Dual Tableau for Propositional Logic with Identity

Abstract

In this chapter we consider propositional logic with identity, referred to as SCI (Sentential Calculus with Identity) introduced in [Sus68]. It is two-valued as the classical logic, but it rejects the main assumption of Frege’s philosophy that the meaning of a sentence is its logical value. Non-fregean logics are based on the principle that denotations of sentences of a given language are different from their truth values. SCI is obtained from the classical propositional logic by endowing its language with an operation of identity, \(\equiv \),and the axioms which say that formula φ ≡ ψ is interpreted as ‘φ has the same denotation as ψ’. Identity axioms together with two-valuedness imply that the set of denotations of sentences has at least two elements. Any other assumptions about the range of sentences or properties of the identity operation lead to axiomatic extensions of SCI. In general, the identity operation is different from the equivalence operation, that is two sentences with the same truth values may have different denotations. If we add (φ ↔ ψ) ≡ (φ ≡ ψ) to the set of SCI axioms, then we obtain the classical propositional logic, where the identity and equivalence operations are indistinguishable. In this way the Fregean axiom can be formulated in SCI. Some extensions of SCI are known to correspond to modal logics S4 and S5 and to the three-valued Łukasiewicz logic (see [Sus71a]). Decidability of the logic SCI is proved in [Sus71c].