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Proofs (cont’)

  • 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

  • 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

  • Soundness Theorem:

    If φ is provable from Δ, then Δ logically entails φ.

  • Completeness Theorem:

    If Δ logically entails φ, then φ is provable from Δ.


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