Double Logic DL and Existential Graphs Gamma-DL

Abstract

In this paper, the deductive system for double propositional logic (DL) and gamma-DL
existential graphs are presented. It rigorously proves the consistency of the DL and that the DL theorems
correspond exactly to the valid existential graphs of gamma-DL. When the language of DL is
restricted to the language of classical propositional logic (CL), the restriction associated with gamma-
DL coincides with the valid existential graphs of Charles Sanders Peirce’s alpha system. It turns out
that intuitionistic propositional logic (IL) theorems are DL theorems; furthermore, when the language
of DL is restricted to the language of IL, the restriction associated with gamma-DL coincides with the
valid existential graphs of Arnold Oostra’s intuitionistic alpha system. As a consequence, it is inferred
that gamma-LD has as particular cases, the alpha existential graphs of LC and LI. Finally, in DL, the
Aristotelian definitions of truth and falsity are derived, with which the ability of DL to solve a version
of the liar paradox, where LC and LI fail, is illustrated.