Again one could use truth tables to define the semantics of first order formulae but ...
Let ∀A ∈ ℕ (A + 1 > A) be a formula
A |
A + 1 > A |
1 |
true |
2 | true |
… |
In order to determine the truth value of ∀A ∈ ℕ (A + 1 > A) one needs to consider all rows in above truth table.
Truth tables are not suitable to determine the semantics or reason about first order formulae.
We will use a denotational semantics to define the formal semantics of a first order formula.
Formulae ×State ↦ Bool
A State is the union of
State : (Vare ↦ Val) ∪ (Varb ↦ Bool)
where Vare ∩Varb = ∅.
We will use σ0,σ1,σ2,… to denote states and Σ to denote the set of all possible states.
Let P be a Boolean variable and A be an integer variable then σ0 s.t. σ0(P) = tt and σ0(A) = 5 is a state.