Question

I find the negation introduction rule which I learned at university a bit confusing to reason out and think that “a, b=>¬a / ¬b” makes more sense as it means that if b implies something which is not true, then b is itself not true. I can’t seem to find an example of where the usual rule is more useful than the one I would like to use. Is there a reason why “b=>a, b=>¬a / ¬b” is used as a rule?

Answer 1

OK, I think I have a pretty rigorous argument which validates said replacement.

Let’s say that we need to introduce a negation on P. So using the usual inference rule, we prove
P => Q
P => ¬Q
and thereby prove ¬P.

Let’s say that there is no way to derive both Q and ¬Q if P is not assumed. But then from P we can derive Q /\ ¬Q which will allow us to derive anything, including the negation of a tautology.

So we can prove ¬P using the proposed rule by doing something like this:

1.  |P                 Assumed
... |...
10. |Q
... |...
20. |¬Q
21. |Q /\ ¬Q           /\ introduction on line 10 and 20
22. |¬(A => A)         Derived from line 21 using contradiction lemma
23. P => ¬(A => A)     => introduction on lines 1-22
24. A => A             Anything implies itself (a tautology)
25. ¬P                 ¬ introduction on line 23 and 24

So using tautologies we can always use the proposed rule of inference.

In other words, if you can use the usual rule of inference to introduce a negation, you can use the proposed rule of inference too.