H*-algebra Semantics for Propositional Logic, Probability Theory, and Quantum Theory

We show that unital $H^*$-algebras provide a unified algebraic semantics for propositional logic, probability theory over finite sample spaces, and finite-dimensional quantum theory. More specifically, we show that key objects and rules in each theory - such as propositional languages, atomic propositions, evaluations, events, densities, probability functions, orthogonal projectors, density operators, the Born rule, and L\"{u}ders rule - can be abstractly characterized using the structure of $H^*$-algebras. For propositional logic specifically, we identify two distinct but isomorphic $H^*$-algebraic representations: the $n$-fold tensor product of $\Z^2$ and $\Z^{2^n}$ (where $n$ is the number of atomic propositions). We show that the former represents propositional languages and the latter represents truth table semantics. The existence of isomorphisms between the two structures allows us to give truth table representations a formal mathematical definition. The $H^*$-algebraic perspective we develop in this paper makes transparent the deep structural analogy between propositional logic, probability theory, and quantum theory: they differ not in the algebraic equations governing their key objects, but in the commutativity and scalar structure of the underlying $H^*$-algebra.