We can, we can, we can carry on trying to prove this till we're blue in the face but, um, er, some of it is not provable as far as I can see.
Notice (importantly) that open formulas are assertible as axioms and provable as theorems in SQML.
The difference between the two notions of provability: "provable in a given formal system S" and provability by any correct means — manifests itself as a consequence of Gödel's Second Incompleteness Theorem, as follows.
Aesthetically, there aren't many things as instantly provable".
But Judge Hernandez did allow that a single post published on Christmas Day in 2010 charging all manner of criminal conduct could be read as containing "provable assertions of fact". A one-day trial took place on Nov. 29, and after deliberating for 75 minutes, the jury awarded Obsidian $1 million and Mr. Padrick $1.5 million.
In fact, Gödel's incompleteness theorem indicates that the principle is false when □nA would be interpreted as being provable in a reasonable proof system, which, however, is certainly not what Brouwer had in mind.
In any event, disproving must be understood as 'establishing inconsistent,' not as 'establishing not provable' (or 'establishing not valid').
Indeed, let \ \Box F\) be interpreted as F is provable and \(t : F\) as t is a proof of F in a given formal theory \(T\), e.g., in Peano Arithmetic \(\mathsf{PA}\).
In particular, the term is used for such procedures for finding whether in a particular logistic system, logical calculus, or formal mathematical system some given "well-formed formula" (generated in accordance with established formation rules) is or is not provable as a theorem of the system.
