I move on to do the next question. But I bump into trouble again.
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.
(6) If Γ,ϕ⊨ψ and Γ,¬ϕ⊨ψ then Γ⊨ψ
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