## Monday, September 14, 2009

### Paraconsistent Negation and Subcontrariety (Repost)

This is a repost, slightly revised, from January 2008.

With what is usually called classical negation, (p & ~p) 'explodes'; that is, combining a proposition with its negation implies everything (a principle that is also called ex contradictione quodlibet). Hartley Slater famously argued that paraconsistent negation, a form of negation for which this explosion does not happen, is really just a subcontrariety operator and therefore to be distinguished from the more robust negation in which (p & ~p) is a genuine contradiction. Slater uses this to argue that really there is no such thing as paraconsistent logic; others, Béziau comes to mind, have taken the same point to argue that even classical logic has paraconsistent elements. In any case, one response to Slater I've occasionally come across (in discussions, for instance) is to argue that there is no significant sense in which paraconsistent negation is subcontrariety, or that, at most, there is just a vague analogy between paraconsistent negation and subcontrariety. But it's easy enough to show that some paraconsistent negation really is nothing other than subcontrariety.

As I've noted before, it is possible to handle all of standard propositional logic in terms of categorical propositions if you make two assumptions:

(1) Propositions are terms;
(2) The universe of discourse is singleton.

As it turns out, it is (2) that makes negation function classically for this propositional logic. You can create a system where negation is paraconsistent if you reject (2) and add the following two assumptions to the list:

(2') The universe of discourse is plural;
(3') The default interpretation is particular.

Thus, 'p' is interpreted as +/D/+P (something in the domain is P); ~p is interpreted as +/D/-P (something in the domain is not P). With (2) universals can be inferred from particulars; which means that these two cannot both be false (as particulars) and cannot both be true (as implying universals), and therefore are contradictory. When we reject (2) however, the particulars no longer warrant inference to their universal counterparts; we no longer have a term logic of one individual. Thus (p & ~p) can be true with no ill effects; the negation is an opposition of quality between two particular propositions and therefore subcontrariety.

This is not, I think, the only possible way to get paraconsistent negation, so it does not show that Slater's argument applies to every such way of getting paraconsistent negation. I confess myself skeptical on this, in fact. But it is important to realize that

(a) this negation is paraconsistent;
(b) this negation is literally subcontrariety (albeit in a limit case of the logic), not merely analogous to it.

This point is related to another. I mentioned that Béziau and others have taken Slater's interesting comment about subcontrariety in another direction, arguing that you can have paraconsistent negations in classical logics. For instance, the not-necessary operator is read as paraconsistent, and its place on the modal square of opposition clearly shows it to be related to subcontrariety (see the PDF here, for example). Suppose we were to take the above categorical approach to propositional logic, the one that accepts (2'). This breaks from standard propositional logic and allows paraconsistent negation. But I haven't said much about how propositions should be interpreted in this approach, except to say that they are particular and in a non-singleton universe. Suppose we understood them in this way: the domain of discourse is possible worlds or, if you prefer, possible states of the world. Then the paraconsistent negation turns out to be nothing other than the modal version that others have suggested.

Very similar things could be said if you were looking for a paracomplete logic; paracompleteness bearing much the same relation to contrariety as paraconsistency to subcontrariety.