# Correctness of the antecedent rule in sequent calculus

1. Oct 24, 2012

### matts0

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:
$\frac{\Gamma \phi}{\Gamma^' \phi}$ 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.

Last edited: Oct 24, 2012
2. Oct 24, 2012

### Preno

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.

3. Oct 24, 2012

### matts0

OK. Thanks a lot.
But it is still a little hard for me to understand that.
Is there any actual case for that?