Introduction to Logic and to the Methodology of Deductive Sciences

Description

referenced in Bochénski, O.P.'s A Precis of Mathematical Logic and this M.SE question regarding if vs. iff:

pt 1 "Elements of Logic. Deductive Method.", §II. "On the Sentential Calculus":

10. Equivalence of sentences

We shall consider one more expression from the field of sentential calculus. It is one which is comparatively rarely met in everyday language, namely, the phrase “ if , and only if ”. If any two sentences are joined up by this phrase, the result is a compound sentence called an EQUIVALENCE. The two sentences connected in this way are referred to as the LEFT and RIGHT SIDE OF THE EQUIVALENCE. By asserting the equivalence of two sentences, it is intended to exclude the possibility that one is true and the other false; an equivalence, therefore, is true if its left and right sides are either both true or both false, and otherwise the equivalence is false.

The sense of an equivalence can also be characterized in still another way. If, in a conditional sentence, we interchange antecedent and consequent, we obtain a new sentence which, in its relation to the original sentence, is called the CONVERSE SENTENCE (or the CONVERSE OF THE GIVEN SENTENCE). Let us take, for instance, as the original sentence the implication:

(I) if x

is a positive number, then 2 x

is a positive number;

the converse of this sentence will then be:

(II) if 2 x

is a positive number, then x

is a positive number.

As is shown by this example, it occurs that the converse of a true sentence is true. In order to see, on the other hand, that this is not a general rule, it is sufficient to replace “2 x

” by “ x2

” in (I) and (II); the sentence (I) will remain true, while the sentence (II) becomes false. If, now, it happens that two conditional sentences, of which one is the converse of the other, are both true, then the fact of their simultaneous truth can also be expressed by joining the antecedent and consequent of any one of the two sentences by the words “ if , and only if ”. Thus, the above two implications—the original sentence (I) and the converse sentence (II)—may be replaced by a single sentence:

x

is a positive number if, and only if, 2 x

is a positive number

(in which the two sides of the equivalence may yet be interchanged).

There are, incidentally, still a few more possible formulations which may serve to express the same idea, e.g.:

from: x

is a positive number, it follows: 2 x

is a positive number, and conversely;

the conditions that x

is a positive number and that 2 x

is a positive number are equivalent with each other;

the condition that x

is a positive number is both necessary and sufficient for 2 x

to be a positive number;

for x

to be a positive number it is necessary and sufficient that 2 x

be a positive number.

Instead of joining two sentences by the phrase “ if , and only if ”, it is therefore, in general, also possible to say that the RELATION OF CONSEQUENCE holds between these two sentences IN BOTH DIRECTIONS, or that the two sentences are EQUIVALENT, or, finally, that each of the two sentences represents a NECESSARY AND SUFFICIENT CONDITION for the other.

transl. of O logics matematycznej i metodzie dedukeyjnej

This is a translation and revision, with a considerable amount of added material, of a book published in Polish in 1936 and in German in 1937. Besides being a clear and readable exposition of the elements of modern logic, it is so arranged that it may be used as a textbook in an introductory undergraduate course in logic and axiomatics. Features not present in the previous edition include the introduction of logical symbols, the truth table method, the calculus of relations and many new exercises.
The book starts with a discussion of variables, sentential functions and quantifiers, followed by a short description of the calculus of elementary propositions, the truth table method and rules of inference. The next three chapters deal with the theory of identity, the calculus of classes and the calculus of relations, including the notions of cardinal number, function and one-to-one correspondence. The sixth chapter, on the deductive method, is perhaps the most significant in the book. It introduces the notion of an axiom system and of a formalized deductive theory, and such concepts as consistency and completeness of systems. The author presents here his view of the importance of methodology (metamathematics) as an independent subject. The last four chapters should be of particular interest to mathematicians. The ideas and methods of the first part of the book are illustrated here by applying them to the actual construction of mathematical theories based on various axiom systems. Four different postulate systems for ordered abelian groups are compared, and finally two different systems of postulates for the real numbers are considered.
The book itself is not a formalized deductive theory of logic, but rather, as its title indicates, an introduction to logic. A good feature is the large collection of exercises, many of them from the field of mathematics. There seems to be no good reason for the author's use of the terms sentential function and sentential calculus in place of the well established terms propositional function and propositional calculus. Although emphasis is placed on the view of Leśniewski that definitions are not mere abbreviations but must conform to rules of definition, this point is not made very clear. In general, however, the book is a model of clarity. It fills the need for an easily intelligible and up-to-date treatment in English of those topics in logic which are of greatest importance for mathematics. Reviewed by O. Frink