Institute of Philosophy
of the Russian Academy of Sciences




  Logical Investigations. Vol. 9. – M.: Nauka Publishers, 2002. – 320 p. ISBN 5-02-013231-4
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Logical Investigations. Vol. 9. – M.: Nauka Publishers, 2002. – 320 p. ISBN 5-02-013231-4

This paper establishes the relation that holds between the temporal past and future and the emergence of uncertainty. This uncertainty involved in argument naturally leads to non-classical logic. A paradox hereby conceived consists in that the logic of uncertainty may be presented as a fragment of classical logic, which is demonstrated in what follows.

We present for the first time a detailed description of the life and work of I. E. Orlov (1886 - 1936) who is well-known as one of the pioneers of relevant logic, and whose interests touched a wide variety of fields of know¬ledge, from philosophy to chemistry and music theory. We show that the socio¬political climate of the1920s and 1930s exerted a significant influence on the style and content of this scholar's work. We theorize that this climate deter¬mined, to a considerable degree, the evolution of Orlov's interests and also his fate.

The article deals with some difficulties which arise in normal modal predicate systems with identity (based on standard semantics of possible worlds). In such systems we can derive some theorems which, under the intended interpretation, are intuitively unacceptable. One of them says: every true statement of identity is necessarily true, or, in other words, there are no true contingent statements of identity. Such theorems seem more unacceptable when we interpret the necessity operator epistemically as ‘someone knows that’ It is shown that difficulties of such kind do not arise in partial epistemic modal systems with identity (based on semantics of partial possible worlds), and no assertions of the above-mentioned kind are provable in them.

We present and discuss the fact that the well-known modal logic S5 and classical first-order logic are paraconsistent logics.

It seems that some English sentences naturally invite irreducibly multiple application of the method of supervaluations. In this paper, I use the machinery and ideology of Game-Theoretical Semantics (GTS) to investigate this phenomenon and trace some of its implications, both on the logical and linguistic sides.

The main results: solving the old algorithmic problems of tabularity and coincidence of normal modal logics with given tabular logic. The problem of tabularity is undecidable; the problem of coinciting is decidable only when given tabular logic is inconsistent. There are other results; open problems are discussed.

We are going to discuss certain systems (K4.G and K4.Grz) of modal logic that are of special interest in connection with the study of the notions of provability in Peano Arithmetic. K4.G (respectively, K4.Grz) is the result of adjoint a modal version G of the second incompleteness theorem (respectively, the formula Grz) to the modal system K4.

The basic notion of quasi-matrix logic is a notion of quasi-matrix. A set (Q,G,qf1,...,qfs) is a quasi-matrix. Q and G are non-empty sets. Q  G; qf1,...,qfs are quasi-functions. This logic had been created to describe connections between statements containing notions “necessity”, “possibility”, “contingency” and some others meaning as factual (physical, ontological) modalities. The main systems of four-valued and three-valued quasi-matrix logic are presenting in [1-6]. Quasi-matrix logic has been applied to the fields beyond logic (theory of notion, philosophical categories, theory of argumentation, etc) and in the other parts of logic as well. These fields are para-consistent logic for dubitable information, logic of propositional аttitudes, three-valued and five-valued logic of norms.

We suggest an intuitionistic variant of the famous W. Quine’s NF which we call NFI and which is equiconsistent with NF.

Logical analysis of the existence notion is closely related to treating individual descriptions as the type of denoting expressions. It is impossible to construct true statements about physically nonexistent objects, about mytho-logical and literary persons in frame of extensional theories of individual descriptions, particularly in Russell's theory. However if existence of any object is understood as known or believed existence of it, then it is easy to ascribe in epistemic modal contexts to statements in question value "truth". It is necessary to revise Russell's contextual definitions of individual descriptions under such approach and to demand fulfilling so called epistemic existence and uniqueness condition by descriptions to make them genuine singular terms.

We set out an intensional semantics for pure positive syllogistic language. According to it each term Q denotes not a set of individuals but a concept d(Q) considered as a non-empty set of positive or negative characters. We define a function * on concepts, which assigns to every concept  the con-trary concept *: pi *  pi * and pi *  pi , where pi is a positive character and pi is a negative character. SaP means that d(P)  d(S), SeP means that d(P)*  d(S)  . We prove that this semantics is ade-quate to syllogistic system with the following axiom schemes: (MaP&SaM)  SaP, (MeP&SaM)  SeP, SeP  PeS, SaS, SiP  SeP, SoP  SaP.

The main issue of the paper is the realisation of Alternative (0) within the logic of sense and denotation (LSD). The structural analysis of meaning and the notion of logical (semantical) competence of subject are sug¬gested as the approaches for solution, that provide epistemic character for the interpretation of LSD.

Leibniz worked out for the assertoric positive syllogistics an arith¬metical semantics with ordered pairs of mutually prime natural numbers as admitted values of term-variables. In [5], [14] was proved that the arithmeti¬cal semantics gives the adequate understanding of the Łukasiewicz's formal Systems L of syllogistics. In set-theoretical (extensional) semantics of the for-mal Systems B and Sm, in contrary to L, the empty value for term-variables is admitted and the propositions SaP and SeP means accordingly S P and S  P = (in B), S & SP and S  P = (in Sm). In this article is defined Leibniz-style arithmetical semantics with arbitrary natural numbers as admit-ted values of term-varibles for these systems.

A new notion of quasi-artificial object is introduced here. This is inspired by Computational Linguistic considering Natural, Formal and Quasi-natural languages. One member of this classification is obviously lost. Quasi-artificial objects as opposed to purely artificial ones are constructed from natural origins by purposeful and often formal transformations and actions. UNL language expressions can be viewed as examples here. Programming, technical engineering and creative thinking phenomena are considered from the point of view of this philosophical opposition.

The classical sentential logic which enrichment with help of truth¬fulness and falsehood operators is proposed in this paper. It is possible to express both semantical and non-semantical laws of contradiction and excluded middle in this logic. We adopted three groups of axioms in this logic: 1) axioms of the classical logic for formulas that are prefixed by truthfulness and falsehood operators; 2) axioms that express truth conditions for implication and 3) axiom that expresses the bivalence principle. This logic is generalized by extending of definition domain of truth and falsity predicates to any symbolic expressions universe of logic language. All axioms except bivalence principle is generalized to this universe too. So we get the symbolic expressions logic.

A propositional logic LAP with semantics of descriptions of state is constructed. For LAP a three valued characteristic matrix and Gentzen-type sequent calculus are presented. A theorem that LAP is paracomplete logic is formulated and a translation from the calculus ClP (which is a formalization of the classical propositional logic) to the LAP is described.

It is observed that fragments with only one monadic predicate letter of such logics as QK, QK4, QT, QS4, QGL, QGrz, and others are undecid¬able. It is proved that fragments with only one monadic predicate letter of QGLsem and QGrzsem (the sets of semantical consequences on Kripke frames of QGL and QGrz correspondently) are not recursively enumerable.

The main result: the provability problem of constant formulas is PSPACE-complete for modal logics K, K4. Some closed questions are discussed.

An attempt is done to represent possible worlds as individual models of knowledge inherited to some abstract knowledge subjects. The logic of those is not fixed and is chosen by knowledge subjects themselves. This becomes possible because of constructing two-leveled related semantics in frameworks of which any formula of object language is not valid in all words. In order to accept some thesis as the law of logic we need to postulate it.

In the paper principles of specifying the notion of analytical truth of judgment are considered. Non-standard approach to explication of that notion is proposed which is based on singling out two types of interpretation along with introduction of admissible possible realizations (semi-models) of language.

I describe a generalized truth-value space of constructive logic based on the concept of a generalized truth value. This space is organized by a specific algebraic structure  a trilattice which is a lattice with three partial orderings, representing respectively an increase in information, truth and con¬structivity. See also the counterpart paper [47].

The paper is continuation of the early published work (cf. Logical Investigations, vol. 6, 1999). Semantics of the system of non-non-fregean (metaphorical) logics is proposed and some metamathematical results are obtained (soundness and completeness theorems are among them). It turns out that the systems of such a kind allow to give a first order treatment of Routley-Griffin’s notion of relative identity. Then a pure metaphorical (non-suszkean) system of logics is proposed and metaphorical situational ontology is developed being the extension of the Wolniewicz’s situational ontology. Final remarks concern the issues of the translation of systems proposed into Leśniewski’s ontology.

We introduce a primitive recursive (PR-) realizability for predicate formulas, based on PR-realizability for arithmetic formulas, introduced by S.Salehi in 2000.The different cases from Kleene’s recursive realizability are  and , in which the recursive functions associated with, are restricted to primitive recursive. It is proved, that the set of PR-realizable predicate formulas are non-arithmetic. The similar results obtained for sets of PR-non-refutable and PR-realizable sequences.