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ABSTRACTS
1. Note on the V. Smirnov's scientific activity: work and life
(1931-1996), Alexander S. Karpenko
2. List of selected publications of Vladimir A. Smirnov
3.1.1. A Dynamic Semantics for Inconsistency-Adaptive Logics,
Dideric Batens
Inconsistency-adaptive logics have a dynamic proof
theory. The stage of a proof determines which formulas are derivable;
formulas derivable at some stage may become underivable at a later stage,
and vice versa. The stage-independent `final derivability' relation
refers (in a precise way) to a stage at which dynamics has come to an end.
Usual semantic systems for inconsistency-adaptive logics are static. Their
semantic consequence relation corresponds to final derivability. It follows
that the outcome of the dynamics at the proof theoretic level is determined
beforehand. If the dynamics is genuine, it must have a semantic
characterization. The block approach enables us to devise such a semantics.
3.1.2. On a Partially Interpreted Logic,
Michael Bezhanishvili
In his well-known article On a three-valued logical calculus and its application to the
analysis of contradiction D.Bochvar has indicated a way how the first
order predicate calculus can be extended without type restrictions avoiding
the rise of logical and semantical antinomies. Such an approach requires to enrich the
object language so that it would formally express meaninglessness of each
paradoxical sentence in it. But the study of modalities of knowledge and
belief shows that epistemic logic also can give such possibility and it can
be used to avoid the antinomies (without introducing type limitations) in
case, if it will be constructed not on the base of classical logic, i.e. if
no pure classical tautology will be valid in it.
The aim of the present paper is to consider such epistemic first order
predicate logic and to state some of its peculiarities. The corresponding
system E4 will be described here semantically.
3.1.3. On Foundations of Mathematics: Some Modern Problems and Achivements,
Albert G. Dragalin
The aim of the lecture is to give a retrospective of some modern directions
of investigations in the area of foundations of mathematics. We try to show
that these investigations are nowadays far more rich than the traditional
exposition as the old dispute between logicism, formalism and intuitionism.
3.1.4. Quantification in Intuitionistic Logic with Provability Smack,
Leo Esakia
A modified quantifier extension Q+HC of the Heyting propositional calculus HC
presented here is inspired, on the one hand, by the provability interpretation of the
Intuitionistic logic (via Goedel's modal translation and Solovay's arithmetical completeness
theorem). On the other hand, by the investigation (K.Fine, Reasoning with Arbitrary
Objects), demonstrating how Arbitrary Objects lead naturally to the correct constraints on
the rules of universal generalization and existential instantiation. On the last page of this
book K. Fine points out: ``In the light of this breakdown, logicians have experienced some
difficulty in setting up a resonable system for intuitionistic logic with a rule of existential
instantiation''. We now present (and try to justify) an amendment to the standard quantifier
extension QHC of the Heyting propositional calculus HC.
3.1.5. A Maximal Lattice of Implicational Logics,
Alexander S. Karpenko
The problem about the finding of the unified foundation for the classification of implicational
logics was raised by V.A.Smirnov in his article Formal inference, deduction theorems and
theories of implication. He suggested to classify implicational logics (1) in relation of
the form of deduction theorem and (2) based only on structural rules. V.A.Smirnov pays attention
to the very important problem, namely, that both method of classification do not include
classical logic. In the first case the deduction theorem which takes place for an implicational
fragment Himpl of intuitionistic propositional logic H is true also for an
implicational fragment TVimpl of classical logic {\bf TV}. And then in this case no
distinction between Himpl and TVimpl logics is made. In the second case it
does not exist any structural rule which would provide the transition from Himpl
to TVimpl. This transition is usually realized due to the admission of Pearce's law
((p implies q) implies p) implies p. But there exists no structural rule corresponding to
this formula.
Our proposal of resolving the problem consists in yielding such logical
construction which comprises implicational logics in question. Moreover,
application the simplest operations to the construction would allow to
generate new logics and even infinite classes of logics.
3.1.6. The Law of Assertion and the Rule of Restricted Permutation,
Alexander Kron
3.1.7. Explicit and Implicit Definability in Modal and Related Logics,
Larisa Maksimova
We consider various versions of the Beth definability property for
propositional normal modal logics, and also for superintuitionistic and
relevant logics. We discuss interrelations of these properties, and find
their algebraic equivalents in case of modal and superintuitionistic logics.
3.1.8. Incomplete Proofs and Program Analysis,
N.N.Nepejvoda
Logical reformulation of programming problems (namely program analysis)
is considered here. We can state here only results and their semi-formal
interpretation. The main formalisms used are constructive logics and
incomplete proofs in these logics.
3.1.9. Belief Revision and Doxastic Commitment,
Krister Segerberg
A complex is a pair (V, T) where V and T are theories in some given language and with respect to
some given logic L. There is an operation * on complexes; it is the only primitive operation
studied here, although of course further operations can be added. Informally, T is the old
notion of a belief set and V the new notion of a commitment set - a set of
doxastic commitments - while * is a revision operation. In this presentation, doxastic
commitments are treated as irrevocable.
3.1.10. The Weakness of Logical Equivalence,
Paul Weingartner
In this note I want to show with some examples that the notion of
logical equivalence (of classical logic) is rather weak: Important concepts
are not invariant with respect to translation in the sense of logical equivalence.
3.1.11. Mathematical Aspects of Modal Logics: Achievements, Tendencies,
Problems, Alexander Chagrov, Michael Zakharyaschev
The paper analyses the development of modal logic in the last 25-30 years. It is based on the
material collected by the authors in the book Modal Logic and the chapter Advanced
modal logic written for the second edition of the Handbook of Philosophical Logic.
3.1.12. Functional Algebraic Models for Non-Classical Set Theory,
V.Kh.Khakhanian
We suggest in the present short note a general method of construction of models for
intuitionistic set theory.
3.1.13. Deductive Logic Course Supported by Interactive Proof Search
Software, A.Novodvorsky, A.Smirnov
In 1991 the first version of interactive proof search support system
DEDUCTIO was developed by the authors. It allowed to describe as data some
logical calculus and supported interactive proof search for them. Professor
V.Smirnov proposed to use DEDUCTIO for teaching logic and started the
process of preparaing a course, integrated with the software system. During
this work V.Smirnov and V.Markin prepared ant tought courses in logic,
A.Smirnov and A.Novodvorsky developed new version of DEDUCTIO program,
specific for teaching logic.
3.1.14. On V.A.Smirnov's Systems RA and RAO, V.M.Popov
The calculi RA and RAO were constructed by Professor V.A.Smirnov. These calculi are well founded
from the view-point of proof theory. Here I shall consider a number of results
obtained for the sentential parts of the calculi called as RA and RAO.
3.1.15. Normalized Inference and Deduction Theorem,
E.A.Sidorenko
The introduced notion of the normalized inference from hypotheses allows to formulate the
deduction theorem, which is adequate for any theory T containing as its theorems:
(1) A implies A, (2) A implies B implies .C implies A implies .C implies B,
(3) (AT implies B) implies B$ (where AT is any theorem of T).
3.1.16. Yet Another Semantics for First-Degree Entailment,
Dimitry V.Zaitsev
Typically the semantics for first-degree entailment (FDE) is connected with the Impossible
Possible Worlds (theories, epistemic situations, state descriptions, set-ups and so on)
Assumption (IPWA). The core idea of this assumption, namely to admit inconsistent and
incomplete assignments, is the foundation for all different semantical approaches to FDE.
3.2.1. Psychologistic Solution of Antinomies,
Andrzej Grzegorczyk
Semantical antinomies seem to be consequences of antipsychologistic
paradigm adopted by logicians at the beginning of 20 Century. Rejecting
this paradigm yields a chance of solving antinomies. Antinomies impair
our whole science. Hence ``solving'' means: such a description of the
intellectual situation which exhibits science as consistent. Of course,
individual knowledge of some human beings may remain inconsistent, but we
want to have an interpretation of individual inconsistency which shows that:
if the individual knowledge of a given human being X is inconsistent, then
this is because X neglects an important methodological principle.
Such a solution of the antinomy of self-liar may be presented (as in the
following), and shows the analogy between the antinomy of self-liar and the
antinomy of barber.
3.2.2. Connectionism and the Relation of Individual and Common
Knowledge, Veikko Rantala
3.2.3. Partially Interpreted Logical Constants,
Gabriel Sandu
3.2.4. The Semantics of Uncertainty, A.M.Anisov
The uncertainty is given by the cardinal number of the set of possible
worlds not less than two, differing in pairs in the interpretation of at
least one predicate symbol. Relationship of the accessibility on the worlds
is absent. In general, by the uncertainty of a statement the situation is
meant, where the statement is true in some worlds and false in the others.
This simple semantic idea leads, however, to some unexpected consequences.
3.2.5. Combined Semantics for Monadic Deontic Logic with Non-Classical
Negations, I.A.Gerasimova
3.2.6. Semantics of the Restricted State-Descriptions Sets for Quasi-Matrix Logic,
Yu.V.Ivlev
I propose quasi-matrix logic for description of the characteristics of factual (ontological) modalities.
3.2.7. Formal Reconstruction of Traditional Syllogistic with Singular and Negative Terms,
Vladimir I.Markin
I set out the formal reconstruction of traditional singular negative syllogistic by means of modern logic.
3.2.8. Lukasiewicz's Three-Valued Logics and Falsehood Logic FL4,
Sergey A.Pavlov
In this note we consider the correspondence 1) between falsehood logic FL4 and logic of Belnap;
2) between sublogics FL3N of falsehood logic FL4 and three-valued logics of Lukasiewicz and
Kleene.
3.2.9. An Approach to Non-Standard Semantics and Some Problems of the Foundation of
Logical Systems, E.D.Smirnova
In modern logic the appearance of most diverse logical systems makes the problem of their
foundation especially acute. We do not consider it right to treat all kinds of logical laws and
structures on one and the same level. It seems to us rational to distinguish two kinds of such
presuppositions. And then two types, two levels of logical laws appear, correspondingly. We
consider it reasonable to subdivide the laws of logic into two types. The first type depends on
the definite ontological assumptions, that is, on assumptions referring to the object of
discourse. The laws of the second kind do not depend on the limitations imposed on the universum
of discourse. They depend only on our notions of truth, falsity, logical entailment and so on.
3.2.10. Situations and Events: Non-Fregean Approach to V.A.Smirnov's Combined Logics,
Vladimir L.Vasykov
3.2.11. Ontological Necessity and Apodictic Syllogistic,
E.K.Voishvillo
It is a well-known fact that there were many attempts to express necessity (N) in Aristotelian
modal syllogistic (where moduses with apodictic conclusion from apodictic and assertoric
premises are admitted, i.e., the principle of the weakest premise is omitted). However, always,
as the author knows, all such explications are based whether on certain ad hoc functions,
which are equal to Aristotelian necessity, or on use expressions of the kind NP(x), where
the sense of N is presupposed to be already defined. I will present an explication of N for
categorical propositions, which yields the above-mentioned moduses of Aristotelian modal syllogistic.
BULLETIN OF THE SECTION OF LOGIC