I honestly have no clue as to where to start with this question. My TA told us to think of it parts:

1) Consider what having a set unified with 'not P' means.
2) What this union being unsatisfiable would mean.
3) What this would mean for the relationship of 'if Q then P' with Gamma.

We barely touched this module yet so I don't know what to start off with. Any help would be greatly appreciated!


The fact that Γ ∪ {¬P} is unsatisfiable, implies that Γ ⊨ P.

This is proved using the definition of the relation of logical consequence ( ) :

every truth assignement (or model) that satisfy the formulae in Γ will also satisfy P.

To say that Γ ∪ {¬P} is unsatisfiable means that there is no truth assignment that satisfy it; thus, every truth assignment that satisfy Γ will not satisfy ¬P, i.e. will necessarily satisfy P.

And this in turn means : Γ ⊨ P.

Now we apply the tuth table for the conditional () :

Q → P is true either when Q is false or P is true.

Thus :

every truth assignment that satisfy P will also satisfy Q → P,

and in conclusion :

every truth assignment that satisfy Γ will also satisfy Q → P,

i.e. :

Γ ⊨ Q → P.

  • So to show Γ ⊨ Q → P, provide a truth table for Q → P when P = false? – K.Wong Oct 17 '16 at 17:09
  • But to show a set unsatisfiable, the consequent would have to be false wouldn't it? – K.Wong Oct 17 '16 at 21:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.