Is it easier to prove something wrong than it is to prove something right?

Question

Constantly I am faced with questions of whether something is the right choice or the wrong choice and I am forced to choose. Often, when faced with a problem, I feel that there is a correct answer and a wrong answer, and generally the correct answer has a few answers that are right and the wrong answer has an infinite (or higher number) of choices that are wrong.

Example: Defining everything a tree is not would be more useful than defining everything a tree is, because there are more wrong answers, and via process of elimination I could more accurately depict a tree knowing what it's not than trying to define what a tree is.

I understand that absolute certainty is not possible, but there does seem to be a infinite number of wrong answers and a limited amount of correct answers (or at least a greater number of wrong answers than right answers). This idea I am trying to explore and I would greatly appreciate guidance on this.

Answer 1

This in general depends on what you import into your notion of proof.

I'll give a couple of examples below:

(a) from math: given a typical background logic, it is not really easier to prove or disprove any given proposition. For example, it is not clear that disproving P = NP, or Goldbachs, is easier than proving it. More generally, by Godels incompleteness theorems, we have that there exist undecidable sentences in any reasonable theory- sentences that cannot be proved or disproved.

(b)from science: one can construe a naive popperian view. Suppose for example that we have a theory T, and some experiment E and we have that T predicts not E. We have just falsfied T, hence disproved it. On the other hand, for a naive verificationist, suppose we have some theory T', experiment E', and T' predicts E'. Then, we have just verified T'. Under these views (first passes!), it is easier to prove something wrong than to prove it right. Of course, Popper's original criticism was that this verification does not constitute a proof of T'. However, adopting a more sophisticated view- such as holism- allows us to modify the background theory T in response to evidence, making it equally hard to inductively show whether T was ever "right" or "wrong" in the first place. Of course, this is all assuming scientific realism (although a similar argument could be made for the instrumentalist but not the anti-realist, likely).

(c): decisions- OP mentions choice. Now, it is possible to be in a game in which you have far more "right" choices. Consider for example when one reaches the tipping point in a game such as chess or go. So long as one doesn't fumble spectacularly, most choices will lead to a win state (and one can prove so, presumably, although such a proof would be beyond our current computational powers).

The upshot of the above is that proof is not in general harder than disproof. Further, it is not clearly the case that there are always more wrong answers than right answers (since, given LEM + bivalence, we can, of any question of form : "P(x)?" we can ask "~P(x)?"). But this is wandering off track.

Answer 2

I think a better way to state this might be something like this:

It is as a general heuristic easier to prove a set of sentences inconsistent (when it is inconsistent) than it is to prove the consistency of a set of sentences (when it is consistent)

Why is this? Well, one simple account is to point out that a set of sentences is inconsistent if there is a contradiction within it - that there is some subset of sentences that together imply each other’s negation. On the other hand, the set is only consistent if no contradictions exist - that is that you can take any subset of sentences (including the set itself) and produce a logically coherent model where they are all true.

This latter kind of activity is a universal statement about everything in the set. On the other hand, it only takes one existing pair of mutually incompatible sentences to show inconsistency.

I’ve not actually thought this through in detail, but the complexity difference of the canonical algorithms appears to be something like the gap between NP vs Exp - if you could zip right to the inconsistency you’d be done relatively quickly, but you need to do your homework for consistency even with the help of a time-travelling supercomputer.

You can often render both in polynomial time using Boolean algebra models, so perhaps the point is a bit silly, but it does seem like the difference is intuitively reasonable!

Answer 3

In science this is related to the notion of falsifiability: a hypothesis can be proven wrong by discovering new data or performing new experiments; however, it cannot be proven correct, since we do not have access to all the possible data and haven't performed all the imaginable and unimaginable experiments.

In other words, it is impossible to prove that something is right, but it is generally possible to prove that something is wrong (if it is indeed wrong).

This finds its reflection in how statistical hypothesis testing is conducted: one postulates as null-hypothesis, a hypothesis that one wants to disprove (with high probability), and judges as negative test result as the desired outcome. In laymen life this finds its expression in, e.g., negative test result for Covid infection (which is usually a good thing).