Helmut, List: No, you had it right the first time. In intuitionistic/synechistic logic, we can derive double negation (nested cuts/ovals) from implication (scroll), but we cannot derive implication (scroll) from double negation (nested cuts/ovals). Implication is primitive, single negation is derived from it as the implication of falsity or absurdity, and double negation is derived from it as a composite of two single negations.
Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Thu, Apr 8, 2021 at 12:15 AM Helmut Raulien <[email protected]> wrote: > > Supplement: Oh-oh, I have mixed it completely up. Every implication is a > double negation, but not every double negation is an implication, so double > negation is the superclass, and thus more primitive than implication. > Just like every kangaroo is a vertebrate, but not every vertebrate is a > kangaroo, so vertebrate is the superclass, and more primitive. The first > vertebrate is said to have been a fish-like creature with a single > spine-bone. > Jon, List, > > if we distinguish at all between implication and double negation, we are > in the field of nonclassical logic. In intuitionalistic logic, every > implication is a double negation, but not every double negation is an > implication. So implication is the superclass, and double negation the > subclass. > the folllowing I derive from Stanley N. Salthe´s paper > "Salthe12Axiomathes": In a subsumption (classification), the subclass has > inherited the classifying traits from the superclass, and then developed > additional traits, that distinguish it from other subclasses of the same > superclass. So the subclass is more complex, less primitive than the > superclass. So implication is more primitive than double negation, and > double negation is derived from implication. But is single negation derived > from double negation, or is single negation even more primitive than > implication, and is just the combination of two negations derived from > implication? > > Best, Helmut >
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