This paper develops a theory of propositional identity which distinguishes necessarily equivalent propositions that differ in subject-matter. Rather than forming a Boolean lattice as in extensional and intensional semantic theories, the space of propositions forms a non-interlaced bilattice. After motivating a departure from tradition by way of a number of plausible principles for subject-matter, I will provide a Finean state semantics for a novel theory of propositions, presenting arguments against the convexity and nonvacuity constraints which Fine (2016, 2017a,b) introduces. I (...) will then move to compare the resulting logic of propositional identity (PI) with Correia’s (2016) logic of generalised identity (GI), as well as the first degree fragment of Angell’s (1989) logic of analytic containment (AC). The paper concludes by extending PI to include axioms and rules for a subject-matter operator, providing a much broader theory of subject-matter than the principles with which I will begin. (shrink)