Semantics of wffs Given an interpretation l=< D, lo> and an assignment o, I(o (A), the value of A with respect to o in l. for each wff is defined as follows I(p(o)=lo(p)if p is a propositional constant I(p)o=o(p) if p is a propositional variable T(Pt1…tn)(σ)=o(P")(t1)()…T(tn)(a) if Pn is an n-ary predicate constant, and t1,…, tn are terms T(P"t1…tn)(σ)=σ(P")(t1)(σ)…(tn)(σ if Pn is an n-ary predicate variable, and ,……, t are terms. Logic in Computer Science -p 6/18
Semantics of Wffs Given an interpretation I =< D, I0 > and an assignment σ, I(σ)(A), the value of A with respect to σ in I, for each wff, is defined as follows • I(p)(σ) = I0(p) if p is a propositional constant. • I(p)(σ) = σ(p) if p is a propositional variable. • I(Pnt1 · · ·tn)(σ) = I0(Pn)I(t1)(σ)· · · I(tn)(σ) if Pn is an n-ary predicate constant, and t1, · · · ,tn are terms. • I(Pnt1 · · ·tn)(σ) = σ(Pn)I(t1)(σ)· · · I(tn)(σ) if Pn is an n-ary predicate variable, and t1, · · · ,tn are terms. Logic in Computer Science – p.6/18
T()(A) I( A(o)=I(A)(o)if A is a wff I(AVBo)=I(A(oVI(B)(o)if A and B are wtrs If a is a wff and x is an individual variable if there exists d∈D such that I(A)(olc/d t otherwise Logic in Computer Science -p 7/18
I(σ)(A) • I(∼ A)(σ) = ¬I(A)(σ) if A is a wff. • I(A ∨ B)(σ) = I(A)(σ)∨I(B)(σ) if A and B are wffs. • If A is a wff and x is an individual variable, I(∀xA)(σ) = F if there exists d ∈ D such that I(A)(σ[x/d]) = F T otherwise Logic in Computer Science – p.7/18