# Philosophy of Logic

In fact, the term “philosophical logic” is very vague, contradictory and does not have a single use. Various experts in mathematics, in symbolic logic and philosophy itself, philosophical logic is understood in different ways, but rather, in its way. Even if it is understood as a special scientific discipline, it is impossible to determine its subject, the boundaries of application and methods. Confusion occurs between the terms “philosophical logic” and “philosophy of logic.” Often, one substitutes for another, although these are two different areas of research.

The term “philosophical logic” appeared in the English-speaking logico-philosophical literature and was widely used in the 50-60’s 20 century. On the one hand, the crisis in the foundations of mathematics (the discovery of paradoxes in set theory and the restrictive theorems of A. Tarski and C. Goedel) required a deep understanding of the very conceptual apparatus of logic. On the other hand, the emergence and rapid development of nonclassical logics, primarily modal logic, attracted the wide attention of logicians with a philosophical orientation, the area of research that became known as the “philosophy of logic” emerged. For logic-mathematicians, the philosophy of logic is the development of set theory and the corresponding questions about the mode of formation of sets and the nature of the number. The discovery of paradoxes in set theory and, in particular, Russell’s paradox (see The Paradox of Logic) raised the question of the nature of mathematics itself. Logicism tried to define the basic concepts of mathematics in logical terms (G.Frege in 1884 and B.Rassel in 1903). This is not only a technical, but also a philosophical problem. In this sense, the grandiose construction was undertaken by N. Whitehead and B. Russell in the Principia Mathematica was unsuccessful. And although there are no paradoxes in their logic-mathematical theory, it was impossible to derive the existence of infinite sets from purely logical axioms. Intuitionism, as another answer to the discovery of paradoxes, raised fundamental questions about the difference between the finite and the infinite, the difference between potential infinity and the actual. There was a problem of the existence and justification of evidence, as well as the problem of the status of classical logical laws. All this is a philosophical problematic. The formalistic program of D. Hilbert (see Formalism) also provoked a lively philosophical discussion, especially the problem of finitism.

In fact, the preceding refers more to the philosophy of mathematics than to the philosophy of logic, but the task of philosophical comprehension of the application of logic to the solution of various problems of mathematics remains. A convincing example here is K. Goedel’s restrictive theorems on the incompleteness of sufficiently rich theories (1931), which indicate that there is not and in principle can’t be an adequate formalism covering all mathematics. The philosophical consequences of these results are discussed to this day and attracted the attention not only of professional logicians, but also of philosophers, methodologists, and amateurs who have no concept of logic. To this should be added a philosophical discussion on the thesis of Church-Turing.

It is interesting that the philosophy of logic was taken up by mathematicians, who received deep results in it (G. Frege, B. Russel, W. Quinn, R. Carnap, etc.). Quinn in 1940 published a book titled “Mathematical Logic”, and in 1970 – under the title “Philosophy of Logic”, which by logic is understood as a systematic study of logical truths, and under the philosophy of logic – a tool for analyzing natural language. The book contains the following sections that Quinn refers to the philosophy of logic: “Meaning and truth” (the problem of utterances and sentences, statements like information, the theory of the meaning of linguistic expressions, truth and semantic agreement); “Grammar” (recursive task of grammar, categories, revision of grammar goal, names and functors, vocabulary criterion, time, events, verbs, propositional settings and modality); “Truth” (definition of truth according to Tarski, paradoxes in the object language, connection between semantic and logical paradoxes); “Logical truth” (in terms of structure, in terms of the model, in terms of substitution, in terms of proof, in terms of grammar); “The scope of logic” (problem of identity, set theory, quantification); “Deviant (deviant) logic” (this means nonclassical logic, primarily multi-valued logic, intuitionistic logic, branching quantifiers); “Foundations of logical truth” (the place of logic, logic and other sciences).

Thus, Quinn concentrated his work around the main problem in the philosophy of logic: what is the truth? The question is so sacred that it has sounded for 2000 years. However, only with the development of symbolic logic, namely, beginning with the works of A. Tarski (1936), was first given a semantic definition of truth for a large group of formalized languages, and at the same time, the boundaries of such a definition are indicated. In fact, the scope of the philosophy of logic is much broader. The theory of the propositional form as statements about certain positions of things (things) in the world, the theory of logical and semantic categories, the theory of reference and predication, the identification of objects, the problem of existence, the doctrine of prepositions, the relation between analytic and synthetic judgments, the problem of scientific law, ontological assumptions in logic, and much more. And even such questions, seemingly purely logical, refer to the philosophy of logic: the essence and general nature of the relation of following or logical deduction between any statements or sets of utterances, the meaning of logical connectives, the informativeness of logical laws, the significance of fundamental theorems obtained in symbolic logic, and in The connection with this is a thorough analysis of such concepts as “computability”, “solvability” “provability” and again “truth.”

In contrast to the philosophy of logic, originally the philosophical logic was called modal logic, i.e., logical analysis of such philosophical concepts as “possibility” and “necessity.” Historically, these two concepts, especially since Aristotle, attracted the constant attention of philosophers, and with the development of symbolic logic, it became possible to analyze the specified modalities and their relationships by precise methods. The same thing happened to philosophical concepts like “future” and “past”. With the development of modal logic, new types of modalities began to fall into the sphere of logical research: time, modal-temporal (not a mechanical connection, but the synthesis of modal and temporal operators), physical or causal, deontological, epistemic, etc. With the release in English in 80 th gg. “Handbook of Philosophical Logic” in 4 volumes summed up some of its development. The second and third volumes are nothing more than the consideration of various nonclassical logics and, of course, such as modal logic, time, multivalued, intuitionistic, relevant, etc. And in general a whole series of new logical theories arise, such as logic of the theory of quantum mechanics, the logic of existence, the logic free from existential assumptions, the logic of binding and permitting (legal and ethical contexts), the logic of actions, teams, assessments, intentions and preferences, the logic of knowledge, faith, belief, doubt, perception, prediction the logic of questions, formal ontology, and so on. However, only with the advent of the semantics of possible worlds (see Possible worlds of semantics) in 50’s (S.Kanger, S.Kripke, A.Priyor, J.Hintikka) it became possible to conduct a logical analysis of many central philosophical concepts: along with the indicated modalities also such as “knowledge”, “faith”, “perception”, “commitment” etc.

Let us pay attention to the fact that each of these logics has its philosophy of logic, and therefore also the philosophical problems listed above, because the definition of the truth of the formula, the logical sequence, the notion of utterance and the meaning of logical operations in most logics are different. Also, each philosophical logic has its additional philosophical problems. For example, in modal logics, such are the problem of reference, cross-identification, i.e. identification of objects in various possible worlds, and in connection with this a problem of quantification arises. In many-valued logics, there is the most complex philosophical problem of interpreting the set of truth values, usually expressed by numbers: rational, natural, integer, real. Many philosophical problems are raised by intuitionistic logic, for example, the existence of two heterogeneous and irreducible classes of semantics: Kripke’s realizability and models.

Philosophical logic has a linguistic and technical apparatus richer and, more importantly, more flexible than symbolic logic, which allowed us to begin to analyze and reconstruct purely philosophical problems, and even fundamental ones such as the problem of fatalism and free will, determinism and randomness, time and asymmetry time, existence and omniscience of God, etc.

In general, the concept of philosophical logic is contradictory. On the one hand, these include all those logical studies that are not purely mathematical and seemingly irrelevant to symbolic logic, understood by many logic philosophers as “playing symbols.” On the other hand, the modern development of modal logic, temporal, intuitionistic and especially multivalued, and some others, is nothing but sections of symbolic logic: the same methods of symbolization and axiomatic methods of construction and, most importantly, in many respects the same purely technical tasks and problems. It is instructive here to construct new sets of theories by nonclassical logics that are purely philosophical in origin, namely, multi-valued, modal, relevant, paraconsistent set theories. It is worth emphasizing that there is something that unites such trends in modern logic, as symbolic logic, philosophical logic, philosophy of logic, nonclassical logic. This refers to the fundamental philosophical question of the late 20th century: what is logic?

Finally, in 90’s there is another term that has a direct relationship to the topic of our consideration, namely, “logical philosophy.” Since 1993, the magazine Logic and Logical Philosophy began to appear in Poland. To define what “logical philosophy” is even more difficult than what is philosophical logic. Most likely, this is all where you can apply logic in any of its forms. Therefore, works from the field of philosophical logic, and from the field of symbolic logic, come here.

Since 1972, under the auspices of the international association of symbolic logic, the most famous journal in the field of philosophical logic is now published – “Journal of Philosophical Logic”.