When I was reading The Beginning of Infinity, one of the passages that stood out to me was about contradictions (emphasis mine):
Since theories can contradict each other, but there are no contradictions in reality, every problem signals that our knowledge must be flawed or inadequate. Our misconception could be about the reality we are observing or about how our perceptions are related to it, or both. (18)
There are no contradictions in reality. This statement struck me as deeply profound, but I couldn’t figure out how Deutsch knew it was true. What if there is some contradiction in reality? What if two of our theories contradict each other, and they are both true? What if no matter how much we search, we’ll never find a theory that reconciles the contradiction, because the contradiction is embedded in the fabric of reality itself?
I reached out to Deutsch on Twitter for help:
All knowledge is conjectural. In particular, logic doesn't have a basis, let alone a 'sure' one.
— David Deutsch (@DavidDeutschOxf) September 9, 2020
His answer was rather unsatisfying. Deutsch mostly critiqued my phrasing, “on what basis can we be sure”. In critical rationalism, we are never “sure” of anything, and our knowledge does not have a “basis”. This was a fair point, but for me the question still stood: how did we know (regardless of whether we were “sure”) that there are no contradictions in reality? Is this an empirical claim? A metaphysical claim? Is “there are contradictions in reality” a bad explanation?
Months after finishing his book, I think I’ve figured out what Deutsch meant. It’s not as profound as I initially thought. “There are no contradictions in reality” is really just a reflection of our definition of “contradiction”. A contradiction is something that is logically impossible, and logic is the study of necessary truth, so “logically impossible things” are just the set of things that we currently believe could not be instantiated anywhere in reality.
Note that there are many things in reality that seem contradictory, but end up being noncontradictory upon closer inspection. One is the waveparticle duality of quantum phenomena, which seems like a paradox but is resolved by the manyworlds interpretation of quantum mechanics.¹ Another apparent contradiction is the notion of infinitesimals, which were initially derided as nonsensical (e.g. by Bishop Berkeley) but were later formalized in the hyperreal number system and nonstandard calculus.
If it turned out that there really was a contradiction in reality, then our understanding of logic and contradictions would have to be revised—clearly what we thought was a contradiction was not actually a contradiction (because it exists in reality, and contradictions cannot exist in reality!). A contradiction is, by definition, something that cannot be.²
What is the methodological implication of all this? It’s exactly what Deutsch was saying in his original quote. When we find an apparent contradiction, we should see that as a flaw in our knowledge. We should try to resolve the contradiction, either by coming up with a new theory, or finding sources of experimental error, or (more dramatically) revising our understanding of what counts as contradictory (by, e.g., formulating a new logic or number system). But we never take the contradiction to be part of reality itself.
Notes

Caveat: I don’t really know what I’m talking about with this example – just going off my understanding of Deutsch’s work (e.g. Beginning of Infinity, Chapter 11).

You might be thinking: if contradictions cannot exist, how can you even be talking about them? See the addendum to my other piece.