Paradoxes and truth in a logic of sentential operators

We show how to extend any classical (first or higher order) language with the sentential quantifiers and operators.  Graph-based semantics coincides with the classical one for the classical sublanguage. The self-referential capabilities allow to express paradoxes, which arise due to unfortunate definitions of operators. They do not ential explosion, differing thus from the contradictions. For such an extension of FOL, a complete reasoning system is obtained by adding to Gentzen's LK two rules for the sentential quantifiers.

For any given structure, an operator defining the semantics can be introduced, allowing to define a nontrivial truth operator. This extension entails unrestricted Convention T and compositionality axioms. It is also semantically conservative, so -- unlike in the standard setting with the arithmetized syntax and truth predicate - every consistent theory can be augmented with the truth operator without introducing any inconsistency.  A general form of such an extension (not relative to any structure) is under investigation.

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The location of the seminar changed to room 5450.