Deductive Systems in Traditional and Modern Logic

Dear Colleagues,

I have the intention of launching a Special Issue of Axioms devoted to (1) the presentation of some new deductive systems, modified known systems and little-known systems with their specifics, intuitive foundation, and methodological or metamathematical properties (2) comparison of systems of the same scientific discipline with each other; as well as (3) a collective review of current and updated systems of the same type, taking into account discussed below all aspects and explaining their functioning in the past, present, and future (deductive systems will be understood as deductive theories and distinguished from their syntactic characterization by appropriate ordered pairs or triples).

Contemporary understanding of science, as a science of a high degree of exactness, requires treating it as a deductive theory (a deductive system). Loosely speaking, such a theory (system) is a set of its language sentential expressions that includes the set of all its expressions which are derivable (are deducible) from some expressions of the set by means of deduction (inference) rules, i.e. which are its consequences and have a proof on the basis of the expressions of the set. The feature of deductive systems (theories) is the deducibility and provability of their theorems. These systems are built using a method of deduction employing (a) the axiomatic method or (b) the natural deduction method (Jaśkowski-Słupecki-Borkowski, Gentzen or semantic tables). Method (b) leads to natural deduction systems, and the most often used method—(a)—leads to axiomatic systems and to the presentation or characterization of logical and mathematical theories as axiomatic deductive systems. Methods (a) and (b) may be used to build some other scientific disciplines such as physics, chemistry, sociology, philosophical and psychological sciences, information sciences, discursive sciences, computer science and some technical sciences.

Deductive sciences have not always been and are not always immediately built as axiomatic systems. Depending on the degree of methodological precision, three of their forms are distinguished: pre-axiomatic, non-formalized axiomatic, and formalized axiomatic. As we know, the pre-axiomatic form was typical of arithmetic and geometry, and later set theory, and probability theory and its axiomatization was carried out only in the 19^th century, while such mathematical theories as the Boolean system and theories of groups, rings, fields were immediately built as formalized axiomatic ones. Logical systems (calculi) and theories based on them and constructed by means of a deduction method are most often formalized axiomatic systems, but they can also be considered as formalized natural deduction systems.

Formalized axiomatic systems have a tradition originating from G. Frege (1891, 1903), but the first axiomatic system (non-formalized) in the history of science—as it was disclosed by Jan Łukasiewicz in his famous monograph on Aristotle’s syllogistic (1951)—was Aristotle’s syllogistic. J. Łukasiewicz initiated the construction of the first systems of syllogistic corresponding to the contemporary requirements, and thus the formalized axiomatic systems. He conducted formalization of syllogistic on two levels, using, for the first time, in addition to the usual axiomatic method by means of proof, a new axiomatic method by rejection proof—the so-called axiomatic rejection (or refutation) method. He, and continuators of his ideas (mainly J. Słupecki and the co-workers from his research circle), applied this method to the bi-level formalization of some classical and non-classical logical deductive systems of sentences or names to define two disjoint sets of language expressions of the given system: the set of all its theses (theorems); asserted, accepted, intuitively true expressions (called the assertion system); and the set of all the other non-accepted, or intuitively false, rejected expressions of the system (called the rejection or the refutation system). In this way, the bi-aspectul formalization of deductive systems provides some new inspiration to build different sciences.

Prof. Dr. Urszula Wybraniec-Skardowska
Dr. Alex Citkin
Guest Editors

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