HINTIKKA'S TRANSFORMATION RULES

HINTIKKA, JAAKKO.

Identity, Variables, and Impredicative Definitions + Vicious Circle Principles and the Paradoxes. (In Journal of Symbolic Logic, Volume 21, Number 3, 1956 + Volume 22, Number 3, 1957).

(No place), The Association for Symbolic Logic, Inc., 1956 + 1957. 8vo. Both entire issues present, both in original printed wrappers. Volume 21, 3 w. a small loss to lower corner of front wrapper, minor loss to lower capital and a tear to lower front hinge, no loss. Upper corner of least few leaves and back wrappers creased (w. tear to wrapper). Volume 22, 3. w. minor loss to capitals. Both issues internally near mint. Vol. 21, 3: pp. 225-248 (entire issue: pp. 225-336); Vol. 22, 3: pp. 245-249 (entire issue: pp. 225-336).

First printing of these two important, but for long overlooked, articles, which together constitute Hintikka's attempt to cope with Wittgenstein's elimination of identity as proposed in the "Tractatus". With the translation rules that Hintikka here put forward, he is the first to try to carry out Wittgenstein's suggestions systematically.

The Finnish born philosopher and logician, Jaakko Hintikka (born 1929), Professor of Philosophy at the University of Boston, is generally accepted as the founder of formal epistemic logic and of game semantics for logic. He has contributed seminally to the fields of philosophical and mathematical logic, philosophy of mathematics and science, language theory and epistemology. Independently of Evert Willem Beth he discovered the semantic tableau, and he is famous for his work on game semantics and logical quantifiers. In 2005 Hintakka was awarded the Schock Prize in logic and philosophy, the philosophical equivalent to the Nobel prize, "for his pioneering contributions to the logical analysis of modal concepts, in particular the concepts of knowledge and belief ".

In the 1950'ies Hintikka took it upon himself to follow Wittgentein's suggestion of elimination of identity suggested in the "tractatus", and in the two offered articles, he succeeds in constructing a logic without identity. The main point of the two connected articles is to show that variables can be used in two ways. One way does not exclude coincidences of the values of different variables (inclusive interpretation of variables), the other does (exclusive interpretation of variables) and can be either weakly or strongly exclusive. He now claims that in the "Tractatus" Wittgenstein adopts the weakly exclusive interpretation of variable and then proves that the weakly exclusive quantifiers are able to express everything that the inclusive quantifiers plus identity can express, and without a sign for identity, - for the first time systematically supporting Wittgenstein's claim that identity is not an essential constituent of logical notation.
"There are a number of references to the exclusive interpretations of variables in current logical literature. An exclusive reading of variables was, in effect, suggested by Ludwig Wittgenstein in "Tractatus logico-philosophicus. As far as I know, however, no one has previously tried to carry out his suggestions systematically. Several misconceptions seem to be current concerning the outcome of an attempts of this kind. Carnap expects radical changes in the rules of substitution. If I am not mistaken, however, at least one form of the exclusive interpretation may be formalized by making but slight alterations in the axioms and/or in the transformation rules of the predicate calculus. Also I hope to say that it is not correct to say (as Russell has done) that Wittgenstein tried to dispense with the notion of identity. What a systematic use of an exclusive reading of variables amounts to is a new way of coping with the notion of identity in a formalized system of logic. Under the most natural formalization of the new interpretations, the resulting system is equivalent to the old predicate calculus (with identity): every formula of the latter admits of a translation into the former, and vice versa." (Vol. 21, Nr. 3, p. 228).

"A deviation from standard English. Recent discussion serves to bring out, amply and convincingly, the utility of observing the ordinary correct use of words and phrases for the purpose of clearing up philosophical problems. In this paper, I shall endeavour to show, by means of an example, that the reverse method may have its interest, too. " (Vol. 21, Nr. 3, p. 225). "This note is a sequel to the previous paper of mine which was entitled "Identity, variables, and impredicative definitions" and published in this JOURNAL, vol. 21 (1956, pp. 225-245. That early paper served to call attention to the dependency of the set-theoretic paradoxes on the interpretation of the variables that may occur in the critical "abstraction principle". (Vol. 22, Nr. 3, p. 245).

Besides these two articles, the two issues also include other important articles within logic, e.g. Quine, "Unification of Universes in Set Theory" and Symonds and Chisholm "Inference by Complementary Elimination".

Order-nr.: 36635