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I am now reading "Intermediate Logic" by David Bostock. I am stuck by a question. It asks:

Show that the basic principle for negation (Γ,¬ϕ⊨ iff Γ⊨ϕ) can in fact be deduced from:

If Γ,ϕ⊨ψ and Γ,¬ϕ⊨ψ then Γ⊨ψ

The principles of "Assumptions", "Thinning" and "Cutting" have been previously introduced in the book.

My wonder is: how can one deduce a bi-conditional statement from a conditional statement?

Thank you!

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  • There is something missing.... What is Γ,¬ϕ⊨ ? Jul 27 '20 at 7:31
  • I think Γ,¬ϕ⊨ means Γ and ¬ϕ are inconsistent. Jul 27 '20 at 8:09
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    As written in p.12, Γ⊨ is to mean: There is no interpretation in which every formula in Γ is true. Jul 27 '20 at 8:10
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Assume the principle If Γ,ϕ⊨ψ and Γ,¬ϕ⊨ψ then Γ⊨ψ and assume Γ,¬ϕ⊨.

From the last one, by Thinning on the right: Γ,¬ϕ⊨ϕ.

By Assumption: Γ,ϕ⊨ϕ.

Thus, applying the principle above, we have:

Γ⊨ϕ.

Thus:

If Γ,¬ϕ⊨, then Γ⊨ϕ.


Similar for the other part: if Γ⊨ϕ, then Γ,¬ϕ⊨ϕ, by Thinning, and Γ,¬ϕ⊨¬ϕ by Assumption.

Thus, Γ,¬ϕ is inconsistent.

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  • So we can just show If Γ⊨ϕ, then Γ,¬ϕ⊨ by assuming Γ⊨ϕ without having to consider the principle If Γ,ϕ⊨ψ and Γ,¬ϕ⊨ψ then Γ⊨ψ ? By the way, your answer helps me a lot! Thank you! Jul 27 '20 at 9:25
  • @HowardLeong - Exactly. Jul 27 '20 at 9:26
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I move on to do the next question. But I bump into trouble again. It asks: Show that version(a) of the disjunction principle implies version(b), given a suitable principle for negation.

version(a) Γ,ϕ∨ψ⊨χ iff Γ,ϕ⊨χ and Γ,ψ⊨χ

version(b) Γ,ϕ∨ψ⊨ iff Γ,ϕ⊨ and Γ,ψ⊨

Some principles for negation:

(1) Γ,¬ϕ⊨ iff Γ⊨ϕ.

(2) ϕ,¬ϕ⊨

(3) ϕ,¬ϕ⊨ψ

(4) ¬¬ϕ⊨ϕ

(5) ϕ⊨¬¬ϕ

(6) If Γ,ϕ⊨ψ and Γ,¬ϕ⊨ψ then Γ⊨ψ

My trial: I have referred to your strategy in solving the previous question. But at best I can only arrive to these four sentences (Given (a) and assuming Γ,ϕ∨ψ⊨ and assuming Γ,ϕ⊨):

(*) If Γ,ϕ∨ψ⊨, then Γ,ϕ⊨χ and Γ,ψ⊨χ

(**) If Γ,ϕ⊨ and Γ,ψ⊨, then Γ,ϕ∨ψ⊨χ

(***) If Γ,ϕ∨ψ⊨, then Γ,ϕ∨ψ⊨χ

(****) If Γ,ϕ⊨ and Γ,ψ⊨,Γ,ϕ⊨χ and Γ,ψ⊨χ

Can I just claim that given the above four sentences, then (b)? However, it seems to me not really clear that (b) follows from them without further explanation. Is something missing? Or one should think completely otherwise? Thank you

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