## Language, truth and logic in mathematics.(English)Zbl 0894.03001

Jaakko Hintikka Selected Papers. 3. Dordrecht: Kluwer Academic Publishers. x, 247 p. (1998).
This book consists of a number of essays by Jaakko Hintikka (some with co-authors), most of them published elsewhere recently. In most, if not all, of them one of the fundamental concepts in the foundations of mathematics and logic is critically examined, resulting in a new perspective which is worth further study.
For instance, in essays 1-3 it is argued that the expressive power of traditional first-order logic is too restricted and that the true logic is what the authors call (Information) Independence Friendly (IF) logic. The latter logic contains expressions of the form $(\forall x)(\forall z)(\exists y / \forall z)(\exists u / \forall x)S(x,y,z,u)$ in which the slash in $$(\exists y / \forall z)$$ indicates that the value of $$y$$ does not depend on $$z$$, but only on $$x$$. This first-order IF formula is equivalent with the second-order (traditional) formula $$\exists f \exists g \forall x \forall z S(x,f(x),z,g(z))$$, but cannot be expressed by a traditional first-order formula. Traditional first-order logic is a special case of IF logic, which is a conservative extension of the first.
Tarksi-type truth-definitions are not applicable to IF first-order logic. The reason is that Tarski’s truth-definition presupposes compositionality, a principle violated by quantifier-independence. Instead, a game-theoretical characterization of truth is available for IF first-order languages. Truth of a sentence $$S$$ in a model $$M$$ means that there is a winning strategy for the initial verifier in the corresponding game $$G(S)$$ with respect to $$M$$. The rules for making moves in a semantic game in Hintikka’s IF logic are the same as in traditional logic, except that in IF-logic incomplete information is allowed. Frege’s logic presupposes complete information. Hintikka claims that a truth-predicate for (arithmetical) IF first-order languages can be formulated in that same language. Also that IF-logic does not allow a complete axiomatization. Another important point is that the game-theoretical semantics of IF first-order languages is independent of all questions about sets and their existence. As Hintikka puts it: model theory of IF first-order logic becomes part of logic itself, and is not a part of mathematics.
In essay 4 it is argued that the semantic incompleteness of IF logic results in a different interpretation of Gödel’s incompleteness result. Since traditional logic is semantically complete, Gödel could deduce from the deductive incompleteness that elementary arithmetic is descriptively incomplete, allowing non-standard interpretations. Since Hintikka’s first-order logic is semantically incomplete, Gödel’s incompleteness theorem no longer excludes the possibility of a descriptively complete axiom system for elementary arithmetic, only allowing standard interpretations.
In essay 5 the consequences for Hilbert’s philosophy of mathematics are studied. Essays 6-8 discuss the contrast between standard and non-standard interpretations of higher-order logics. An alternative concept of computability is formulated in essay 9, and its impact on Church’s thesis is discussed. IF logic constitutes in a natural sense a logic of parallel processing, which is the topic of essay 10. Many of the topics in these essays are also discussed, sometimes more extensively, in J. Hintikka’s book: The principles of mathematics revisited (1996; Zbl 0869.03003). A review of this book by W. Hodges has appeared in J. Logic Lang. Inf. 6, 457-460 (1997).
Summarizing, Hintikka is challenging most, if not all, of the traditional well-established views on the foundations of mathematics and logic. A most exciting enterprise, which certainly needs and deserves further study.
Contents: 1. “What is elementary logic? Independence-friendly logic as the true core area of logic” (pp. 1-26) [in: K. Gavroglu et al. (eds.), Physics, philosophy and the scientific community, 301-326 (Dordrecht, Kluwer) (1995)]; 2. (with Gabriel Sandu) “A revolution in logic?” (pp. 27-44) [Nord. J. Philos. Log. 1, 169-183 (1996; Zbl 0891.03001)]; 3. “A revolution in the foundations of mathematics?” (pp. 45-61) [Synthese 111, 155-170 (1997)]; 4. “Is there completeness in mathematics after Gödel?” (pp. 62-83) [Philosophical topics, Vol. 17, No. 2, 69-90 (1989)]; 5. “Hilbert vindicated?” (pp. 84-105) [Synthese 110, 15-36 (1997)]; 6. “Standard vs. nonstandard distinction: a watershed in the foundations of mathematics” (pp. 106-129) [in: J. Hintikka (ed.), From Dedekind to Gödel. Essays on the development of the foundations of mathematics, 21-44 (1995; Zbl 0841.03001)]; 7. “Standard vs. nonstandard logic: higher-order, modal, and first-order logics” (pp. 130-143) [in: E. Agazzi (ed.), Modern logic, 283-296 (1981; Zbl 0464.03001)]; 8. (with Gabriel Sandu) “The skeleton in Frege’s cupboard: the standard versus nonstandard distinction” (pp. 144-173) [J. Philos. 89, 290-315 (1992) (a postscript has been added)]; 9. (with Arto Mutanen) “An alternative concept of computability” (pp. 174-188) [not previously published]; 10. (with Gabriel Sandu) “What is the logic of parallel processing?” (pp. 189-211) [Int. J. Found. Comput Sci. 6, 27-49 (1995)]; 11. “Model minimization – an alternative to circumscription” (pp. 212-224) [J. Autom. Reasoning 4, 1-13 (1988)]; 12. “New foundations for mathematical theories” (pp. 225-247) [in: J. M. R. Oikkonen et al. (eds.), Logic colloquium ’90, Lect. Notes Log. 2, 122-144 (1993; Zbl 0794.03017)].

### MSC:

 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03A05 Philosophical and critical aspects of logic and foundations 00A30 Philosophy of mathematics 00B10 Collections of articles of general interest 00B60 Collections of reprinted articles 03B60 Other nonclassical logic