Hintikka set

In mathematical logic, a Hintikka set is a set of logical formulas whose elements satisfy the following properties:

1. An atom or its conjugate can appear in the set but not both,
2. If a formula in the set has a main operator that is of "conjuctive-type", then its two operands appear in the set,
3. If a formula in the set has a main operator that is of "disjuntive-type", then at least one of its two operands appears in the set.

The exact meaning of "conjuctive-type" and "disjunctive-type" is defined by the method of semantic tableaux.

Hintikka sets arise when attempting to prove completeness of propositional logic using semantic tableaux. They are named after Jaakko Hintikka.

Propositional Hintikka sets

In a semantic tableau for propositional logic, Hintikka sets can be defined using uniform notation for propositional tableaux. The elements of a propositional Hintikka set S satisfy the following conditions:

1. No variable and its conjugate are both in S,
2. For any [math]\displaystyle{ \alpha }[/math] in S, its components [math]\displaystyle{ \alpha_1, \alpha_2 }[/math] are both in S,
3. For any [math]\displaystyle{ \beta }[/math] in S, at least one of its components [math]\displaystyle{ \beta_1,\beta_2 }[/math] are in S.

If a set S is a Hintikka set, then S is satisfiable.

Sources

• Smullyan, R. M. (1971). First-Order Logic (Second printing ed.). Springer Science & Business Media. pp. 21, 26–27. ISBN 978-3-642-86720-0.