Institute of Philosophy
of the Russian Academy of Sciences

  Logical Investigations. Vol. 7. – M.: Nauka Publishers, 2000. – 318 p. ISBN 5-02-008359-3
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Logical Investigations. Vol. 7. – M.: Nauka Publishers, 2000. – 318 p. ISBN 5-02-008359-3

Development of logic at the end of 20th century generated such questions as «What is logic?» or «What is logical system?». The aim of this paper is to find out and analyze the phenomena that led to such unexpected questions. Firstly, that is a continual variety of logical systems; secondly, an extension of logical system as a result of its restriction; thirdly, an embedding (translation) of richer logical systems into weaker systems; fourthly, a ten-dency to the investigation of classes of logical systems in the fashion of cer-tain constructions; fifthly, a completion of algebraization of logic; sixtly, the demands of computer revolution, etc. The result of logical concretization of two mathematical constructions - a closure operator introduced at the begin-ning of this century and a concept of category appeared at the middle of the century - was the beginning of investigations of different arrays of logical systems in a form of some lattice or category construction. Logic apparently loses the quality of science concerned with correctness of reasoning (and this is the phenomenon which expresses a crisis of modeling of truly human logic) and becomes the science about constructions that have an extremely abstract logical nature. In this sense logic is transformed just in metalogic with its new role. And the main point is that Logic comes out of limits of what is properly logical. Content of the paper: 1) Introduction; 2) Closure operator and deduc-tive systems; 3) Logical comprehension of continuum; 4) An extension of classical logic as a consequence of its restriction (translations and embed-dings); 5) Lattices of theories and logics; 6) Lattices of calculi and other con-structions; 7) Constructions called as «category» and «topos»; 8) Closure operator and category together; 9) Algebraization of logic; 10) In a search of logical system. Bibliogr.: 50-60 pp.

The article contains the analysis of Markov's never published investigations connected to Kleene's realizability, and is based on fragmentary data, any comprehensive written account being lacking. The following two Markov's problems are formulated: 1) Is there a possible link between so called Markov's principle and the completeness of the intuitionistic propositional calculus relative to Kleene's realizability? 2) Does the notion of Kleene's realizability enjoy the so called «disjunction property»?

There is a continuum of so called «superintuitionistic logic». Usual they are studied from a purely logical point of view. But many logi¬cally different calculi coincide when used in applied theories. Some results about «applied classification of logical calculi» are presented here. Arith¬metic are considered here as semi-formal system with -rule. Arithmetic with finite-valued superintuitionistic logics are classical. Arithmetic with a logic of linear chains in classical. Minimal arithmetic-based theories are defini-tionally equivalent to intuitionistic. хА(х)  хA(х) implies classical logic in any theory with two different elements. x (A(х)уВ(х, у))  ху(A(х)В(х, у)) implies (ABС)  (AB)  (AС) in any theory with two different elements.

A translation from the calculus Cl which is a formalization of the implicative fragment of classical logic to the calculus Int which is a for¬malization of the implicative fragment of intuitionistic logic is constructed. This translation is such that for all formulas A the following condition is satisfied: a formula A is deducible in Cl iff the translation of A is deducible in Int.

Some questions about strict implication in modal system S3 are discussed. The first theorem shows that strict implication formulas have the property of equivalent replacement; as well known the logic S3 has not the full theorem of equivalent replacement. The second theorem of the paper is connected with an open question [4] on description of one variable strict implication fragment of S3: this fragment contains at least 29 non-equivalent formulas; in the author's paper [2] the number was 24.

In the paper Lambek's type implicative deductive systems and expo¬nential categories for them are considered and some metamathematical theo¬rems are proved (deduction theorem, functional completeness). Then we for¬mulate both sequential implicative deductive systems and exponential multi¬categories for which the cut elimination theorem is proved. Finally sequential deductive implicative metasystems and dual deductive systems and categories are discussed.

We define the relation of negative equivalence on the class of non-trivial extensions of minimal logic as follows. Logics are negatively equivalent if they define the same negative consequence relation or, equivalently, if they have the same class of inconsistent sets of formulas. We point out the least logic in any class of logics with fixed intuitionistic and negative counterparts and prove that each of such logics is closed under the rule (  )  . We prove also that negative counterparts of extensions of negative logics can be treated as theirs logics of contradictions.

A sequential axiomatization for the set of all formulas which are dedusible in Smirnov’s quasi-minimal calculus is presented.

It is known how to present every deduction in the {!, I}-free Classical Multiplicative Linear Logic as (the result of an obvious translation of) a deduction in the intuitionistic MLL. We extend the result to the language with I and give short proofs which do not use proof nets

We describe an augmentation of Intuitonistic propositional Logic by a modal operator admiting provability interpretation (Box-as-Proof modality). We do not intend to give a systematic survey, but were a short selection of attractive (algebraic, relational, topological and categorical) features of the modalized Heyting Calculus. We discuss also an enrichment of the Heyting Calculus by temporal modalities Always and Before, extending the expressive power of the Calculus.

The paper deals with the Hao Wang calculi of partial predicates PP and EP. It is shown that an analogue of the Craig interpolation theorem holds in PP and EP.

An algorithm of quadratic (or near) complexity that searches for a common signal in a given set of sequences (in particular DNA sequences), i.e. a system of similar words of a fixed length and satisfying some certain condi¬tions, has been developed.

In this article the axioms are found for the earlier-built three-value semantics of uncertainty. The resulting axiomatic system is non-contradictory and complete with this semantics. In the basis of the axiomatic system there is the algorithm of converting the formulae of the initial language into the auxil¬iary one that eliminates the uncertainty symbol.

Classical logic can not be applied in some philosophical reasonings, especially in those which contain contradictory, antinomical, paradoxical (true and false), senseless (neither true nor false), nonproved statements. So the problem of applicability of the classical logic and other logics to various statements may be important. Let L1 and L2 be two defined in usual way propositional logics such that the language for L1 is a sublanguage for L2. Logic L1 with its connectives {C1, …, Cn} is called applicable to wff A in L2, (symbolically: Ap(L1{C1, …, Cn}, A, L2)) iff for any theorem T in L1 each formula Ti obtained by substitution of A for all occurences same variables in T is deducible in L2. The conditions of applicability of propositional classical logic will be defined in the languages of the intuitionistic logic, Łukasiewich's logic Ł3, Kleene's logic, enriched by full equivalence. The conditions of applicability of propositional classical logic, Łuka¬siewich's logic, Kleene's logic will be defined in the language of the logic FL4 with falsehood operator.

Method of analytic tableaux for implicative and positive relevant logics is developed. It is based on certain modification of the method of analytic tableaux elaborated by Beth, Hintikka, Smullyan and Fitting. This modification includes a simple signing of negative occurences of subformulae in a formula and corresponding definition of closed analytic tableau. The method proposed is used to construct decidable propositional relevant calculi without negation, namely the systems RA and RApos.

This paper deals with logic of anaphoric update operator. It is well known that the diverse variants of dynamic semantics are efficient way to interpretation of anaphora. At the same time an anaphora itself may be considered as update operator with property of eliminating. This operator acts within the limits of the epistemic attitudes of the recipient of phrase and determines the meaning of anaphoricaly defined names. It is possible to express the static meaning of anaphor’s operator by means of epistemic logic by use the methods of van Eijck and de Vries’s update logic.

The main aim of the paper is to formulate the deduction theorem which would be valid for any logical theory T closed by MP. The using of MP in a consequence B1,..., Bm of inference B from hypotheses Г in a theory T is said to be normalized iff for each member of the consequence Bi (i  m), obtained from Bk and Bl (k,l  i) by MP, the follow¬ing conditions are satisfied: (a) if Bk is the major premise of MP and has a form Bl Bi, then it precedes the minor one Bl ; (b) there is no any member Bk (l  k  i) of the consequence between Bl and Bi except members which are result of MP with the same minor premise Bl ; (с) Bl doesn’t depend from any hypotheses preceding the major premise Bk. Definition. A normalized inference of B from hypotheses Г = {A1,...,An} (n  0) (symbolically: A1,..., An В) in a theory (calculus) T is said to be such a finite consequence of propositions (formulae) B1,..., Bm (m  1) which satisfies the following conditions: (1) The last member of the consequence Bm is coincided with B, and for any Bi (1  i  m): (a) Bi is one of the hypotheses Г; or (b) Bi is a theorem of T; (c) Bi is obtained from two of previous members of the consequence by MP under its normalized using; or (d) Bi is a conjunction of two of previous members of the consequence (by rule of adjunction: RA). (2) The formula Bm depends from each member of the consequence B1,..., Bm. A consequence B1,..., Bm is said to be normalized one iff it fulfils the definition. Any inference B from hypotheses Г may be normalized. Let B1,..., Bm be a normalized consequence of inference B from hypotheses Г in a theory Т. A hypothesis Bi is said to be blocked for deduc-tion iff there is at least one of the following blocking conditions: (b1) Bi is the first hypothesis of consequence and it be not used as minor premise; (b2) after step Bi the rule RA is used; (b3) after step Bi there is a result of MP, where the minor premise is a T-theorem and the major one depends from hypothe¬ses; (b4) after step Bi there is a result of MP, which itself is used as minor premise of MP where major premise precedes all hypotheses from which the minor one depends. Let’s Гb be a list of the all the blocked for deduction hypotheses, and A1,..., An be a list of all hypotheses unblocked for deduction standing at the same order in which they occur in the consequence. The expression Гb,A1,..., An В is a normalized writing of the normalized inference В from Г. Deduction theorem. If the consequence of formulae B1,..., Bm is a normalized inference В from Г in a D-theory T and Гb,A1,..., An  В is the corresponding normalized writing of Г В, then any statement Гb,A1,..., Ai-1 Ai .....An В (1  i  n) is valid in T. Adequacy theorem. A statement ГA В is valid in a D-theory T iff there exist a normalized consequence of inference B from hypotheses Г and A and it is corresponded by Г,AВ.

Questions about effective variants of deduction theorem for normal modal logics are discussed. Some proof of an external deduction theorem for minimal normal modal logic K - a formula  is deducible from  in K iff the formula *   belongs to the dynamic logic - is given.

In the paper a relation between logic and philosophy is considered. The issues of foundations of logical systems on the basis of semantics of different types are investigated. On the other hand a role of logic in solving fundamental philosophical problems such as foundations of theoretical knowledge and constructing theoretical pictures of the world is studied.

The logical structure of relativity-like theories is discussed. Some difficulties, arising herein, are pointed.

The aim of this article is to show that Aristotle's apodeictic syllogistic is a fragment of a one-place predicate extension of the propositional modal system T.

The aim of the paper is to provide a formalization of Nikolai Vasiliev's imaginary logic and to establish its metatheoretical relation to quantified many-valued logic. Syllogistic type calculus IL which axiomatizes the set of imaginary logic laws is formulated. The natural translation of affirmative, negative and «indifferent» (contradictory) statements of the imaginary logic into the language of quantified three-valued logic is offered. It is proved that IL system is embedded into three-valued quantified Lukasiewicz logic.

The aim of the paper is to provide a formal reconstruction of Nikolai Vasiliev's idea of n-dimensional logic - logic with n quality types of categorical propositions. According to Vasiliev n-dimensional logic is the generalization of Aristotelian syllogistic (2-dimensional logic containing the two quality types of propositions - affirmative and negative) and of the Imaginary logic (3-dimensional logic containing the three quality types of propositions - affirmative, negative and contradictory). Natural semantics of n-dimensional logic and the adequate formal system are formulated.

The paper regards main types of modal notions used in biology. Explanations of the meaning of the main modal terms is given and the way of usage of suggested quasi-matrix logical systems in scientific research is demonstrated.

The purpose of this paper is to show the possibility and the usefulness of temporal qualification of normative propositions. Such the device allows us to make more exact the relation of deontic alternativeness and the concept of deontic consistency. We offer to modify the so-called “standard” models. We use a temporal structure that is discrete, branching “to the left” and infinite in both directions. The system of tense-logic based on this structure has been proved to be complete and decidable one. So the temporal interpretations of deontic operators are expected to be adequate ones.

The article provides foundation for the thesis of Bolzano being one of the first in the history of logic and mathemathics who asserted the necessity of using only direct proofs in scientific texts. The autor is going to dewelope the foundation of the Bolzano's priority as the founder of constructivism in logic.

The paper revaluates Kant's argument against treating the existence notion as a predicate. The author agrees with Kant, Russell, Carnap, and Quine's point of view that existence contexts are expressed with the help of logical quantifiers. At the same time the author believes existence predicate to be modal (meaning epistemic modalities) and it helps us to construct true assertions about physically nonexistent objects impossible for example in Russell's theory of definite descriptions.