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Correctness of the antecedent rule in sequent calculus

  1. Oct 24, 2012 #1
    Hi. I have a question on the correctness of the antecedent rule in sequent calculus when I read the book "mathematical logic" written by H.-D. Ebbinghaus etc.
    The rule says:
    [itex]\frac{\Gamma \phi}{\Gamma^' \phi}[/itex] if every member of Γ is also a member of Γ' ( Γ⊂ Γ' ,where Γ and Γ' are formula sets and Φ is a formula)
    and the correctness has been showed in the book (Γ'⊨Φ). So basically it means if the sequent in the numerator is correct, then we have sequent in the denominator being correct.

    But since Γ'⊨Φ means that every interpretation which is a model of Γ' is also a model of Φ, what if we have Γ' = Γ ∪ ¬ Φ, then there shall be no interpertation that is a model of Γ' and Φ at the same time. Then how is it correct?
    I think I have misunderstandings in some part, but I still don't know where it is.

    Thanks in advance.
     
    Last edited: Oct 24, 2012
  2. jcsd
  3. Oct 24, 2012 #2
    If there is no model of Γ, ¬Φ, then trivially each model of Γ, ¬Φ is also a model of Φ.

    Note that what you call "the antecedent rule" is normally called Weakening.
     
  4. Oct 24, 2012 #3
    OK. Thanks a lot.
    But it is still a little hard for me to understand that.
    Is there any actual case for that?
     
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