A statement that implies both and is called a false statement, and anyone can prove anything at all from a false statement, as we all too frequently observe on the political front these days.

There is however a reasonable way of handling boundaries, for instance, as illustrated by the circumference of a region in a venn diagram, and that is by means of differential logic.  I’ve been tortoising my way toward the goal line of explaining all that, and it’s going a bit slow, but there’s a gentle introduction at the other end of the link below, if you wish to achilles ahead.

• Differential Propositional Calculus • Part 1

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science