Regarding your question about a paradox, what would qualify to you as "an actual paradox"? What is the definition against which you try a contender? Free free to look it up in a dictionary, but that probably won't help you generate a definition that makes "This statement is false" non-paradoxical. Note also that the etymology of paradox is "beyond strange", so historically the bar for qualifying is simply to be a idea or combination of ideas that is remarkably strange or surprising.

> most sentences are neither true nor false. Nothing interesting has a probability of 0.000 or 1.000.

I'll start by observing that surely you're talking about propositions, not sentences, nor utterances. Or at least you ought to be.

But more significantly, I'll note that most propositions are either true or false (under a given interpretive framework), but that as epistemologically-unprivileged observers, we must assign empirical propositions probabilities that are higher than 0 and lower than 1. Propositions like "I am a fish" or "You hate meat" or "If Rosa hates meat then Alexis is a fish" are either true or false, under any given set of meanings for the constituent words (objects, predicates, etc). I'm curious what probability you think applies to propositions like "2 + 2 = 4" and "All triangles have 3 sides" and "All triangles have less than 11 sides". I think there are very many interesting propositions that differ from these only in degree of complexity (e.g. propositions about whether or not certain code, run on certain hardware, under certain enumerable assumptions about the runtime, will do certain things).

Based on your very strange claim that all interesting sentences have non-zero non-unity probability, perhaps you're saying that you find theorems uninteresting, and moreover are only interested in statements of empirical belief, such as "I put the odds of the sun failing to rise tomorrow lower than one in a billion." In that case, I cannot imagine what statement interest would qualify as a paradox, except perhaps insofar as some empirical statements of belief are "beyond strange".

"This sentence is false" is a paradox under pretty much everyone's notion of a paradox.

> I'm curious what probability you think applies to propositions like "2 + 2 = 4" and "All triangles have 3 sides" and "All triangles have less than 11 sides".

Those are great examples, thanks. All true, and there's nothing interesting about them.

"All triangles have 3 sides" might be an uninteresting triviality, but "The sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse in a right-angled triangle" is neither trivial nor uninteresting and yet it has a probability of 1.

I love this example the most. It's the exception that proves the rule.

If you need to dig this hard to find something interesting with a probability of 1, that's pretty good evidence that the vast majority of interesting statements are not of the true/false variety.

Although I don't find it interesting, I am open-minded enough to ... embrace.. the .. uh.. diversity of the world, that allows some people, to find that interesting.

Pythagorean Theorem is "digging hard" and "not interesting"? Mind explaining?

The language that contains all Turing Machines that halt on all inputs is not decidable.

Or

e^(iy) = sin(y) + i * cos(y)

Are those uninteresting trivialities to you?

Yes, exactly. And "this code has a mathematical error in it" is often interesting, often non-trivial, and often has probability 1 (and often probability 0).

And these things are exactly the sort of thing that "differ from [trivialities about triangles] only in degree of complexity".

Note that "all triangles have 3 sides" is probably an axiom, but "all triangles have less than 11 sides" is a trivial theorem.