How to Reason with Four-valued Definitional Set Theory
1. There is a "ground level" composed of statements in first-order logic that can be reasoned about in the usual manner.
2. The axiom of set comprehension is interpreted as allowing arbitrary definitions of sets (rather than asserting the actual existence of sets). A general logic of definitions would allow us to define any sort of object with any sort of definition, and reason about it from there; in this case, though, I only am worried about defining sets.
3. The definition of a set S can be "for all x: x is an element of S if and only if [statement]" for any statement (in the full language, not just the first-order part).
4. If "x is an element of S" can be derived from the definition of S plus any relevant information on x, then it is so. Similarly for "x is not an element of S". "Relevant information" here means the definition of x if x is a defined set, plus all information existing at the first-order level. Notice that the definition of S might imply both "x is an element of S" and "x is not an element of X", or neither of these.
That's it! Unless I've missed something. But, anyway, one thing that is very specifically not included is a way for contradictions (or anything) to leak over from the level of defined objects to the "ground level" of normal logic.
Oh, I suppose this should be added:
5. Statements of the full language can be reasoned about with some sort of intuitionistic paraconsistent logic.
This should compound statements to be constructed; for example, if two different statements about sets are true by definition, then "statement1 and statement2" should be treated as a true sentence. I think an OK way of doing this would be to allow the Gentzen-style [Edit: oh, I meant Prawitz-style!] introduction/elimination rules for and, or, forall, and exists, skipping over the rules applying to implication and negation. But, don't take my word for it!
Now, the system is definitely not referentially complete. It lacks the ability to refer to its own possible truth-value combinations. I'd need to add a nonmonotonic operator allowing the logic to conclude that a a statement was not true-by-definition and not false-by-definition, which of course gain credibility as attempts to prove a statement true/false fail. This gets somewhat confusing, because false-by-definition is distinct from not true-by-definition. Anyway, soon I'd be adding in the full machinery of a logic of "implies". So, the four-valued set theory doesn't seem like a complete foundation. Still, it is interesting!