Formal semantics of Propositional Logic

Formal (denotational) semantics of Propositional Logic:

Propositions ×State ↦ Bool

Propositions: set of all possible propositions
Bool: set of semantic Boolean values {tt,ﬀ}
State: a ‘snapshot’ of the semantic values of the propositional variables in a formula

Example 3.

proposition: P ∧ (P ∨ Q)

state σ: semantic value of P is tt and of Q is ﬀ

State in the truth table corresponds to the ﬁrst two elements in a row.
The semantic value of the proposition w.r.t. a state is the last element in a row of the truth table.

A state is a mapping State from the set of propositional variables Var^b to the set of Boolean values Bool ≜{tt,ﬀ}.

State : Var^b ↦ Bool

We will use σ₀,σ₁,σ₂,… to denote states and Σ to denote the set of all possible states.

Example 4.

Let σ₀ be a state such that

σ₀(P) = tt

σ₀(Q) = ﬀ

2024-08-02