Hey Neuro,
Well propositional logic (
a la Aristotle & co) only allows the non-logical notation (the variables, if you will) to stand in for entire propositions; so we could only really represent the statement (assuming it is a proposition, which is debated) as "S" or something simple like that. I think our other option would be something like
Premise1: ~Premise1
Where "~" is just the notation for "not".
Predicate logic (
a la Russell & co) allows us to break things down a bit more and talk about the objects within the propositions, but since the object in this case
is the proposition we have a problem. Maybe a logician will have a better idea than mine but the closest thing I can think of is to say
Premise one: ExFx
"E" in predicate logic just means "there exists" (it should be a back-to-front E but I'm being lazy). "x" is just "something". F isn't part of the logical language; we use it to stand in for a predicate. So I'm suggesting maybe we let "F" mean "asserts than ExFx is false". So ExFx would mean "there exists some
x such that
x has the property of asserting 'ExFx is false'". But I feel like this is cheating.
Either way, my point is that logical symbolism is something we've made up and we can reinvent it for our purposes; one way or another we can express anything in a formal language. These different languages are useful for different things, and they differ in their scope, complexity, completeness, consistency, power, and so on. I don't see any need to confine ourselves to one formal language. This isn't to say that logic cannot express truth or falsehood; but rather that those truths belong, if you like, to the "universal grammar" rather than to the logic itself. Something we haven't tapped into, and perhaps can't* - Mossling seems to make a similar point, in a more poetic way, on page 4. And regarding the OP, I see no reason to believe that the world "conforms" to those truths, any more than it "conforms" to the fact that Jupiter is bigger than Earth. If that makes sense. I have problems in general with the necessity/contingency distinction, because I don't think it can possibly - to take it on its own terms - be supported with evidence.
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* With a nod to Godel: any axiomatic system capable of generating Peano arithmetic cannot be both complete and consistent. A consistent system will always entail a statement which it cannot prove - structurally similar to the liar paradox. Our access to truth through reasoning seems, in this way, to be inexorably limited. But in other ways, we already knew that.
With a nod to Tarski and Dennett: to understand the deep, foundational structure of our thought, we should have to be smarter than what that very structure allows. Or as Wittgenstein put it: "‘in order to draw a limit to thinking, we should have to think both sides of this limit."