[FOM] Why inclusive disjunction?

[Neil Tennant]

On Thu, 11 Jan 2007, Richard Heck wrote:

> Thanks, Neil. But in my book, the absurdity sign is a zero-place
> connective. What I had in mind---perhaps this was obvious---had
> something to do with harmony and all that, and, since the absurdity sign
> is as much in need of introduction and elimination rules as anything
> else, any worries one had about mixing connectives in rules will surface
> with it, too. I'm not, to be sure, saying such worries can't be
> resolved. Just asking whether they would need to be.

Let's treat the absurdity sign not as a symbol in the object language, but
as a "punctuation marker" within proofs. Then it needs no introduction or
elimination rules, but can feature in the framing of such rules for
genuine connectives, such as negation, the Sheffer stroke, exclusive
disjunction, and joint denial. Taking this route also helps resolve
problems of irrelevance, by the way!---the Absurdity Rule (ex falso
quodlibet) finds no place in the resulting system.

Neil Tennant