Logic (Greek – the science of thinking, from? – word, speech, reason, reasoning) – the science of the laws, forms and methods of intellectual (intellectual) cognitive activity. Since the work of the intellect is always carried out in a linguistic form, research in the field of logic is directly related to the study of various kinds of linguistic constructions from the fulfillment of certain cognitive functions by them. Language, in this case, is regarded as an instrument of cognition, i.e. as a means by which information about the world is fixed, this information is transformed and the world around us is studied.
At present, logic is a branched and multifaceted science that contains the following main sections: the theory of reasoning (in two versions: the theory of deductive reasoning and the theory of plausible reasoning), meta-logic and logical methodology. Studies in all these areas at the current stage of the development of logic and are mainly carried out within the framework of logical semiotics.
In the latter language, expressions are considered as objects that are in the so-called sign situation, which includes three types of objects – the very language expression (sign), the object designated by it (the meaning of the sign) and the interpreter of signs. In accordance with this logical analysis of the language can be conducted from three relatively independent points of view: the study of the logical syntax of the language, i.e. the relationship of the sign to the sign; The study of the logical semantics of the language, i.e. the relation of the sign to the object designated by it; and the study of logical pragmatics, i.e. relationship of the interpreter to the sign.
In logical syntax, the language and the logical theories based on it are studied from their formal (structural) side. Here we define the alphabets of the languagesof consistent theories, define the rules for constructing from the alphabet characters various complex language constructions – terms, formulas, conclusions, theories, etc. The syntactic division of a set of linguistic expressions into functors and arguments, constants and variables is carried out, the concept of a logical form of expression is defined, the concepts of a logical subject and a logical predicate are defined, various logical theories are constructed and methods of operating in them are constructed.
In logical semantics, language and logical theories are studied from the content side of them. Since language constructs not only designate something but also describe something (they make sense), in logical semantics they distinguish between the theory of meaning and the theory of meaning. In the first, the question is asked which objects denote signs and how they do it. Similarly, in the theory of meaning, the question of what is the semantic content of linguistic expressions is solved and how they describe this content.
In semantics, all language expressions, depending on their meanings, are divided into classes, called semantic categories. These are the following categories – sentences and terms. Proposals are divided into the narrative – affirming the presence or absence in the world of some situation (such sentences are called statements), interrogative – expressing the question and motivating – expressing imperatives. Terms, in turn, are divided into descriptive (names, predicators, objective functors) and logical.
Logic terms are particularly important for logic as a science since the whole procedural aspect of our intellectual work with information is ultimately determined by the meaning of these terms. Logical terms include bundles and operators. Among the first, the predicating links are “is” and “are not” and are propositional (logical connectives): unions – “and” (“a”, “but”), “or” (“either”), “if, then” phrases – “it is wrong that”, “if and only if” (“if and only then”, “necessary and sufficient”) and others. Among the latter, the statement generates – “all” (“every”, “any”), “some” (“exists”, “some”), “necessary”, “possible”, “accidental,” etc. and name-forming operators are “a lot of objects such that”, “the object that” and others.
The central concept of logical semantics is the concept of truth. In logic, it is subjected to a careful analysis, since without it it is impossible to interpret in a clear form the logical theory, and therefore, to investigate and understand it in detail. It is now obvious that the powerful development of modern logic has been largely determined by the detailed development of the notion of truth. Another important semantic concept is closely connected with the notion of truth – the concept of interpretation, ie, the procedure of attributing, by means of a special interpretive function, to linguistic expressions the valuesassociated with a certain class of objects, called the universe of reasoning. A possible realization of a language is a strictly fixed pair , where U is a universe of reasoning, and I is an interpreting function that maps the names of elements of the universe, n-local predicators – sets of ordered n-ok elements of the universe, n-local subject functors – n-local functions that map the n-th elements of the universe to the elements of the universe. Expressions relating to formulas are put in correspondence with two values- “true” or “false” – in accordance with the conditions of their truth.
A variety of possible implementations can be associated with the same class of sentences. Those implementations in which each sentence in the set of sentences T, assumes the value “truth”, is called a model for G. The concept of the model is especially studied in a special semantic theory – the models of the theory. In this case, different models of different types are distinguished: algebraic, set-theoretic, game theoretic, probability-theoretic, etc. The notion of interpretation has the most important meaning for logic, because it defines two central concepts of this science – the concept of logical law and logical sequencing.
Logical semantics is a substantial part of logic, and its conceptual apparatus is widely used for theoretical justification of certain syntactic, purely formal constructions. The reason for this is that the aggregate content of thought is divided into logical (expressed by logical terms) and concrete (expressed in descriptive terms), and therefore, when we separate the logical form of expressions, we abstract, generally speaking, not from any content. This distraction, i.e. consideration of the formal side of thoughts, is only a way of isolating in a pure form the logical content of them, which is explored in logic. This circumstance makes Kant’s understanding of logic unacceptable as a purely formal discipline. On the contrary, logic is a deeply meaningful science in which each logical procedure receives its theoretical justification through meaningful considerations. In this connection, the term “formal logic” in its application to modern logic is inaccurate. In the true sense of the word, one can speak only of the formal aspect of research, but not of formal logic as such.
When considering certain logical problems, in many cases it is also necessary to take into account the intentions of the interpreter, which uses linguistic expressions. For example, consideration of such a logical theory as the theory of argumentation, dispute, discussion is impossible without taking into account the goals and intentions of the participants in the dispute. In many cases, the methods of polemics used here depend on the desire of one of the disputing parties to put their opponent in an uncomfortable position, to confuse him, to impose on him a certain vision of the problem under discussion. Consideration of all these issues is the content of a special approach to the analysis of language – “logical pragmatics.”
The most fundamental section of logic is the theory of deductive reasoning. At present, this section in its hardware (syntactic, formal) part is represented in the form of various deductive theories – calculi. The construction of such a device has a double meaning: first, theoretical, since it allows us to distinguish a certain minimum of the laws of logic and forms of correct reasoning, from which it is possible to justify all other possible laws and forms of correct reasoning in a given logical theory; secondly, purely practical (pragmatic), since the developed apparatus can be used and used in the modern practice of scientific knowledge for the exact construction of specific theories, as well as for the analysis of philosophical and general scientific concepts, methods of cognition, and so on.
Depending on the depth of the analysis of statements, the propositional calculus is distinguished and quantifier theories-predicate calculi. In the first analysis of the reasoning is carried out to the exclusion of simple sentences. In other words, in the propositional calculus, we are not interested in the internal structure of simple sentences. In predicate calculi, analysis of reasoning is carried out taking into account the internal structure of simple sentences.
Depending on the types of quantifiable variables, predicate calculi of different orders are distinguished. Thus, in the predicate calculus of the first order, the only quantifiable variables are individual variables. In the predicate calculus of the second order, variables for properties, relations, and objective functions of different terrain are introduced and quantified. Correspondingly, third and higher order predicate calculi are constructed.
Another important division of logical theories is associated with the use of the presentation of logical knowledge of languageswith different categorical grids. In this connection, one can speak of theories constructed in the languagesof the Frigate-Rasslau type (numerous variants of the predicate calculus), the syllogistic, or algebraic (various algebras of logic and algebra of classes-Boolean algebra, the Zhegalkin algebra, the de Morgan algebra, the Hao-Wang algebra, and others). For many theories constructed in languageswith different categorical grids, their mutual translatability is shown. Recently, in logical research, the category-theoretic language, based on a new mathematical apparatus, category theory, is actively used.
Depending on the method of constructing conclusions and proofs applied in logical theories, the latter is divided into axiomatic calculi, natural deduction calculi, and sequential calculi. In axiomatic systems, the deduction principles are given by a list of axioms and inference rules that allow one to pass from one of the proved statements (theorems) to other proved assertions. In systems of natural (natural) derivation, the deduction principles are given by a list of rules that allow one to pass from one hypothetically accepted statement to another statement. Finally, in sequential calculi, the deduction principles are given by rules that allow one to pass from some deduction statements (they are called sequents) to other statements about deductibility.
The construction in the logic of this or that calculus forms a formal line of logical research, which it is always desirable to supplement with meaningful considerations, i.e. construction of the corresponding semantics (interpretation). For many logical calculi, such semantics are available. They are represented by meaning of various types. These can be truth tables, so-called analytical tables, beta tables, various kinds of algebras, possible worlds of semantics, descriptions of states, etc. On the contrary, in the case when the logical system is initially constructed semantically, the question arises of formalizing the corresponding logic, for example, in the form of an axiomatic system.
Depending on the nature of the utterances, and ultimately on the types of relations of things that are studied in logic, logical theories are divided into classical and non-classical. At the heart of such a division is the adoption in the construction of the appropriate logic of certain abstractions and idealizations. In classical logic, for example, the following abstractions and idealizations are used: a) the principle of two-valuedness, according to which every utterance is either true or false, b) the principle of extensionality, ie, permission for expressions that have the same meaning, their free replacement in any contexts, which means that in classical logic they are only interested in the meaning of expressions, not in their meaning, c) the principle of abstraction of actual infinity, which allows us to talk about essentially non-constructive objects, d) the principle of existentiality, according to which the universe of reasoning must be a non-empty set, and each proper name must have a referent in the universe.
These abstractions and idealizations forms the point of view, the foreshortening, under which we see and evaluate the objective reality. However, no set of abstractions and idealizations can fully embrace it. The latter is always richer, more mobile than our theoretical constructs, which justifies the free variation of the original principles. In this connection, a complete or partial rejection of any of these principles leads us into the field of nonclassical logics. Among the latter distinguish: many-valued logic, in particular probabilistic and fuzzy, in which there is a rejection of the principle of double-valued; intuitionistic logic and constructive logic, in which reasoning is examined within the framework of abstraction of potential feasibility; modal logics (aletics, temporal, deontic, epistemic, axiological, etc.), relevant logic, paraconsistent logic, logic of questions in which statements with non-extensional (intensional) logical constants are considered; logic, free from existential assumptions, in which there is a rejection of the principles of existentiality, and many others.
This shows that logic as a science that provides a theoretical description of the laws of thinking is not something once and for all given. On the contrary, each time with the transition to the exploration of a new field of objects requiring the adoption of new abstractions and idealizations, when new factors that influence the process of reasoning are taken into account, this theory itself changes. Thus logic is a developing science. But what has been said demonstrates something more, namely, that the inclusion in the logic of a certain theory of the laws of thinking is directly related to the adoption of certain ontological assumptions. From this point of view, logic is not only a theory of thinking, but also a theory of being (ontology theory).
An important part of the modern logic is meta-logics. The latter explores various problems related to logical theories. The main questions here are about the properties that logical theories possess: consistency, completeness, the existence of permissive procedures, the independence of the original deductive principles, as well as the various relationships between theories, etc. In this sense, the meta-logic is, as it were, the self-reflection of logic about its constructions. All metatheoretical studies are conducted on a special metalanguage, which is a natural language enriched with special terminology and metatheoretical deductive means.
The logical methodology is another section of modern logic. Usually, the methodology is divided into the general scientific methodology, in which the cognitive techniques applied in all fields of scientific knowledge, as well as the methodology of individual sciences: the methodology of deductive sciences, the methodology of empirical sciences, as well as the methodology of social and humanitarian knowledge, are studied. In all these sections, logical methodology participates as a specific aspect of the study. Thus, in the general methodology, the study of cognitive techniques, such as the elaboration and formulation of concepts, the establishment of their types and various methods of operating with conceptual constructions (division, classification), the definition of terms, etc., are among the logical aspects.
Particularly great success has been achieved in the field of the methodology of deductive sciences. This was due to both the construction of the logic itself in the form of a deductive apparatus, and the use of this apparatus to justify such a deductive discipline as mathematics. All this required the development of essentially new cognitive methods and the introduction of new methodological concepts. In the course of the work carried out here, it was possible, for example, to generalize the concept of a function, that it passed into the category of general methodological, theoretical-cognitive concepts. We now have the opportunity to consider not only numerical functions but also functions of any other nature, which made it possible to make a functional analysis of the language the leading method for studying linguistic expressions. It was possible to work out with all thoroughness and rigor such important methods of cognition as the method of axiomatization and formalization of knowledge. For the first time, it was possible to define theoretically-proof (deductive) methods of cognition in a clear and, most importantly, diverse form, to develop the theory of expressibility and definability of some terms through others in theories, to define the concept of a computable function in various ways.
At present, the logical problems of the methodology of empirical sciences are actively developed. This area includes research on the construction and testing of hypotheses (in particular, the hypothetical-deductive method), the analysis of various types of plausible reasoning (induction and analogy), measurement theory. Here interesting results were obtained on the relationship between the empirical and theoretical levels of knowledge, procedures for explanation and prediction, operational definitions. Various models of empirical theories are designed to clarify their logical structure.
Among the general methodological-logical principles are those laws and principles of cognition, which are investigated within the framework of dialectical logic. In many cases, they act as some warning signs about how we can meet with surprises in the way of knowing. In the field of methodology of empirical, as well as social and humanitarian knowledge, it is of great importance to distinguish between absolute and relative truth; in the field of historical knowledge, the demand for the coincidence of the historical and the logical becomes essential, which in fact means the usual requirement of the adequacy of cognition transferred to the sphere of historical disciplines. Recently, attempts have been made to construct deductive systems in which individual features of dialectical logic are formalized.
For millennia, logic was an obligatory discipline of school and university education, i.e. performed its general cultural task – the propaedeutics of thinking. Modern logic fully retained this didactic and educational-methodological function. However, the recent development of a powerful apparatus of modern logic has allowed it to become an important applied discipline. In this connection, we point out the significant use of logic in the field of mathematics (metamathematics), linguistics and informatics. Studies in these fields of knowledge have had a decisive influence on the formation of the most modern logic so that one can speak of the mutually enriching influence of these disciplines. Recently, logical problems actively penetrate into other spheres of knowledge – jurisprudence, ethics, aesthetics, etc. All this points to the ongoing process of knowledge legislation, which will grow with time.