A question on consistency in propositional logic.

[Mathelogician]

Hi everybody!

We have a theorem in natural deduction as follows:
Let H be a set of hypotheses:
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H U {~phi) is inconsistent => H implies (phi).
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Now the question arises:

Let H={p0} for an atom p0. So H U{~p0}={p0 , ~p0}.
We know that {p0 , ~p0} is inconsistent, so by our theorem we would have:
{p0} implies ~p0.
Which we know is impossible.(because for example it means that ~p0 is a semantical consequence of p0).

Now what's wrong here?
Thanks

 

[Ackbach]

Well, your theorem or schema is negating the phi, which you're not doing. In your example, you should end up with {p0} implies p0. No doubt Evgeny can correct any mistakes I just made.

 

[Evgeny.Makarov]

Well, your theorem or schema is negating the phi, which you're not doing. In your example, you should end up with {p0} implies p0.

You are right. If we apply the theorem to H U{~p0}, then phi from the theorem is p0. Therefore, the theorem concludes that {p0} implies p0.

 

[Mathelogician]

Oooooooops!