http://planetmath.org/encyclopedia/PeircesLaw.html

Jon Awbrey

But I’m just not familiar enough with the papers to say much without getting back up to speed sadly.

Lukasiewicz used an unfamiliar prefix notation, one that is good for some technical reasons, but not easily read, written, or understood. He did trouble to invent a conditonal, which *almost* works, but doesn’t quite.

I prefer the notation and some of the concepts of C. I Lewis, who attempted to make a comparison with his modal logic and that of Lukasiewicz, but wound up rejecting it, partly because of the inferior conditional, and partly because he, like the intuitionists, held tightly to the principle of the excluded middle. This similarly creates a subtle flaw in his modal logic, so that his theory is forced to use comparatively difficult and cumbersome methods of proof and semantics.

Lukasiewicz apparently overlooked the possibility, which is implicit in his definitions and axioms, that he could define a strict or exact conditional. In practical terms, that’s a huge oversight. I can hardly blame his logic for having a poor reputation, because without a strict conditional, it’s crippled to the point of paraplegia. With it, any superlative sufficient to describe the difference looks like hyporbole.

So I’m definitely sympathetic to what you say limited primarily by not nearly as much knowledge of Lukasiewicz as I should. (And honestly I haven’t spent much significant time on formal logic since the 90′s although I hope to remedy that one of these days)

I don’t blame you for being skeptical. Back when I was just trying to figure out why three valued logic was a mostly useless curiosity and not a *real* logic, I never dreamed that the results would be so far-reaching and extensive.
I would express the concepts you have given as
Peirce (not P) = ~[]P (not necessarily P)
Intuitionism, (not P) = ~P (not possible P) and from this draw some quick conclusiona about how these negations ought to behave. For instance, Double negation is not equivalent to a given proposition in either case, but there is an interesting symmetry.

Nevertheless, (to go all Mormon about logic) none but those who arouse some particle of interest sufficient to experiment with it will ever be enlightened by it. If no one is interested here and now, I’ll go away and try someone else, somewhere else, some other time.

The idea that there is one set of rules for everything seems difficult to accept.

I expected that simpler and more general methods of logic that are capable of reproducing, reconciling, and explaining the results of various seemingly incompatible systems of logic would be of some interest to someone. I was naively overoptimistic. What I’ve found instead is resounding indifference.

It seems to me that once you have a sound theory, the applications to such things as computation, geometry, and so forth all fall much more easily into place, although I’ll admit that I haven’t found those sufficiently interesting to compare the different approaches. I’m primarily interested in the comparison of different systems of logic.

The problem I’ve seen with intuitionism is that there is a subtle flaw in its foundations which, although it doesn’t invalidate it, makes it much more difficult and cumbersome to apply and interpret than it needs to be.