We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic
. The language of
is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side,
is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for
known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for
and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.
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