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Metalogic- atomic components in common for truth-functional entailment

  1. Oct 18, 2009 #1
    Metalogic-- atomic components in common for truth-functional entailment

    I'm trying to prove, assuming [tex]\Gamma [/tex] is a set of sentences of SL-CN which truth-functionally entails P, that ~([tex]\Gamma[/tex] is consistent & P is not truth-functionally true (ie entailed by the empty set) & [tex]\Gamma[/tex] 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 [tex]\Gamma[/tex] 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 [tex]\Gamma[/tex] t-f entails P.
    Hyp p. reductio that ([tex]\Gamma[/tex] is consistent & P is t-f indeterminate & [tex]\Gamma[/tex] 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 [tex]\Gamma[/tex], 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
     
  2. jcsd
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