I've been working on a little diagram that illustrates several points, only a couple of which I'll mention in the first post. I post this here because I want to improve upon my diagram; hopefully you the reader can provide useful feedback.
The gray area represents what you can prove based on the number of assumptions you make. If it's a white region, that means it is not provable but something can be non-provable for a couple of reasons: it's false or it's true but unprovable.
If you assume nothing other than the ambient logical axioms (identity, noncontradiction, and law/axiom of excluded middle), the only true statements are tautologies. This is represented by region 1.
On the other extreme, if you assume a statement and that statement's negation (obtained by slapping "NOT" in front of the statement), then this is a contradiction. Assuming a statement and its negation leads us to conclude (through some mathematical tomfoolery) that ALL grammatically-correct utterances are TRUE and provable!! That means, for instance, that all negations of statements are also true and provable. Everything is both true and false simultaneously... This is only if we assume a statement and its negation. So we can't make too many assumptions or else all grammatically-correct utterances are true and false. This corresponds to regions 5, 4, and part of 1.
The "interesting" cases are between when you assume more than nothing but less than assuming a pair of mutually exclusive statements.
The gray area represents all statements that are provable from the assumptions, except for region 5.
More to follow..
