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# Logical Investigations. Vol. 8. – M.: Nauka Publishers, 2001. – 320 p. ISBN 5-02-013115-6

After having read the majority of publications on the problem of time, which are numerous, we acquire a grievous feeling that it is being mistreated. The issues discussed are often other than time itself but quite different phenomena although related to it in certain ways. Here we intend to focus discussion right on the phenomenon of time as it is without substituting it by other problems no matter how important they are. Eight fundamental properties of time are being considered in this paper.

The present paper presents a logic that allows for the abnormal behaviour of any logical constant and for the ambiguous behaviour of any non-logical constant, but nevertheless offers an interpretation of the premises that is as normal as possible.

Hao Wang [10] suggested two extensions of predicate calculus permitting partially defined predicates. He formulated two calculi of partial predicates PP and EP. Interpretation of implication of these calculi depends on the relations between all possible values of its parts. Therefore, the notion of a formula is restricted in PP and EP as implication is not iterating in them. A.Rose [9] constructed an independent system of axioms for propositional fragment of PP with finite number of axioms and proved its completeness by means of a syntactic criterion of provability in it. N.M.Ermolaeva [4] has done the same for propositional fragment of EP applying semantical criterion of the validity. In the present article we give the Rose style proof of complete¬ness of the propositional fragment of EP introducing a syntactic criterion of provability in it, and on the ground of a modified relational semantics we formulate extensions of PP and EP permitting iteration of implication.

The aim of this paper is to try to characterize classical propositional logic (CPL) with the notion of mathematical structure. We start by justifying this approach. We recall the importance and significance of the notion of structure in mathematics and in logic. We explain the idea of a general theory of logics based on structures, Universal Logic. CPL is not one structure but a class of equivalent structures, CPL-structures. We survey a series of structures that can be considered as CPL-structures. The main problem is to find a notion of equivalence which permits to gather into a whole this multiplicity. We show in particular that the modern concept of equivalence of structures, based on the notion of expansion by definition and isomorphism, is not adequate to define a satisfactory notion of equivalence that will define the class of CPL-structures. An alternative definition, postmodern equivalence, is introduced. It appears that this tentative of characterization of the class of CPL-structures is not only relevant for Universal Logic, but also for the general theory of mathematical structures, since the case of CPL-structures shows the insufficiency of the modern concept of equivalence between structures.

The problem of eliminability of Axiom of Choice from the metatheory of propositional modal logics is dicussed. Using of the language of modal formulas is proposed. Three examples of applications of this language are given: the Blok’s theorem on the degree of incompleteness; a logic without immediate predecessors; the finite axiomatizability of tabular logics.

The purpose of this paper is to review some ideas from Dunn and Meyer (1997) and Dunn (2001) and "write them large". The first paper showed how to represent combinatory algebras using ternary frames, and the second paper showed how to do the same for relation algebras. It should be pointed out that these representations both fall under the general heading of "gaggle theory", as developed in a series of papers beginning with Dunn (1991), in which an n-ary operator is represented using an n+1-placed relation. These results were presented in a somewhat mathematical fashion, although philosophical motivations were introduced as well. The present paper will focus on these philosophical motivations and clarify and extend them. In addition it will point to some future research directions in connection with Pratt's dynamic logic and Hoare's logic of programming.

The purpose of this paper is a restitution of a longstanding notion of weak-transitivity and a modal system, which may be identified with the set of formulas valid in all weak-transitive Kripke models. The modal system has a finite models property and can be axiomatized quite simple: it is the smolest normal logic to contain all instances of the formula p p p. The modal system is an especially interesting as the logic of all topological spaces provided that the limit-operation, a fundamental topological notion, is treated as the diamond modality.

Quasi-matrix logic is based on generalisation of classical logic principles: bivalency (propositions take values from the domain {t (truth), f (falsity)}); consistency (a proposition can not have both the values); excluded middle (a proposition necessarily has some of these values); identity (in a complex proposition, a system of propositions, an argument one and the same proposition has one and the same value from the domain {t, f}); matrix principle - logical connectives are defined by matrices. As a result of generalisation we have quasi-matrix logic principles: the principle of fourvalency (propositions take values from the domain {t n, tc, f c, f i}); consistency: can not have more than one value from {t n, tc, f c, f i}; the principle of excluded fifth; identity (in a complex proposition, a system of propositions, an argument one and the same proposition has one and the same value from the domain {tn, tc, f c, f i}); the quasi-matrix principle (logical terms are interpreted as quasi-functions). Quasi-matrix logic is a logic of factual modalities.

There analogies of substructural logics in Gilbert'’s form are considered. The main emphasys is made on Łukasiewicz’s infinite valued logic Ł and logic RM. Since some Gilbert'’s substructural logics are extensions of classical logic then the problem of explication of the notion of “substructural logics” arises.

The family E(J) of extensions of Johansson's minimal logic J can be divided into two disjoint parts: paraconsistent logics and superintuitionistic logics. We apply the results and methods found in our study of super-intuitionistic logics to paraconsistent logics. We obtain the full description of paraconsistent extensions of the logic JE' containing J, which have the interpolation property CIP or the projective Beth property PBP. In addition, we find some necessary condition for a logic in E(J) to have PBP or CIP.

We present a formal realization of Leibnitz' idea of intensional interpretation of the traditional syllogistic. We set out an intensional semantics and prove that it is adequate to Lukasiewicz' syllogistic. The core idea of this semantics is to associate with each term of a categorical statement not a set of individuals but a concept considered as a non-empty and non-contradictory set of positive or negative characters and to treat syllogistic constants as denoting intensional relations between concepts. According to this approach, ‘Every S is P’ means that S contains all characters from P, ‘Some S is P’ means that S and P don't contain contrary characters (positive and negative).

In [4] and [1] the set of formulae rejected in the Lukasievwicz`s caluculus of positive syllogistic was axiomatized. In this article we construct analogous adequate calculi of rejected formulae for the systems of positive syllogistic of Slupecki [3], Shepherdson [7; the system B] and Smirnov [5].

In this article a new proof of consistency of the classical formal number theory is given. The proof is performed in the Kleene-Nelson style.

Phenomenon of informalizability is studied from two points of view. First of all, as a phenomenon of humanitarian thinking. Secondly it is treated as one of the levels in the hierarchy of knowledge and skills following pr. Beloselsky-Belozersk

There are two stages in V.A.Smirnov's constructing of combined calculi of propositions and events. In [5] a number of two-level combined calculi are constructed where external level of propositions corresponding to an abstract part of logic is varied as well as their internal level corresponding to ontological presuppositions. At the same time the event terms p differ from propositions p and acts of assertion are not iterated. In his later work [6] V.A.Smirnov points out that an act of assertion itself is an event and the question of iteration arises. Referring to ideas laying in the grounds of von Wright's logic TL in [7] V.A.Smirnov proposes the generalized combined logic of propositions and events. V.A.Smirnov compares the language of combined logic of propositions and events OCM with the language of von Wrights logic of truth TL and shows that for each T-sentence corresponding formula of OCM can be constructed and vice versa. Since von Wright's logics of truth are similar to the logic with operators of truth and falsehood [3.4] (these are compared in [4]) it is interesting to compare the last one with combined logic of propositions and events. V.A.Smirnov formulates a number of combined calculi and logic of truth TL in sequent form. It is formulating the logic with operators of truth and falsehood FL4 also in sequent form.

A translation from the calculus Int which is axiomatisation of the intuitionistic propositional logic to the calculus Int+ which is axiomatisation of the positive fragment of Int is constructed.

The question about recursive enumerability of modal predicate logics defined by non first-order definable classes of Kripke frames is considered. It is proved that logics of classes of frames without infinite ascending chains are not recursively enumerable.

Kant’s claim that time is the form of inner sense is well-known. The following question is discussed: what is (if is) the form of time?

It is defines such the notion of the normalised standard inference from hypothesis that the deduction theorem in general form being relevant for every arbitrary calculus, including the empty one. This is reached without any change the notions of inference. The deduction theorem itself consist conditions of its adaptation.

W. Ackermann [1,2] constructed a logical system with unrestricted comprehension principle. Here the propositional fragment of this logical system is considered. This logic is a weakening of the classical logic based on an informal interpretation of the implication as derivability in an unspecified deductive system. A Kripke-style semantics for this propositional logic is pro¬posed and completeness theorem is proved.

A non-standard approach to semantics of intensional systems is developed. The semantics for intensional system IPL (Intensional Predicate Logic) is constructed. The pecularities of IPL: (1) any expression including intensional predicates and operators has an intension as well as an extension; (2) intesional contexts differ from extensional mainly by ascription of specific values to intensional predicates (operators) and, what is more important, by a way of their combination with arguments; (3) an intension of any complex extensional expression is a function of intensions of its compounds; (4) an extension of any complex intensional expression is a function of functor’s extension and intensions of its arguments; (5) unlike Montague's method, this approach allows to construct an intensional logic as first-order system. Logical aspect of analysis of intensional contexts is important for us. A key to the puzzle of these contexts can be found just there not in behavior of proper names. The core idea of this approach is that semantical analysis of intensional contexts presupposes, first of all, identification of peculiarities of their logical structure. Proposed approach discovers peculiarities of semantics of intensional contexts. It gives the key for comprehension Kripke's “puzzle of belief contexts”.

Getting started from ideas of N.A.Vasiliev and exploiting some conceptions of G.Frege, V.A.Smirnov introduced several combined calculi of sentences and events consisting of two parts: the abstract (external) logic depending on epistemological assumptions and the empirical (internal) logic depending on ontological ones. Early the author proposed to approach algebra of events as the discursive system of S. Jaśkowski (cf. [5]). One more interesting possibility would be an exploitation of an S4.2-modal algebra instead of an S5-modal algebra for discursive logic as an algebra of events. As it was shown by R.Goldblatt [4] in the Diodorean interpretation of modality where the operator of necessity  is read as “it is now and always will be the case that” time would be modelled by the four-dimensional Minkowskian geometry that forms the basis of Einstein’s special theory of relativity. In this case “event” y coming after event x just in case a signal can be seen from x to y at a speed at most that of the speed of light (i.e. y is in the causal future of x). Passing to the S4.2-modal algebra of the “histories” (subsets of events or causal paths) we thus obtain a combined calculus of sentences and histories. The same would be done in a more abstract way if we consider an algebra of the histories as a complete orthomodular lattice following to W.Cegla's approach (cf. [2]). For both formulations of combined causal logic of Minkowski spacetime a semantic of (event) bundles and semantic of possible worlds is built and some metamathematical results are obtained.

In [1] and [2] the background for pure theory of relevant entailment was built: ideological considerations were axiomatized and supplied with algebraic semantics. In this paper I will prove the adequacy for both relational and combinatory semantics of TE.