> I think it's certainly possible that a model built on propositional logic does not cover the whole of reality.
Not only is it possible, this is the normal state of affairs. Until we have a theory-of-everything, none of our models cover the whole of reality.
But all of our models rest on a foundation of certain basic assumptions. Among these are that (some) statements about reality can be classified as true or false, and that these are mutually exclusive possibilities. Not all statements can be classified as true or false (e.g. Euclid's fifth postulate) but for the ones that can, if it is true then it is not false, and if it is false then it is not true.
If you deny that, then the entire edifice of human intellectual achievement really does crumble into dust.
I don't see how that would necessarily mean that all science and math go poof; if some phenomena satisfy our axioms, they would continue to satisfy analysis based on those axioms. If some phenomena do not, they would be excluded from our analysis.
We already hypothesize at least one such event: the Big Bang. Scientifically, it's nonsensical to reason about what happened "before" it to cause it; but we also seem to have a ton of evidence that it actually happened.
I'm not trying to be woo-woo new-agey here BTW. I don't think it's wrong to consider the limits of scientific analysis.