# Metalogic- atomic components in common for truth-functional entailment

1. Oct 18, 2009

### nonvestigial

Metalogic-- atomic components in common for truth-functional entailment

I'm trying to prove, assuming $$\Gamma$$ is a set of sentences of SL-CN which truth-functionally entails P, that ~($$\Gamma$$ is consistent & P is not truth-functionally true (ie entailed by the empty set) & $$\Gamma$$ has no atomic components in common with P).

I broke it down into two cases of P not being t-f true; being t-f false and t-f indeterminate. In the false case this is easily in conflict with $$\Gamma$$ being consistent while entailing P, as it must also entail ~P if P is t-f false.

case 2) P is t-f indeterminate
Assume $$\Gamma$$ t-f entails P.
Hyp p. reductio that ($$\Gamma$$ is consistent & P is t-f indeterminate & $$\Gamma$$ has no atomic components in common with P).

I reason that since the empty set doesn't entail P, P has to be coming from something in $$\Gamma$$, but given the structure of the language it can't without sharing an atomic component. My teacher said it has nothing to do with induction, so I think I might be barking up the wrong tree. Also I don't know how to show this other than checking every case of the structure of P, which doesn't have to be atomic, which WOULD be induction. HALP!

Stephanie