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Proofs (cont’)
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If there exists a proof of a sentence φ from a set Δ of premises and the standard axiom schemata using Modus Ponens, then φ is said to be provable from Δ, written as
Δ ⊢ φ
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There is a close connection between provability and logical entailment (⊧):
A set of sentences Δ logically entails a sentence φ if and only if φ is provable from Δ
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Soundness Theorem:
If φ is provable from Δ, then Δ logically entails φ.
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Completeness Theorem:
If Δ logically entails φ, then φ is provable from Δ.
