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# HINTIKKA, JAAKKO(1929–)

The logician and philosopher Jaakko Hintikka was born in Vantaa, Finland. Receiving his doctorate from the University of Helsinki in 1956, he was a junior fellow at Harvard University from 1956 to 1959, a research professor at the Academy of Finland, and a professor of philosophy at the universities of Helsinki, Stanford, Florida State, and currently Boston University.

Hintikka developed semantical logical methods and uses them in philosophy. He advocates applying mathematical logic, especially model theory, in philosophy, most notably to questions in philosophy of language, but also to the study of Aristotle, Immanuel Kant, and Ludwig Wittgenstein. His main contributions in logic are those of model set, distributive normal form, possible-worlds semantics, and game-theoretic semantics.

A critical view of the Tarski truth definition led Hintikka to the concept of a model set as a more constructive approach to semantics. A model set has enough information to build a canonical term model in which sentences belonging to the set are true.

A model set is a set S of first-order formulas without identity (for simplicity), with negation in front of atomic formulas only, in a countable vocabulary, and containing possibly new individual constants, such that:

1. No atomic sentence φ satisfies both φ H and ¬φ H
2. If φ ψ H, then φ H and ψ H
3. If φ ψ H, then φ H or ψ H
4. If xφ (x ) H, then φ (c) H for some constant c
5. If xφ (x ) H, then φ (c) H for all constants c occurring in H

A sentence has a model if and only if it is an element of a model set. Attempts to build a model set around the negation of a sentence form a tree, known as a semantic (or Beth) tableau. Infinite branches of this tree are model sets for ¬φ. If the tree has no infinite branches, it is finite and can be considered a proof of φ in the style of Jacques Herbrand and Gerhard Gentzen. Model sets came to play a central role in Hintikka's other work, such as distributive normal forms, possible-worlds semantics, and game-theoretic semantics.

Distributive normal forms, first introduced in monadic predicate logic by Georg Henrik von Wright, are defined as follows: Let Ani (x 1, , x n ), i K n list all atomic formulas in a finite relational vocabulary (without identity, for simplicity), and the variables x 1, , x n . If F is a formula, let [F ]0 = F and [F ]1 = ¬F. Let C 0,n i (x 1, , x n ), i I 0, n list all possible conjunctions j [Anj (x 1, , x n )]ε(j ) where ε runs through all functions K n {0, 1}. Let C m +1,n i (x 1, , x n ) i I m +1, n list all possible formulas

where J I m,n +1.

If a 1, , a n satisfy Cm,ni (x 1, , x n ) in a model M and b 1, , b n satisfy Cm,ni (x 1, , x n ) in a model N, then Cm,ni (x 1, , x n ) codes a winning strategy for player 2 in the m -move Ehrenfeucht-Fraïssé game starting from the position {(a 1, b 1), , (a n , b n )}.

Every first-order sentence ϕ of quantifier rank m is logically equivalent to a unique disjunction of formulas of the form C m,oi . This disjunction is the distributive normal form of ϕ. The process of finding the distributive normal form of a given sentence cannot be made effective. Intuitively, one pushes quantifiers as deep into the formula as possible.

Distributive normal forms can be used to systematize definability theory, such as the Beth definability theorem, the Craig interpolation theorem, and the Svenonius theorem, and to systematize infinitary logic, emphasizing formal aspects more than the game-theoretic approach by Robert Vaught.

In the logic of induction Hintikka used distributive normal forms to give, in contrast to Rudolf Carnap, positive probabilities for universal generalizations. He developed a theory of surface information to support a thesis of the nontautological nature of logical inference, with applications to Kant's analytic-synthetic distinction.

Hintikka's formal definition of possible-worlds semantics, or model systems, for modal and epistemic logic is based on his concept of model set, unlike Saul Kripke's approach, which uses actual models as possible worlds.

A model system (𝒮, R ) consists of a set 𝒮 of model sets and a binary alternativeness-relation R on 𝒮 such that:

1. If ϕ H 𝒮, then ϕ H.
2. If ϕ H 𝒮, then there exists an alternative H 𝒮 to H such that ϕ H.
3. If ϕ H 𝒮 and H 𝒮 is an alternative to H, then ϕ H.

A set S of formulas is defined to be satisfiable if there is a model system (𝒮, R ) such that S H for some H 𝒮. A formula ϕ is valid if its negation is not satisfiable. Hintikka applied possible-worlds semantics to epistemic logic, deontic and modal logic, and the logic of perception and to the study of Aristotle and Kant. (See Hintikka [1969] for a summary of his theory of possible-worlds semantics. Hintikka's 1962 book is well-known outside of philosophy, most notably in the study of artificial intelligence and theoretical computer science.)

Game-theoretic semantics has its origin in Wittgenstein's language-games, Paul Lorenzen's dialogue games, Ehrenfeucht-Fraïssé games, and Leon Henkin's game theoretic interpretation of quantifiers. The semantic game of a sentence ϕ in a model M is a game between myself and nature about a formula ϕ and an assignment s. For ϕ = ϕ 1 ϕ 2, nature chooses ϕ i . For ϕ = ϕ 1 ϕ 2, I choose ϕ i . Then we continue with ϕ i and s. For ϕ = xψ (x ), nature chooses s, which agrees with s outside x. For ϕ = xψ (x ), I choose such s. Then we continue with ψ (x ) and s. For negation, we exchange roles. For ϕ atomic, the game ends. I win if s satisfies ϕ in M, otherwise nature wins.

Game-theoretic semantics became Hintikka's tool for analyzing natural language, particularly pronouns, conditionals, prepositions, definite descriptions, and the de dicto versus de re distinction and for challenging the approach of generative grammar. Sentences like "Every writer likes a book of his almost as much as every critic dislikes some book he has reviewed" led Hintikka to consider partially ordered quantifiers and eventually independence friendly (IF) logic (1996), with existential quantifiers x /y, meaning that a value for x is chosen independently of what has been chosen for y. Thus, the semantic game of IF logic is a game of partial information.

IF logic is equal in expressive power to the existential fragment of second-order logic. The satisfiability of a sentence can still be analyzed in terms of model sets, but not provability. Wilfrid Hodges (1997) gave IF logic a compositional semantics in terms of sets of assignments, and Peter Cameron and Hodges (2001) proved it has no compositional semantics in terms of assignments only. Truth in various structures of mathematics can be reduced to logical consequence in IF logic, as in full second-order logic. IF logic has no negation and is not axiomatizable. This is countered by IF logic having a truth definition in IF logic.

## Bibliography

Cameron, Peter, and Wilfrid Hodges. "Some Combinatorics of Imperfect Information." Journal of Symbolic Logic 66 (2) (2001): 673684.

Hintikka, Jaakko, and Merrill B. Hintikka. Investigating Wittgenstein. New York: Blackwell, 1986.

Hodges, Wilfrid. "Compositional Semantics for a Language of Imperfect Information." Logic Journal of the IGPL 5 (4) (1997): 539563.

### works by hintikka

"Distributive Normal Forms in the Calculus of Predicates." Acta Philosophica Fennica 6 (1953).

Knowledge and Belief: An Introduction to the Logic of the Two Notions. Ithaca, NY: Cornell University Press, 1962.

Models for Modalities. Dordrecht, Netherlands: D. Reidel, 1969.

Logic, Language-Games, and Information: Kantian Themes in the Philosophy of Logic. Oxford, U.K.: Clarendon Press, 1973a.

Time and Necessity: Studies in Aristotle's Theory of Modality. New York: Oxford University Press, 1973b.

The Principles of Mathematics Revisited. New York: Cambridge University Press, 1996.

Selected Papers, Vols. 16. New York: Springer, 2005.