##
**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)].

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)].

Reviewer: H.C.M.de Swart (Tilburg)

### 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 |