It is sometimes held that everything follows from a contradiction. If we abandoned that inference rule,
which kind of proofs, if any, would be impossible? If not impossible, which kind of proofs would be
radically more difficult?
It is not "sometimes held", as if that's an opinion. It is a provable result in formal systems with
true-false dichotomies that a contradiction, since it equates a true and a false statement, entails
anything.In other words, and putting it very roughly, since, in effect, true equals false, then anything
goes. If you do this rigorously and formally in, as I say, a formal system where true and false
statements are the only ones allowed, then it's easy to demonstrate that you can fill in the blanks and
make any statement true... since false statements are, given your contradiction, true. You see? So
you can't "abandon" that "rule", because it isn't a rule. It's a conclusion, an inference, resulting from
the structure of these systems. Now there aresystems with infinite gradations between true and false,
i.e., fuzzy logical systems, and I don't know enough about these to say what the result of equating a
mostly true and a mostly false statement would be.
At any rate, as I say, you can't"abandon" that "rule". If you said that you simply weren't allowing
contradictory statements in your system, then logicians would merely applaud your rigor and continue
with their work. Nothing would change; they might be a bit skeptical that you'd be able to be so
consistent... but there are many people with that degree of consistency in a variety of fields employing
formal systems. It's merely what's expected from a competent theorist.