Traditional Logic
This article deals with the structure of concepts, different from Aristotelian tradition based on the ontology “thing–property”. Nonaristotelian concept structure is analyzed on the example of procedural concepts of the JSMmethod of automatic hypotheses generation. For procedural concepts a refinement and a widening of G. Frege’s triangle are proposed. For the procedural concepts Frege’s triangle formed by intension (content), extension (scope) is supplemented with the procedural proposition which transforms the initial data by means of the intension into the extension. In the paper is also given an example of the violation of the so called “law of the inverse relation” between the scope and the content of a concept for the concepts of the JSMmethod of automatic hypotheses generation. Formulated are the specialties of the construction of nonaristotelian concepts as means of organization of knowledge, not “forms of thought”. The example of concepts representing the JSMreasonings demonstrates the difference of their structure from the understanding of concepts within the aristotelian tradition. The problem of development of ideas into concepts allowing exact characterization is also discussed. Keywords: Aristotelian tradition of concepts, extension (scope), intension (content), J.S. Mill’s inductive methods, JSMreasoning, procedural concepts, G. Frege’s triangle. For two syllogistics (fundamental and traditional) we define nonstandard translations into the predicate calculus. These translations make it possible to treat syllogistic theories as logics of antiextensions of the subjects and the predicates of categorical statements. In compliance with the first translation, SaP means that antiextension of S is included in antiextension of P, SeP means that antiextensions of S and P don’t contain common elements, SiP means that the intersection of antiextensions of S and P is nonempty, SoP means that antiextension of S is not included in antiextension of P. For arbitrary syllogistic formula A, the second translation includes additionally the precondition that antiextensions of all the terms in A are nonempty. It is proved that these two syllogistics are embedded into the classical predicate calculus under the given translations. Keywords: syllogistic, predicate calculus, embedding function, extensions and antiextensions of terms. In the article we propose a new type of models for categorical attributive propositions. Usually the subject and predicate of the attributive proposition are interpreted extensionally as some sets of individuals. Logical relations between subject and predicate are understood as the settheoretic relations between their extensions. We propose a syntactic interpretation of subject and predicate of attributive propositions and interpret them as some formulas of propositional logic. These formulas can be understood as the definitions of common terms. Logical relations between subject and predicate are understood as the logical relations between their definitions. The article contains a proof that the system of fundamental syllogistic is consistent and complete with respect to the proposed semantics. Built interpretation can be generalized to other systems of syllogistic. Keywords: logic, attributive proposition, common term, syllogistic, definition, concept, analyticity, model.
Nonclassical Logic
We consider the class of propositional normal modal logics. The two main concepts related to this class and analyzed in the paper are the finite model property and constant formula. A propositional normal modal logic has the finite model property, if it can defined as the set of formulas true in frames of some set. All “natural” propositional normal modal logics turned out to have the finite model property. In the 60 years it has been observed that in some cases adding to the axiomatics constant axiom remains Kripke completeness, and hence the finite model property. Note (folklore) that using the deduction theorem it can be shown that here as logic can take the minimal normal modal propositional logic K. Under constant formula, the constraction of which does not use variables, that is, the basic formula is the constant ⊥ (false). (Note that in the absence in language the constant can be considered constant formula is a formula that is equivalent to any of substitutional instant; that is, say, the formula p∧&¬p.) The main result of the paper is the definition of a normal modal propositional logic L and a constant formula φ, such that the result of adding to the logic L axiom φ does not have the finite model property. The paper concludes with a short list of open problems. Keywords: normal modal logic, finite model property, constant formula, deduction theorem. We study an expressive power of temporal operators used in such logics of branching time as computational tree logic or alternatingtime temporal logic. To do this we investigate calculi in the firstorder language enriched with the temporal operators used in such logics. We show that the resulting languages are so powerful that many ‘natural’ calculi in the languages are not Kripke complete; for example, if a calculus in such language is correct with respect to the class of all serial linear Kripke frames (even just with constant domains) then it is not Kripke complete. Some near questions are discussed. Keywords: Kripke incompleteness, firstorder logic, computational tree logic, alternatingtime temporal logic, recursive enumerability. We offer a generalization of the wellknown Glivenko’s theorem on doublenegation translation. In “Sur quelques points de la Logique de M. Brouwer” V.I. Glivenko got a result, which is now called Glivenko’s theorem and which establishes the equivalence between a statement that a formula belongs to classical propositional logic and a statement that a doublenegation of this formula belongs to intuitionistic propositional logic. Glivenko’s theorem is an important achievement in the field of research concerning links between logics conducted using the embedding operation. Here we propose a generalization of Glivenko’s theorem and describe a method which is based on this generalization for constructing analogues of the statements that is some special form of Glivenko’s theorem. In this paper we used author’s original sublogics of classical propositional logic. In particular, logic Int_{<ω,ω>} played a principal role (it is, also, a sublogic of intuitionistic porpositional logic). The use of this logic made it possible to give such a generalization of Glivenko’s theorem that covers some extensive (cardinality of the continuum) class of sublogics of intuitionistic propositional logic. Keywords: Glivenko's theorem, classical propositional logic, intuitionistic propositional logic, language L, Llogic, calculus HInt_{<ω,ω>}, Llogic Int_{<ω,ω>}, calculus GInt_{<ω,ω>}, Glivenko's type logic. In this paper four lattices of fourvalued modal logics are considered. Different algebraic structures are the basis for construction, then these structures are gradually extended by the endomorphisms and constant functions. In the first case, the lattice of extensions of Boolean algebra B^{2} is formed, then the lattice of extensions of De Morgan algebra DM4 is constructed. In both cases the different modal logics appear, the properties of which are described and compared. The lattice with tetravalent modal logic TML is considered individually. Finally, the first two lattices are joined and the class of basic fourvalued modal logics is singled out. This class consists of Łmodal system of Lukasiewicz, logic V2 of Sobochiński and von Wright’s truthlogic T′′. Special attention should be paid to the logic Tr, which is functionally equivalent to the logic V2 and occupies a central place in the final lattice. This logic is the only of the aboveconsidered logics which has the Craig’s interpolation property, and besides is an excellent candidate for the role of propositional logic of truth. In conclusion its axiomatization is presented. Keywords: modal logics, lattices of logics, Boolean algebra, De Morgan algebra, endomorphismus, closed classes of functions, Boolean cascades, logic Tr. In [6] the definition of natural implication was introduced. One of the criteria for natural implication is the normality of logical matrix [2, p. 134], a condition sufficient for verification of modus ponens. In this paper two definitions of modus ponens are regarded: in the designationpreserving sense and in the tautologousnesspreserving sense. These formulations are considered as applied to twovalued and threevalued cases. In twovalued case these formulations are equivalent. But in case of threevalued logic we have another situation: they are not equivalent, but the first formulation entails the second, the reverse is not the case. According to that fact, the definition of natural implication is transformed and truth tables for extended class of natural implications are presented. Keywords: threevalued logic, natural implication, modus ponens principle
Logic and Language
According to some philosophers we can distinguish two trends in dealing with (especially natural) language. One of them is older and uses explications that simplify the richness of the language, so that the result of its efforts is an artificial image of language not corresponding to its real shape. The more recent trend tries to capture all the richness of the language together with all its irregularities and is represented mainly by Quine’s and later Wittgenstein’s philosophy. The older trend (I call it analytic group, AG, here) is sometimes criticized as being somehow obsolete while the more recent trend (called here QW group, QW, here) is then evaluated as more promising (more ‘progressive’). I try to show that AG is incomparable with QW because both try to answer distinct questions, solve distinct problems. (A comparison could be realized on the higher level of evaluating the choice of problems itself, which is another topic.) Keywords: sense, meaning, reference, denotation, explication, abstraction. This contribution zeroes in providing a formal tool for modelling reasoning with public announcements. In Section 1 I briefly delineate the idea of a new approach to public announcement interpretation via enthymematic implication presented in [1]. Section 2 contains a preliminary review of different cases representing reasoning with public announcements. Starting out from famous Muddy Children example I distinguish four types of such Leibniz set a problem of the Universal characteristic, but he is extremely focused on the mathematical explanation instead of metasymbolic consideration of the method. The symbols of a metascience are energy maximums and minimums. Creation of a block matrix (by means of the left tensor square) allowed to reveal macrolevel. The alphabet of educational metasymbols solves a problem of polystructural integration of knowledge naturally through their comparison. The genetic table has 4 blocks of the designated and antidesignated pairs of metasymbols which are based on the universal language. Universal language unites various sciences and eras, considering everything from the point of view of Eternity, allowing to expect new results. Keywords: universal characteristic, maximums and minimums, metasymbols, Universal language.
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