2. Show that Σ ⊆ Prop(A) is consistent and complete iff there is exactly one truth...

2. Show that Σ ⊆ Prop(A) is consistent and complete iff there is exactly one truth assignment that satisfifies Σ.

Expert Solution

we say that Σ denotes a set of propositions, that is, Σ ⊆ Prop(A)

If Σ is consistent and Σ ` p, then Σ ∪ {p} is consistent

Proof. Assume the second form of Completeness holds, and that Σ |= p. We want to show that then Σ ` p. From Σ |= p it follows that Σ ∪ {¬p} has no model. Hence by the second form of Completeness, the set Σ ∪ {¬p} is inconsistent.

We say that Σ is complete if Σ is consistent, and for each p either Σ ` p or Σ ` ¬p. Completeness as a property of a set of propositions should not be confused with the completeness of our proof system as expressed by the Completeness Theorem. (It is just a historical accident that we use the same word.) Below we use Zorn’s Lemma to show that any consistent set of propositions can be extended to a complete set of propositions.

=Suppose Σ is consistent. Then Σ ⊆ Σ' for some complete Σ' ⊆ Prop(A)

Colby Messinger answered 7 months ago

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