While looking into a picture-based puzzle, I came across the following expression:
$\Xi \leftrightarrow( T ( \ulcorner \Xi \urcorner ) \rightarrow S )$
$T ( \ulcorner \Xi \urcorner ) \leftrightarrow \Xi$
$|T ( \ulcorner \Xi \urcorner )$
$|--$
$|\ \Xi$
$|\ T ( \ulcorner \Xi \urcorner ) \rightarrow S )$
$|\ S$
$T ( \ulcorner \Xi \urcorner ) \rightarrow S$
$\Xi$
$T ( \ulcorner \Xi \urcorner )$
$S$
It appears to me to be some form of propositional calculus, but I have not been able to find any similar axioms, especially none with the quine quotes. Based on the lines used, I am guessing it is some sort of schematic form of a proof.
I have looked into Hilbert, Gentzen, and Frege; none of these quite match what I am looking at. The expression in and of itself seems fairly simple, though without knowing the framework, I can't understand what role $T$ and $S$ play.
Sorry to be so vague. I have studied only basic logic and proofs in college. If anyone can help me identify what axiom system, type of calculus, or formal system this is, I would greatly appreciate it!
This is a version of Curry's Paradox.
Take "T" to be truth-predicate. Now take Ξ to say e.g. ‘if I am true, then the moon is made of green cheese’. Then it seems that, by stipulation, (i) Ξ if and only if (if 'Ξ' is true, then the moon is made of green cheese). And, by the general disquotational property of the truth-predicate, (ii) Ξ if and only if 'Ξ' is true. Then -- by the given argument -- we can conclude from the truths(?!) (i) and (ii) that the moon is made of green cheese.
You can see the argument set out more clearly at the beginning of Chapter 12 of the notes linked here: https://www.logicmatters.net/igt/godel-without-tears/ (which also explains the connection to Löb's Theorem).
For lots about Curry's Paradox, see here: https://plato.stanford.edu/entries/curry-paradox/