  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, 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 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.
