Classical and non-classical logics: essay on the foundations of logic (Q2785507)

This book constitutes an essay on the foundations of logic. It deals with the nature of logic. Its author, the Brazilian logician and philosopher Newton Carneiro Affonso da Costa, is mainly interested in the relations between reason and logic, as well as on the connections between the rational activity which is reflected by logic, and experience. He has been a very original and creative scientist. He has worked in several fields, as for instance in the foundations of mathematics, in the foundations of classical mechanics and related topics in physics, being internationally recognized as the founder of paraconsistent logic, that constitutes nowadays an important and rich field of research.NEWLINENEWLINENEWLINEIn paraconsistent logics the principle of contradiction is not valid in general, and inconsistency does not lead to trivialization.NEWLINENEWLINENEWLINEIn particular, the development of paraconsistent logic and mathematics, as might be expected owing to their revolutionary character, leads us to reflect on the nature of logic. After the construction of paraconsistent logics, we have to rethink topics like the meaning of negation and contradiction, the role of logic in the fields of mathematics and of the sciences, the significance of inconsistency, etc. From a certain point of view, the book constitutes the author's answer to all these and related questions, especially motivated by the birth of paraconsistent logic.NEWLINENEWLINENEWLINEJean-Yves Béziau, disciple and collaborator of da Costa, translated the Portuguese version of the book into French. In a beautiful Preface he presents the work, discusses paraconsistency and da Costa's ideas, and presents the general content of the book. NEWLINENEWLINENEWLINEIt is supposed that the reader has a good familiarity with contemporary logic. Meanwhile, some technical developments, necessary for the comprehension of the arguments, are presented at the end of the book, in two appendices. These excellent appendices, also written by Béziau, deal with paraconsistent logic and the theory of valuation, another important creation of da Costa.NEWLINENEWLINENEWLINEThe author believes, and justifies his belief, that reason constitutes itself through history, following mainly the contingency originated by the scientifical progress. So, the apparent \textit{a priori} nature of reason appears relative. Reason transforms itself in a constitutive element of the culture of a given epoch, then having social and cultural connotations relatively to its own history.NEWLINENEWLINENEWLINEThe book studies the following problems: the question of the nature of rational and logical activities; the question of the connections between reason and reality; the question of the genre of rational knowledge -- if it is only dicoursive or if there is a rational intuition; the question of the criteria of truth of logical-mathematical judgements, which apparently do not depend on reason itself.NEWLINENEWLINENEWLINEThe author's attitude in dealing with these questions is positive and critical. He sites his researches in the field of the scientifical philosophy, whose object is the theory of science, and whose objective, for him, is a product of history.NEWLINENEWLINENEWLINEThe book consists of an Introduction, 4 chapters and the 2 mentioned appendices.NEWLINENEWLINENEWLINEIn the first chapter, the author analyses the relations among reason, logic and language, and between logic and mathematics. Concerning the relations between reason and logic, he distinguishes two fundamental positions: the dogmatical position, according to which logic and the rational coincide, the essential formal principles of reason constituting the laws of logic, that is essentially unique and practically does not depend on experience; and the dialectical position, for which logic and the rational are never identified and the exercise of reason can be done through distinct logical-mathematical systems, part of logic being based on the interconnection between reason and experience, and so all the possible logics are permissible from the rational point of view. In spite of the ambiguity of the term, da Costa considers that reason is dialectical, evoluting according to the advance of science, its categories being historical. So, reason cannot be \textit{a priori} codified via a fixed logical system.NEWLINENEWLINENEWLINEMathematics is not simply reducible to logic but, for da Costa, one of the merits of the logicist school is to have shown that logic is close to mathematics, due to their objectives and methods. One of the causes of this fact is the fundamental use, by both, of the contemporary axiomatic method and formalization. Due to the axiomatic method reason becomes objective. Nowadays, the tendency to use the axiomatic method, and so the deductive systematization, also appears in the so-called sciences of the real and in the sciences of nature. But the author recalls that in the sciences of the real, the inductive procedures, which constitute a method of discovery, are also used.NEWLINENEWLINENEWLINELogical principles reflect, from a certain point of view, the laws that rule the exercise of reason. But there is no logical-rational activity without the linguistic vehicle, so the logical laws finish to be characterized through language. The author states that besides the syntactical and semantical dimension of language there is its pragmatical dimension, these three levels being essential for the perfect comprehension of the logical mathematical disciplines. It is then evident that pragmatics is pertinent for the study of the foundations of logic and mathematics, and the axiomatic-formal methods can be used to build pragmatical systems, similarly to what is done with the syntactical and semantical levels of language.NEWLINENEWLINENEWLINEIn the section where the relations between language and reason are discussed, the author tries to make evident that the syntactical genres and the rational categories are closely related. In the following, the author formulates three principles which rule reason: the Principle of Systematization -- the reason always expresses itself by means of a logic; the Principle of Unicity -- in a given context the underlying logic is uniquely determined; and the Principle of Adequacy -- the logic of a given context should be the most suitable one to this context.NEWLINENEWLINENEWLINEBut our reason, in order to really work, has to manipulate symbols, to construct languages, etc. As a consequence, there is another central principle, which constitutes the basis of our logical activity, the so-called Constructive Principle of Reason, stated by da Costa as follows: the integral activity of reason presupposes that it possesses a certain constructive power, whose regularities are essentially those of intuitionistic arithmetic.NEWLINENEWLINENEWLINEA consequence of the historicity of reason is its dialectical character. Chapter 1 ends with a discussion about the dogmatical and dialectical conceptions of reason.NEWLINENEWLINENEWLINEChapter 2 is devoted to the non-elementary and heterodox logic. The author intends to show that the traditional classical logic, whatever its presentation is, is not absolute, being dialectizable and so relativizable. He explicates what is, for him, the meaning of the word ``logic'', the consequences of the dialectization of certain fundamental concepts like for instance the concepts of truth and falsity, the relation between logic and semantics, one of the main finalities of logic being the study of logical consequence.NEWLINENEWLINENEWLINEIn a long and fundamental section, the fundamental laws of reason -- the principle of identity, the principle of contradiction and the principle of excluded middle -- are analyzed. The author considers that the three laws, in their usual formulations, are vague and relative, stating that there are no laws of reason, in the sense of traditional logic.NEWLINENEWLINENEWLINELogic, besides its subjective aspect has an objective aspect, ontological. The logical-rational idealization has its roots in the experience, isolating its important properties, schematizing and extrapolating them. The only origin of the logical laws consists of the applications of the principles of systematization, unicity and adequation, having as aim to put experience in order. NEWLINENEWLINENEWLINEDa Costa states its Norm of Relativity: there is practically no principle that cannot be razed. Logical principles depend on experience.NEWLINENEWLINENEWLINEThe important section in which the so-called heterodox logics are treated begins by establishing the limits of the concept of orthodox or classical logic. Details about the distinct heterodox systems are not presented, but only some essential comments on the comprehension of the different kinds of nonclassical logics. The author criticizes Quine's position concerning nonclassical logics.NEWLINENEWLINENEWLINEThe concept of truth constitutes a central category of logic and da Costa is perfect in considering Tarski's definition of truth and his semantical method as one of the most important conquests of contemporary logic. He also calls attention that Gödel's theorems constitute a remarkable point of the evolution of mathematics and logic -- they indicate that the axiomatic method, essential to logic and mathematics, is not sufficient to serve as a basis for formal sciences, so confirming the pragmatic dimension of deductive sciences.NEWLINENEWLINENEWLINEChapter 3, on Hegel's thesis, deals with informal paradoxes and antinomies and discusses their interpretations and solutions from the point of view of paraconsistent logic -- only on account of logical motives, contradictions must not be eliminated.NEWLINENEWLINENEWLINEThe last chapter, on intuition and speech, begins by a discussion on the problem of the intuition in mathematical logic. The formal sciences are based in part on a certain kind of rational and formal intuition, reason being decomposed in two vectors -- the immediate (intuitive) and the mediate (discoursive): the method utilized in scientifical philosophy, concerning the theory of science, is based on reason and the experience of science.NEWLINENEWLINENEWLINEDa Costa finishes the chapter with a section about the historicity of reason, calling to our attention that it is a consequence of the philosophical position of this work that, if we want to understand science, in the philosophy of science it is necessary to consider the historicity of science, that is, its historical character. So, the scientifical position in the philosophy of science, implies to recognize the dialectical reality of science.NEWLINENEWLINENEWLINEThe very good two appendices are written by the translator of the book, \textit{Jean-Yves Béziau}. The aim of both is to present some technical details and supplementary information.NEWLINENEWLINENEWLINEIn Appendix 1, the reader finds a detailed treatment of the known da Costa caclulus \(C_1\), and its first-order extension \(CP_1\), and a section on paraconsistent set theory.NEWLINENEWLINENEWLINEAppendix 2 presents the theory of valuation, a general theory of logic based on the idea of bivaluation. This theory is also a creation and construction of da Costa, who had elaborated a great part of it in 1961.NEWLINENEWLINENEWLINEThe expositions of these appendices, very well prepared and structured, are of a general and descriptive character. At the end of each appendix there is a very good little bibliography relative to its subject.NEWLINENEWLINENEWLINEIn the book, the author insists for several times on the fact that today there are numerous logics, some of which can be regarded as complementary to classical logic (this is the case, for instance, for modal logics), and others of them as rivals of classical logic (this is the case, among others, for intuitionistic, many-valued and relevant logics). Furthermore, nonclassical logic is not uniquely determined. In effect, one can present several arguments in favour of this point of view, related to: Gödel's incompleteness theorems, the semantical incompleteness of higher-order logic, the independence results in set theory and the existence of (``classical'') alternatives to classical mathematics, for instance Solovay's mathematics. Today, therefore, one is faced with a very important philosophical problem: how should we choose a logical system?NEWLINENEWLINENEWLINEThe author's answer is that such a problem is pragmatical in nature. The logical systems are singled out according to the three mentioned pragmatical principles of systematization, unicity and adequacy, similar in intention to Descarte's rules for the direction of spirit. These principles and the constructive principle of reason make up the so to say nucleus of reason. They form the axis around which da Costa discusses the most fundamental topics of logic and its philosophy: the relations between reason and logic; the connections between logic and language; the concept of formal system and the primary character of any axiomatization of logic; the syntactical and semantical dimensions of a language; the distinction between pure and applied semiotics, as well as object language and metalanguage; the intuitionistic conception of mathematics, where the pragmatical level appears essential; the incompleteness theorems; logical foundations of quantum mechanics; the nature of heterodox logics, paying special attention to intuitionistic, many-valued, relevant and paraconsistent logics; the foundations of set theory; truth and falsehood and the thee main theories of truth -- the theories of correspondence, coherence and pragmatic truth; the role of negation and the existence of contradictions and inconsistent theories; the character of mathematics and Platonism. Particular attention is paid to paradoxes. The author's view is that there are legitimate paraconsistent mathematical theories, which are inconsistent though non-trivial (not every sentence is provable), so that there are admissible contradictions, at least at the abstract level of mathematical inquiry. As for actual inconsistencies, true contradictions in the real world, da Costa is agnostic: if they exist, this fact has to be established by some special science.NEWLINENEWLINENEWLINEThe book brings the essence of the author's thought on logic and his ideas relative to some essential and central questions of the philosophy of logic and, in general, of the philosophy of mathematics and formal sciences. It was generated by paraconsistent logic, that da Costa wanted to understand. It is an original book, written by one of the most important scientists and thinkers of our times. It is fundamental and represents a school of thought.