I read something on stack exchange recently about direct proof versus indirect proof (specifically, proof by contradiction). The question was about the merits of each and the top answer was that direct proof is to be preferred.
The reason given was that a direct proof provided more information than a reductio proof. I particularly liked this explanation :
Hi Smith.
Looking at the two examples given in the text you quote, I am not sure the terminology used is correct, i.e. direct/indirect 'proof'.
The indirect method described is argument by elimination - presumably all the presumed possibilities of solution to a problem are listed, and all but one are eliminated for one reason or another, leaving one probable solution. Of course, this works only so long as we assume all possibles are listed and that those listed are indeed possible.
The direct method presumably refers to all other approaches, where evidence is gathered and a single hypothesis covering them is eventually formulated on that basis. Here again, this is good method, but of course there are many pitfalls possible. Also, this method actually uses the method of elimination in a tacit way, when this hypothesis rather than any other is formulated and adopted.
So, my point is that neither of these examples refers to deductive proof, but both rather to inductively-probable 'proof'.
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Hi Avi,
Maybe that example wasn't such a good one because I was referring only to deductive proofs. I don't have time at the moment but will post another example later. What I mean by direct and indirect proof is that the latter uses an assumption but the former doesn't. So direct proofs only use the given premises whereas an indirect proof such as a reductio argument must assume the contradictory of the conclusion, and a so-called conditional proof introduces an assumption in order to end up with an if-then statement.
This was the answer which received the most upvotes in reply to the question : "Are proofs by contradiction (PBC) weaker than other proofs?"
Actually, the simple answer is "no", because when considering deductive arguments the argument is either valid or not, and for the purpose of proving validity PBC is as good as any other method. The issue is how it compares to a "direct" proof (which doesn't use any assumptions) in terms of its usefulness in generating further propositions relevant to the argument. Anyway here is the answer :
Hi Smith. I read your post. I am not a mathematician, so cannot deeply comment on this subject as such.
However, from the purely logical point of view, in deductive logic, ‘p implies q’ and its contraposite ‘not-q implies not-p’ are exactly equivalent, both meaning no more or less than “the conjunction ‘p and not-q’ is impossible” (or, identically, “the conjunction ’not-q and p’ is impossible’).
For example, in syllogistic theory, we use direct reduction to the first figure to validate fourth figure moods, and indirect reduction (reductio ad absurdum) to the first figure to validate second figure moods. See here: http://thelogician.net/FUTURE-LOGIC/Syllogisms-Validations-10.htm. Neither method is better or worse than the other in terms of logical certainty. They are just conveniences.
In inductive logic, things are more complicated as already pointed out in my previous reply to you. Notably, any claim to having an exhaustive list of hypotheses implying a certain phenomenon is, of course, itself a mere hypothesis. Indeed, any claim that this or that is a credible hypothesis for that phenomenon is itself, ultimately, a mere hypothesis. So, every conclusion is in the realm of the merely probable, even if some possibilities are more probable than others.
As regards the mathematical case presented, the apparent contradiction seems easy to resolve (if I understand it correctly). Both p and p+1 could well be primes. The fact that the symbol p is used in both terms does not affect this possibility. That is to say, ‘p+1’ could well be a case of ‘p’ – symbols do not signify fixed values but all values of a kind. So, the two series are really one and the same. They differ only superficially in symbols used.
As regards the claim that, in the said mathematical context, a proof viewed as more ‘direct’ might be more informative or productive than one viewed as more ‘indirect’ – well, ok, that might well happen. But the logical validity of both avenues can still be equal, even so, if the logic involved is sufficiently strict.
I hope that helps.
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