Two Claims I'm Striving to Understand

1) Neils Bohr, famous 20th century physicist (link here):

"The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth."

As his son summarized it, this is a way to classify truths. Profound truths are recognized by the fact that the opposite is also a profound truth, in contrast to trivialities where opposites are obviously absurd. I understand what it is literally claiming, but I fail to see how it could ever be applied, and why many have considered it so powerful.

Let's try applying this maxim to a few potential truths. How about, "everything that goes up must come down." Now, how would you construct the opposite of this truth? Two possible opposites are that either everything that goes down must come up, or that everything that goes up must not come down. Both of these are untrue, so by his dichotomy the statement is a triviality.

For another example, let's take something that Bohr himself worked on, the complementarity principle of quantum physics that a single quantum mechanical entity can either behave as a particle or as wave, but never simultaneously as both. One possible opposite of this statement is that a single quantum mechanical entity must always behave simultaneously as both a particle and a wave. This is demonstrably false, suggesting that by his dichotomy one of the key findings of quantum physics is merely a triviality.

"Going meta" and subjecting this maxim to its own rule also yields many potential opposites. One is that the opposite of a correct statement is a profound statement, but the opposite of a profound truth is a false statement. This is obviously false, and I'm not sure what that implies.

So my issues with this claim are that there is no obvious method for choosing which opposite of many possible ones you should evaluate and that I can't encounter any profound truths beyond maybe some generalization of Newton's third law (can you?). So, why is this famous, and why did Bohr consider it such a useful maxim?

2) Robin Hanson, in the comments to one of his old posts about conspiracies,

I’m struck by how many people think we understand social science well enough to exclude the possibility of a large group keeping a secret. Do these same people accept standard economics?

This was somewhat of a view quake to me when I first read it, because I was (am?) one of those people who excludes the possibility of a large group keeping a secret. My confusion stems from my inability to pinpoint the facet of standard economics that these social science enthusiasts are violating.

Let's presume that our large group contains at least some self-interested individuals. If the secret were important enough, then over time at least one of the members would probably realize that he/she could reap a large economic and potentially social reward by defecting from the group and revealing the secret through some other business venture (i.e., a tell-all book) that would capitalize on the knowledge. Of course, there might be an issue of credibility for the defector. But evidence shouldn't be too hard to gather in the form of tape recorded statements, etc., unless the organization keeping the secret is truly Big Brother. So, what am I missing here?