Math 430: Group Work III
Monday, 2/13
Recall the following versions of Soundness and Completeness: (in the following table,
Γ⊆ L0and ϕ∈ L0)
(version) (what)
Soundness (v.1) If Γ is satisfiable, then Γ is consistent
Soundness (v.2) If Γ `ϕ, then Γ ϕ
Soundness (v.3) If `ϕ, then ϕ
Completeness (v.1) If Γ is consistent, then Γ is satisfiable
Completeness (v.2) If Γ ϕ, then Γ `ϕ
Completeness (v.3) If ϕ, then `ϕ
To recapitulate (v.3): Soundness is saying “every formula provable from no assumptions is
a tautology”, Completeness is saying “every tautology is provable from no assumptions.” We
call Γ the set of “assumptions” not “axioms” so as not to confuse Γ with the set of logical
axioms (LA).
We have already proved several of these true
(version) (proved?) (where)
Soundness (v.1) yes PS2, Problem 5.
Soundness (v.2) yes Lemma 1.27
Soundness (v.3) not yet
Completeness (v.1) yes Theorem 1.37
Completeness (v.2) yes Theorem 1.41
Completeness (v.3) not yet
In fact, S./C.-(v.3) follows from S./C.-(v.2) by considering Γ = ∅.
Here are some related problems:
1. Let θ∈ L0. Prove that θis not a tautology if and only if (¬θ) is satisfiable. [Equiva-
lently: θis a tautology iff (¬θ) is unsatisfiable.]
2. Observe that “ϕ” is equivalent to ϕbeing a tautology. This is because “ϕ” stands
for “Γ ϕ” where Γ = ∅. Moreover, if Γ = ∅, then “(∗) for all ψ∈Γ, ν(ψ) = T” holds
vacuously for all truth assignments,ν. We see this in the following logical structure
of (∗): (a universally quantified conditional where the antecedent is never satisfied)
∀x((xis a formula ∧x∈Γ) →ν(x) = T)
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