MHB A question on consistency in propositional logic.

AI Thread Summary
The discussion centers on a theorem in natural deduction regarding the inconsistency of hypotheses. It highlights a specific case where the set H contains a single atomic proposition p0, leading to the conclusion that {p0} implies ~p0, which is deemed impossible. Participants clarify that the theorem requires negation of phi, and in this case, phi should be p0, resulting in {p0} implying p0 instead. This correction emphasizes the importance of correctly applying the theorem's conditions. The conversation concludes with an acknowledgment of the misunderstanding in the initial application of the theorem.
Mathelogician
Messages
35
Reaction score
0
Hi everybody!

We have a theorem in natural deduction as follows:
Let H be a set of hypotheses:
====================================
H U {~phi) is inconsistent => H implies (phi).
====================================
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
 
Physics news on Phys.org
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.
 
Ackbach said:
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.
 
Oooooooops!
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
Back
Top