This volume presents recent findings in diverse domains of a modern non‐classical logic. It combines reports delivered on the 2nd International Conference Smirnov’s Readings, sections on symbolic logic and philosophical logic (Moscow, 1999).
Liberation of a modern logic from a garment of human reasoning is studied in Karpenko’s paper “Logic at the borderline of millemum”. Author illustrates how a phylum of logic evolved from a closure operator and deductive systems to logical comprehension of a continuum; how classical logics were enriched by translations and embeddings; and how logic was influenced by algebra via lattices of theories and lattices of logics, categories and topoi. In the paper, we find a realistic explanation for the century‐long transformation “proper‐logic → meta‐logic → outer‐logic”.
This theme is continued in Pavlov’s paper “Conditions of an applicability of classical logic in the framework of languages of non‐classical logics”. The problem of applicability arises because a classical logic could not be properly utilized in certain types of philosophical reasoning, e.g. those including contradictory, antinomical, paradoxical or nonsense statements. Various conditions of applicability of classical logic are discussed in relation to languages of intuitionistic logic, Lukasiewich logic, Kleene logic and logic FL4 with falsehood operator.
Lednikov’s contribution “On the existence notion and existential statements” gives new treat to a notion of existence. Despite being usually expressed in logical quantifies an existence context may be studied using epistemic modalities. This could help to construct truthful assertions about physically nonexistent objects.
We know that dynamical seroantics give adequate interpretation of an anaphora. The anaphora itself could play a role of an update operator with a property of elimination. Such operator acts on a domain of episremic attitudes of a phrase and determines a meaning of anaphorically defined names. Thus Mikirtumov, in his paper “Semantics of anaphora as an operator of accomplishing definition” demonstrates that a static meaning of anaphoric operator is epistemically expressed using methods of van Eijck and de Vries update logic.
Two contributions of the volume deal with logics invented long time ago by N.A. Vasiliev. They are an imaginary logic (three‐valued logic with three quality types of propositions: affirmative, negative and contradictory) and n‐dimensional logic (representing n qualities of categorical propositions). In his paper “An embedding of N.A. Vasiliev’s imaginary logic into quantified three‐valued logic” Markin formalizes the imaginary logic and uncover metatheoretical relation of the imaginary logic with a quantified multi‐valued logic. He offers natural translation of affirmative, negative and contradictory statements to language of quantified three‐dimensional logic and proves that the imaginary logic is embedded into three‐valued Lukasiewicz logic. Natural semantics and formal constructs of n‐dimensional logic are offered in Kostyuk’s paper “N‐dimensional logic: modern reconstruction”.
Augmentation of intuitionistic propositional logic by modal operators admiring provability interpretation and temporal modalities always and before is suggested in Esakia’s paper “Synopsis of front on theory”. The contribution looks attractive because it shows a spectrum of algebraic, relational, topological and categorical features of the modalized Heyting calculus.
Karavayev (“On temporal qualification of normative propositions”) designs a complete and decidable system of temporal logic based on modification of a standard model of normative propositions. He uses a temporal structure, which is discrete and two‐side infinite. As a result, he offers adequate temporal interpretations of deontic operators.
Quasi‐matrix logical systems for scientific research are derived from modal notions used in biology in Mev and Mev’s paper “A problem of building up the theory of factual modalities”.
Enormous diversity of the results, published in the book, may be well presented as a brief list of some paper titles. The titles include “Impficatire logics, Lambek’s systems and exponential multicategories” (Vasyukov), “Kant’s model of Goa?” (Bocharov and Yuraskina) “Bolzano as the forerunner of constructivism” (Fyodorov), “Interpolation theorem for Hao Wang’s partial predicate calculus” (Bezhamshvili), “On semantics of Aristotle’s apodictic syllogistic” (Mchedlishvili), “On indiscrepancy of algebraic structures” (Novosyolov), “Sequential axiomatization of quasiminimal logic” (Popov), “Classical Multiplicative linear logic” (Mints and Soloviev), “An embedding of implicative fragment of classical logic into implicative one of intuitionistic logic” (Popov), “Universal deduction theorem” (Sidorenko), “Logics in philosophy and philosophy of logics” (Smirnova), “Logic and the relativity principle” (Samokhvalov), “Analytical tableau for positive logic free from ‘paradoxes’ of material implication” (Bystrov), “Early Markov’s constructivism realizability seroantics” (Nagorny), “An axiomatic calculus of uncertainty” (Anisov), “On applied theories with superintuitionistic logics” (Nepeivoda), and “On negative equivalent extensions of minimal logic” (Odintsov).
The book is a good reading for academics and students. Anyone interested in non‐classical logics and state‐of‐art results in logical sciences will be pleased with the quality, variety and amazing freshness of the contributions.
