Semantics for propositional logic
Following the general method for other formal systems in this book, we must connect the system for propositional logic of the last chapter with the boolean algebras of the chapter preceding it, by using boolean algebras to provide ‘meanings’ or semantics for boolean terms and the symbolic manipulations in the system for proof. As for other logics in this book, we will develop our semantics far enough to present a Completeness Theorem and a Soundness Theorem. The basis of our semantics is the following very simple idea of a valuation.
Definition 7.1 Let X be any set, and B a boolean algebra. A valuation on X is a function f : X → B.
This notion of a valuation is very straightforward, but is enough to give a valuation v: X → B interesting extra structure. A valuation induces a map BT(X) → B defined by evaluating boolean terms over X in the boolean algebra B, using the value v(x) in place of the symbol x from X and the operations in B for all other terms. Formally, this means making the following definition by induction on the number of symbols in a term:
v(¬ σ) = (v(σ))′
v(σ Λ) = (v(σ))Λ(v(?))
v(σ ν?) = (v(σ))ν(v(?))
v(⊥) = ⊥
where the right hand side of each of these is evaluated in B using its boolean algebra structure.