TABLE OF CONTENTS
Igor A. Gorbunov. Logic, Unity in Three Persons
This paper is mainly a review of some well-known facts concerning interconnections between such basic syntactic notions of logic as relation of logical consequence, consequence operator, and the lattice of theories under a logic. In doing so, we seek to provide evidence for the fact that, to define the logic syntactically, it is necessary and sufficient to define one of these three notions: namely, if one of them is defined, it unambiguously determines the other two.
We consider in detail conditions that are both necessary and sufficient to prove the following statement: a closure operator generated by a class of sets of formulas can be interpreted as a consequence operator. To that end, we introduce the notion of a system of sets of formulas forming a lattice of theories. We prove that such a system defines a logic and consider some possible approaches to constructing such systems.
The paper draws attention to the fact that the most popular syntactic definitions of logics (such as sequent calculi, Frege-type calculi, closures of sets with respect to inference rules) can be equally well understood as defining relations of logical consequence, consequence operators and compact elements of the lattice of theories under a logic.
Keywords: relation of logical consequence, consequence operator, lattice of theories, methods of specifying logic, compactness in lattices
Vladimir I. Shalack. Analysis vs Deduction
In the paper, we consider four types of problems that naturally arise in connection with the definition of a logical inference: 1) verifying the proof of: Г⟨A1,...An⟩A; 2) search for interesting consequences: Г⟨...⟩?; 3) search for the proof: Г⟨.?.⟩A; 4) search for hypotheses: ?⟨...⟩A. Modern logic focuses on the problem of finding the proof of statements. Gödel’s restrictive theorems have a direct relation to it. At the same time in real practice, the task of search for hypotheses is much more common. The main part of this work is devoted to the investigation of this problem. A target proposition A is given, and it is required to find the set of premises Г from which it is logically deducible. The choice of suitable premises Г occurs on the basis of the logical analysis of proposition A. We distinguish six different grounds for the selection of these premises: 1) acceptance of explicit definitions for predicate and functional symbols; 2) acceptance of axiomatic definitions for predicate and functional symbols; 3) acceptance of previously proved theorems; 4) acceptance of empirically true statements; 5) acceptance of statements describing the result of some action; 6) acceptance of plausible hypotheses that may be relevant to the problem being solved. In this paper we construct a calculus that formalizes the problem of an analytic search for the justification of a thesis. Two metatheoremes are proved, from which it follows that the constructed calculus really allows us to solve this type of problems.
Keywords: Problem solving, logical reduction, logical inference, search for proof, search for hypotheses, analytical tables, theory of definitions
Alexander A. Belikov. Vojshvillo-style Semantics for Some Extensions of FDE: Part I
In this paper I examine the semantics of semi-generalized state descriptions - a kind of the informational semantics for logic of first-degree entailments (FDE) proposed by E. K. Vojshvillo in the early eighties. A key feature of the approach is to consider state descriptions, which do not satisfy the classic ontological conditions of consistency and completeness that allows to determine a relevant entailment. By relevant entailment we understand such a relation, that is free from the classical paradoxes: A ∧ ∼ A |= B and B |= A ∨ ∼A. I consider well-known extensions of FDE, which are formulated in terms of binary consequence systems: three-valued Kleene logic, three-valued Priest logic and classical logic. The first two of these can be semantically defined using semi-generalized state descriptions: for Kleene logic I use ⊤-generalized state descriptions (consistent but incomplete), for Priest logic I use ⊥-generalized state descriptions (inconsistent but complete). The entailment relation for Kleene logic defined in terms of truth-and-non-falsity preservation from the premise to the conclusion. In turn Priest logic determined by entailment relation defined through the preservation of falsity-and-non-truth from the conclusion to the premise. The paper includes proofs of the corresponding completeness and soundness theorems. In the case of classical logic, we provide only a sketch of completeness and soundness with respect to the semantics of classical state descriptions (consistent and complete). This article is the first part of studies on E. K. Vojshvillo semantics for different extensions of textbf FDE.
Keywords: Klenee logic, Priest logic, first-degree entailments, classical logic, generalized state descriptions
Nikolai N. Prelovskiy. Logical Matrices and Goldbach Problem
The paper considers equivalent formulations of Goldbach conjecture in terms of sets of tautologies in sequences of logical matrices and single logical matrices. The significant part in this consideration belongs to concepts of tautology in a logical matrix, sums and products of logical matrices from sequence Kn+1 of Karpenko matrices. Thus the paper proposes an answer to A.S. Karpenko’s question about possible relations between sequences of logical matrices similar to Kn+1 and an open problem, known as binary Goldbach conjecture: every even natural number n ≥ 4 may be represented as a sum of two prime numbers. The proposition that all finite-valued matrices in the sequence M have tautologies iff the binary version of Goldbach conjecture (G2) is true is proven. Using the properties of matrix product operation, it is proven that the infinite-valued matrix M⊗ has tautologies iff G2 is true. The paper also mentions that the set of tautologies of M⊗ (id est the logical theory defined by M⊗) is equal to the certain theory defined in terms of finite-valued Lukasiewicz logics Ln iff G2 is true. These results were restated in terms of sequences of matrices and their products from a large class of logical matrices. Thus it was found out that G2 has certain ogical aspects, as it is equivalent to existence of defined non-empty logical theories.
Keywords: many-valued logics, logical matrices, tautologies, Goldbach conjecture
Natalya E. Tomova. On Properties of a Class of Four-valued Paranormal Logics
The paper is devoted to the results obtained during the investigation of a class of fourvalued literal paranormal logics, i.e. logic, which are simultaneously paraconsistent and parapcomplete at the level of literals; that is, formulas that are propositional letters or their iterated negations. Paraconsistent logic allows the possibility of operating with conflicting information, parapcomplete logic allows us to build reasoning in conditions of incomplete information. With both types of uncertainty, with both inconsistent and incomplete information, paranormal systems work. In  the class of four-valued literal paralogics obtained by combining isomorphs of classical logic, which are contained in fourvalued logic of Bochvar B4, is considered. As a result, together with the isomorphs themselves, logical matrices that correspond to these logics form a ten-element upper semilattice with respect to the functional embedding of one matrice into another. In this paper we investigate the class of matrices that make up the supremum of the said semilattice. The matrices of this class have interesting functional properties, namely, they correspond to the class of all external four-valued functions. The paper also provides an algorithm for constructing a perfect disjunctive J-normal form of a four-valued external function. As it turned out, there are well-known logics in the literature that are functionally equivalent to the logics of the class in question. For example, one of them is the logic V , which is a formalization of intuitions of N.A. Vasilyev’s imaginary logic of. Thus, we have considered the question of the correlation of all these systems both in the class of tautologies and in the class of valid consequence relations. As a result, it is proved that all systems are equivalent in the tautological class, but they differ in the properties of the consequence relation.
Keywords: four-valued paranormal logics, functional properties, external functions, tautologies, consequence relation
HISTORY OF LOGIC
Valeriy V. Vorobyev. Stephanus Alexandrian Is a “Successor” of Ammonius
In this paper the author analyzed the commentary on Aristotle’s Chapter 9 of “De Interpretatione” by Stephanus Alexandrian (the second half of VI – first half of VII c.) — the philosopher of late neoplatonic school. Stephanus Alexandrian was supposed to be the pupil of Johannes Philoponos who was one of Ammonius Hermiae‘s (435/445 – 517-526) pupils and has not attracted special attention of philosophy historians till now though his philosophical works have survived. Stefanus‘ s commentary is not large and its content is quiet similar to Ammonius‘s commentary.
The corresponding fragment of Stephanus’s text was translated and analyzed. The author marks that Stephanus accepts the so called “traditional” or “standard” interpretation of the problem of “the sea battle tomorrow”. Generally speaking, its meaning consists in that there are differences in defining the truth of tensed propositions. We consider that propositions about past and present events are true or false but propositions about contingent future events have different truth values. Stefanus (followihg Ammonius) introduces the expression “definitely (horismenos) true” to characterize such propositions.
The Stephanus’s text containing the well known “reaper paradox” has been translated as well. This paradox was mentioned by many ancient authors, but it has survived only in the works of Ammonius, Stephanus and one more anonymous author. In Diogenes Laertius‘s edition there is the note which contains the reaper paradox translation. However this translation is very clipped that‘s why it is very misrepresented. Lately the reaper paradox attracts attention of contemporary authors and requires further
Keywords: Aristotle’s “De Interpretatione”, Ammonius, Stephanus, Reaper Argument
Anastasia O. Kopylova. Tensed Propositions In W. Ockham’s Logic
This article presents the reconstruction of W. Ockham’s approach to the analysis of truth conditions of tensed propositions in order to clarify Ockham’s view and to present it in a systematic way. The article focuses on the chapter seven of the second book and chapter seventy two of the first book of the treatise Summa Logicae. One of the points that makes the analysis of Ockham‘s theory of tensed and modal propositions significant is the fact that he rejected the standard scholastic tool of the analysis of modal and tensed propositions — ampliation (ampliatio). Therefore, Ockham had to create his own theory that was based on his general ideas of supposition and predication that were primarily described by him in terms of the present tense. The main aim of this article is to examine why Ockham doesn’t use traditional tool for analysis of the truth-conditions in propositions about Future and Past. In the beginning of the article there is a textual reconstruction of the chapter seven, then there is an examination of the role of subject term and predication rules in this kind of propositions. Subsequently there is a general chart of the analysis of truth conditions in tensed propositions in Ockham’s view. In the article author claims that the ground of the rejection were Ockham’s ontological interests which were presented in his debate with W. Burley. Instead of traditional disjunction Ockham suggests detachment of the two senses of proposition. This idea leads to semantic controversy. Reference to the objects in past and future cannot be reduced to the reference to objects in present. Nominalism and mental language theory leads him to these semantic decisions.
Keywords: ampliatio, suppositio, truth, medieval logic
Yuriy V. Ivlev. The Subject and Prospects of Development of Logic
The article discusses a subject of logic and some prospects for its development. It is argued that logic is the science of thinking. That is, thinking is an object of the science of logic. The subject of logic is a special structure of thoughts and thinking processes which is called (quite unsuccessfully, according to the author of the article) forms of thoughts and processes of thinking. These structures are discovered by partial abstraction from both semantic and substantive meanings of non-logical terms which are included in the language expressions that represent thoughts and processes of thinking. The modern logic differs from the traditional logic in using methods which are similar to mathematical methods — methods of symbolic logic. However, it preserves all achievements of traditional logic which are important for both scientific and everyday knowledge. The logic that is described in some textbooks published in the forties of the last century in the USSR is called surrogate. There are said to be empirical and theoretical levels of research in logic, as well as logic and “as-if-logic”, classical and nonclassical logics. The prospects for the development of “as-if-logic” and the logic itself are under discussion. The usefulness of research in the field of “as-if-logic” is highlighted — there can be created a range of “as-if-logical” systems with some of them being interpreted as actual logical systems itself afterwards. There can be developed new methods for proving metatheorems, which will be applied in proving some results concerning actual logical systems. Two directions are indicated to be prospects for the development of logic — empirical and theoretical researches. Possible applications of quasi-matrix logic in the field of logic as well as in the other areas of cognition are identified.
Keywords: the object of the science of logic, the subject of logic, traditional logic, modern logic, “as-if-logic”
Nikolai N. Nepejvoda. Formalization as the Immanent Part of Logical Solving
The work is devoted to the logical analysis of the problem solving by logical means.
It starts from general characteristic of the applied logic as a tool:
1. to bound logic with its applications in theory and practice;
2. to import methods and methodologies from other domains into logic;
3. to export methods and methodologies from logic into other domains.
The precise solving of a precisely stated logical problem occupies only one third of the whole process of solving real problems by logical means. The formalizing precedes it and the deformalizing follows it.
The main topic when considering formalization is a choice of a logic. The classical logic is usually the best one for a draft formalization. The given problem and peculiarities of the draft formalization could sometimes advise us to use some other logic.
If axioms of the classical formalization have some restricted form this is often the advice to use temporal, modal or multi-valued logic. More precisely, if all binary predicates occur only in premises of implications then it is possible sometimes to replace a predicate classical formalization by a propositional modal or temporal in the appropriate logic. If all predicates are unary and some of them occur only in premises then the classical logic maybe can replaced by a more adequate multi-valued. This idea is inspired by using Rosser–Turkette operator Ji in the book . If we are interested not in a bare proof but in construction it gives us it is often to transfer to an appropriate constructive logic. Its choice is directed by our main resource (time, real values, money or any other imaginable resource) and by other restrictions.Logics of different by their nature resources are mutually inconsistent (e.g. nilpotent logics of time and linear logics of money).
Also it is shown by example how Arnold’s principle works in logic: too “precise” formalization often becomes less adequate than more “rough”.
Keywords: applied logics, formalization, choice of logic
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