e.g. [6], [10] and [11]. Effectively, the interpretations associated with these descriptions shed new light on the problem of logical many-valuedness.

Key notions of Tarski's methodology of deductive systems

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 Tarskis methodology are presented and discussed through, the recent development of the original concepts and ideas.

Many-Valued Logic and its Philosophy

This chapter discusses the most important systems of many-valued logic and the vital philosophical and metalogical problems of many-valuedness. The chapter presents a historical account of the most important systems of many-valued logic and their development in particular, to give an overview of the original motivations and characteristic properties. The chapter thus outlines an early history of the subject and recalls some important justifications, interpretations and explanations of many logical values. Accordingly, some known, although the most justified, approaches to many-valuedness are recalled, and further the authors own approach to the problem of many-valuedness based on the inferential theory of (structural) propositional logics is presented in the chapter.

Degrees of maximality of Klukasiewicz-like sentential calculi

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.

Beyond Three Inferential Values

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.

Referentiality and Matrix Semantics

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.

Logic, Formal Methodology and Semantics in Works of Ryszard Wójcicki

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.

Identity, many-valuedness and referentiality

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].

Helena Rasiowa-a view of the academic trajectory and the influence upon Polish and international scientific community

The death of H. Rasiowa was a big loss for both, the Polish and the international scientific community. The life of Professor Rasiowa was passionately devoted to the development of logic, algebra and computer science and brought the fruits already reckoned among the results of the highest importance. The influence of her personality goes far beyond Poland and the group of logicians, mathematicians and computer scientists.