I'm trying to show something similar to a deduction theorem, where:
{Γ&P}⊨Q ⇒ {Γ}⊨P→Q
We know that {Γ&P}⊨Q. Γ is our set of basic assumptions, P is acting as an auxiliary assumption in my attempt to prove the conditional from Γ.
We can prove that:
P→(Γ&P) [from the functional profile of the conjunction in strong Kleene, and Γ as our set of basic assumptions.]
We can prove that:
(Γ&P)→Q [as we know that {Γ&P}⊨Q (although I am also struggling for this part of the proof as well)]
So, if we have: P→(Γ&P) and (Γ&P)→Q
We should be able to prove that: P→Q
But I am unsure of this proof from (P→(Γ&P))&((Γ&P)→Q) to P→Q
Edit: I have looked at the truth tables and found the following:
The light green represents all lines where P1⊨Γ&P1 (Γ in the table = L) The dark green represents all lines where Γ&P1⊨P2.
It seems that there are two lines (3.2 and 3.3) where Γ&P1⊨P2 but (Γ&P1)→P2 fails.