Exclusion vs Explosion

Francesco Berto (University of Amsterdam)

Various authors (e.g. Parsons (1990), Shapiro (2004)) have objected to dialetheists that they cannot rule things out and express disagreement: their “~A” , or “A is false”, does not rule out the truth of A. Priest (2006) has replied by proposing primitive rejection (not characterized as the acceptance of negation) as an exclusion-expressing device; but qua pragmatic operator, this has expressive limitations. Nor will arrow-falsum, A → ⊥, work as an exclusion-expressive device, for it suffers from a Curry-type revenge.

I here present some thoughts on an exclusion-expressing device based on a primitive, non-logical notion of exclusion between predicates, or the corresponding properties. The idea is to have a predicative operator, “#”, that, taken as input Px, ouputs  its minimal incompatible, #Px (what follows from the having of some property or other incompatible with P). Given a dialetheic-friendly (say, De Morgan) negation ~, we want  #Px ⊨ ~Px (if something is minimally incompatible with being P, then it is not P).  However, ~Px ⊭ #Px: x may fail to be P, without thereby having a feature positively incompatible with P. Such an operator can be used to express exclusion, and it is free from the most obvious revenge (the sentence saying “I am minimally incompatible with truth”). Whether it is robustly revenge-free (i.e., whether a non-triviality proof for a formal theory embedding a rigorous characterization of  “#” can be given), I still have no idea.