That which has no complement is indistinguishable from its complement and therefore contains zero information. But if logic has no informational value, then neither does logical consistency. And if logical consistency has no informational value, then consistent and inconsistent theories are of equal validity.
Rex professes a lack of understanding as to why not having a complement is the same as being indistinguishable. The standard answer, of course, is that since information always restricts (or constrains) a potential by eliminating its alternatives therein, nothing to which informational value can be attached lacks a complement (in some probability space). For example, since observing that something exists is to rule out its nonexistence - existence and nonexistence are complementary states, provided that we conveniently classify nonexistence as a "state" - such observations distinguish existence from nonexistence and thus have positive informational value. On the other hand, that to which no information at all can be attached cannot be said to exist, and is thus indistinguishable. Because this applies to consistency and inconsistency, it also applies to logic and nonlogic.
Rex then states that "'logic' and 'not-logic' is a contradiction, under logic, so if logic admits both 'logic' and 'not-logic' then logic is self-contradictory." Not if it treats nonlogic as something which is excluded by logic in any given model, for example a nondistributive lattice. He then observes that “there's no problem with having non-logical statements, just allowing the entire theory of logic and the theory of not-logic to simultaneously exist in the same model.” Although I see where Rex is coming from, logic and nonlogic can in fact exist in the same model, e.g. a nondistributive lattice, provided that nonlogic does not interfere with logic in that part of the model over which logical syntax in fact distributes, e.g. the Boolean parts of the lattice. That the non-Boolean parts of the lattice approximate poorly-understood relationships among Boolean domains is irrelevant to the value of such "non-logical" models, as we see from the fact that nondistributive lattices permit the representation of real noncommutative relationships in quantum mechanics.