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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Correctness of the antecedent rule in sequent calculus

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