By DAISUKE SAKAI
The first to be reviewed is Tarski’s classical text. This book is available on Amazon as an E-book for a cheap price; but I think that this book should already be in the public domain by now. What is referred to as the “methodology of the deductive sciences” is now known as “metalogic” or “metamathematics,” which is a discipline which concerns what may be called “the logic of logic,” or “the science of logic.”
Metamathematics is now more or less a subfield of mathematical logic. I have managed to complete perusing this book within a month as a self study text, completing every single question in it. I was only once stuck on a problem; it was solved after I asked online for help. The book itself apparently started off as one which was meant for the intellectual layman to read, rather than being a college textbook. Consequently it is the case that the book is rather explanatory than technical.
Apparently there are people who think this causes it to be longwinded. However I think this is likely because they have already been exposed to the topic already, and are using Tarski’s book as a supplementary resource. I have felt that logic itself has stemmed from concrete uses. Perhaps it may be clear to people from the twenty first century, especially those in the field of science and technology, the enormous import of the use of variables and symbolization.
This was not the case back when this book was written, and there is a particular emphasis on the use of variables, which I found quite enlightening. There is a long section on the usage and definition of the symbol of “implies,” which I find quite useful and didactic. It is true that this canon is written almost a century ago. Consequently the terminology is dated, and numerous exercises refer to the field of geometry; to be specific, Euclidian geometry. This kind of mathematics is not dealt in IB mathematics, A-level mathematics or
virtually any western standardized exam.
A useful reference might be “Elements,” by Euclid himself; and glancing through the axioms might be sufficient, but it is recommended to perhaps looking through three or four of the proofs provided by Euclid. The Japanese education however still has a significant (that is, in the sense that it is non trivial) portion of its syllabus the Euclidian geometry. Other than this as long as one has attained a high school level education, there is no prerequisite knowledge required; not even that; only common sense and rigorous and persistent thinking are the qualifications.
For the actual content of the book; as already mentioned, there is an extensive discussion on using variables, which although might seem redundant, may draw attention to things that the modern reader has taken for granted. The discussion of variables organically
develops into a that of quantifiers, from this, relations and interpretations (for which a mathematically inclined person may find expected).
A rigorous but not extensive theory of sets is introduced, referred to as the “theory of classes,” and at the end of the first part of the book, methods of applying logic to proofs are introduced. Models and interpretations are used but not treated as a formal subject of investigation. The second part of the book primarily concerns with proving theorems from axioms, primarily axioms of groups, abelian groups, field axioms, and axioms of the real numbers. It may be taken as an easygoing introduction of (abstract) algebra. There was one point which I noticed however; the group axioms which Tarski propounds are not in fact equipollent (equivalent) to the current definition. That is, by Tarski’s axioms, one can state that the null set is a group. There is an explanation of the concepts of completeness, consistency and decision problems. Being an introductory textbook, no proofs of Godel’s theorems are given. There is no formal discussion of natural deduction, but its methods are mentioned, and implicitly used in the proofs, and the exercises.
The method of studying this book itself is best stated by the author: “I feel that the book contains enough material for a full-year course. Its arrangement, however, makes it feasible to use it in half-year courses as well. If used as a text in half-year logic courses in a department of philosophy, I suggest a thorough study of its first part, including the more difficult portions, with omission of the entire second part. If the book is used in a half-year course in a mathematics department—for instance, in the foundations of mathematics—, I suggest the study of both parts of the book, with omission of the more difficult passages.” The fact that the content does not match entirely with modern syllabi does not at all detract from the fact that it is a very expository and well written text on logic.