ON NON-MONOTONOUS PROPERTIES OF SOME CLASSICAL AND NONCLASSICAL PROPOSITIONAL PROOF SYSTEMS

We investigate the relations between the proof lines of non-minimal tautologies and its minimal tautologies for the Frege systems, the sequent systems with cut rule and the systems of natural deductions of classical and nonclassical logics. We show that for these systems there are sequences of tautologies ψn, every one of which has unique minimal tautologies φn such that for each n the minimal proof lines of φn are an order more than the minimal proof lines of ψn.

NONCLASSICAL LOGICS IN RULE-BASED SYSTEM VERIFICATION PROBLEM

Работа посвящена обсуждению вопросов применимости неклассических логических исчислений к задаче верификации продукционных баз знаний. Рассмотрены возможности некоторых трёхзначных, четырёхзначных, а также нечётких логик. Показано, что хорошим подходом к верификации является использование логик с векторными семантиками в форме VTF-логик. Основанные на них экспертные системы смогут верифицировать свои БЗ без привлечения дополнительных (и внешних по отношению к ЭС) архитектурных элементов.

Paradoxes in scientific cognition and nonclassical logics

The subject of this research is the scientific paradoxes and such means for its resolution as nonclassical logics. The author defends a thesis that paradoxes often stimulate the scientific development. It is demonstrated that most vividly the problem of paradoxes manifested in crises in the fundamentals of mathematics; the attempts for its resolution in many ways contributes to the emergence of nonclassical logics. It is substantiated that nonclassical logics helped to resolve and explain the paraded occurring in scientific cognition. Comparative analysis is conducted on the capabilities of  three-valued “quantum logics” of Garrett Birkhoff and John von Neumann and “logics of complementarity” of Hans Reichenbach. Potential of the three-valued logics of D. Bochvar and nonclassical systems of A. Zinoviev in resolution and explanation of logical paradoxes, as well as importance of temporary logics of G. H. Wright for the philosophy of science is revealed. Special attention is paid  to the paraconsistent logics. The author determines two points of view in understanding of their essence and value for science and philosophy, which juxtaposition shows that none of them fully complies with the actual state of affairs. The main conclusion consists in the statement that paradoxes of scientific cognition should not be assessed just negatively; they also carry a positive meaning: detection of paradoxes in the theory testifies to the need for their elimination, more detailed research and stricter approach towards development of the theory, which in solution of this task can be accomplished by nonclassical logics.

A model of second-order arithmetic satisfying AC but not DC

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 (1979)].

REXPANSIONS OF NONDETERMINISTIC MATRICES AND THEIR APPLICATIONS IN NONCLASSICAL LOGICS

The operations of expansion and refinement on nondeterministic matrices (Nmatrices) are composed to form a new operation called rexpansion. Properties of this operation are investigated, together with their effects on the induced consequence relations. Using rexpansions, a semantic method for obtaining conservative extensions of (N)matrix-defined logics is introduced and applied to fragments of the classical two-valued matrix, as well as to other many-valued matrices and Nmatrices. The main application of this method is the construction and investigation of truth-preserving ¬-paraconsistent conservative extensions of Gödel fuzzy logic, in which ¬ has several desired properties. This is followed by some results regarding the relations between the constructed logics.

The History and Future of Logic Puzzles

This chapter first considers the history of logic puzzles through the contributions of Lewis Carroll and Raymond Smullyan. These two figures are united not only by their love of logic, but also by their conviction that puzzles provide an accessible gateway into the deep ideas of the subject. Their puzzles depend on what philosophers typically refer to as “classical logic.” Although mathematical historians generally credit Aristotle with being the first to undertake the study of logic in a systematic manner, the first to present logic explicitly for recreational purposes was Carroll. Smullyan is the most significant among the authors who have explored the recreational possibilities of propositional logic. The remainder of the chapter considers a possible future development of logic puzzles by investigating puzzles based on nonclassical logics.