The Structure of Paradoxes in a Logic of Sentential Operators

Abstract Any language $$\mathcal {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> of classical logic, of first- or higher-order, is expanded with sentential quantifiers and operators. The resulting language $$\mathcal {L}^+\!$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mspace/> </mml:mrow> </mml:math> , capable of self-reference without arithmetic or syntax encoding, can serve as its own metalanguage. The syntax of $$\mathcal {L}^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math> is represented by directed graphs, and its semantics, which coincides with the classical one on $$\mathcal {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> , 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: $$\mathcal {L}^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math> can express paradoxes but remains free from them. For an expansion $$\mathcal {L}^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math> of any FOL language $$\mathcal {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> , 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.