A question arose concerning one of Peirce’s ways of explaining logical negation.
I commented as follows.
One way of saying “not x” or “x is false” is to say “x implies α” where “α” is taken to mean “any proposition whatever”. This is the hoary old rule of ex falso quodlibet, more lately going under the name explosion principle. It is related to the definition of an inconsistent logical system as one in which every formula is a theorem, and thus in which no line of distinction can be drawn between true and false.
One place where Peirce makes use of this style of negation is in his comments on a logical formula we now call Peirce’s Law.