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Any language L of classical logic, of first- or higher-order, is expanded with sentential quantifiers and operators. The resulting language L + , capable of self-reference without arithmetic or syntax encoding, can serve as its own metalanguage. The syntax of L + is represented by directed graphs, and its semantics, which coincides with the classical one on L , uses the graph-theoretic concepts of kernels and semikernels. Kernels provide an explosive semantics, while semikernels generalize this to situations where paradoxes do not lead to explosion, thus distinguishing them from contradictions. Paradoxes arise only at the metalevel due to specific interpretations of the operators, but they can be avoided: L + can express paradoxes but remains free from them. For an expansion L + of any FOL language L , with the non-explosive semantics, a complete reasoning system is obtained by extending Gentzen's classical sequent calculus with two rules for the sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. The novel semantics and self-referential capabilities seem promising for a further extension of classical logic towards one capable also of consistently expressing its own syntax and truth theory. [ABSTRACT FROM AUTHOR]

Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation. [ABSTRACT FROM PUBLISHER]