Paraconsistency is orthogonal to (consistency XOR inconsistency). Equivalently, paraconsistency is to non-dual as (p XOR not p) = not (p and not p) is to duality. Contradiction tolerance isn't a positive quality; it's defined by the absence of contradiction intolerance. Strongly consistent systems have absolutely no contradictions in them; strongly or strictly consistent systems also have logical explosion proofs when you put a contradiction in the premises. Contradictions in the consequences of a proof can be handled by erasure of one or more of the premises that entailed the contradiction. That's the basis for Reductio ad Absurdum, and the specific form of proof by contradiction where you posit the existence or non existence of a proposition and derive a contradiction to prove that it is specifically not that proposition, and under excluded middle and non-contradiction, show that the negation of the proposition necessarily exists. This forms the basis of many indirect proofs and measurements.

What all this is for is to exclude contradictions from our arguments except as consequences of proofs that erase propositions that prove contradiction. We can weaken consistency in various ways to tolerate contradictions. Basic logic is one example where the structural rules of weakening and contraction are absent as are the axioms of non-contradiction and the excluded middle. Without the ability to arbitrarily copy or erase information, it is difficult to definitely prove anything. Paraconsistent logics with consistent metasystem necessarily prove strictly fewer theorems than their consistent metasystem. Paola Zizzi proved that paraconsistent logics of this kind are subject to a no-self-reference meta theorem for contradictions. Zizzi's logics Lq and Lnq lack all axioms except a pair of propositionl identities for each proposition p_n and its primitive negation p_(n+1); Lq and Lnq also lack all structural rules. The turnstile representing metalogical constructive proofs can't be expressed in the syntax of Lq or Lnq and the metasystem is consistent. Her turnstile admits complex values, but the sum of probabilities is required to equal one, and the sum of the cross products z_0 z_1* + z_0* z_1 = 0. This effectively ensures that the proofs of Lq and Lnq and any quantum computer implemented satisfying it will have its output reduce to classical computations satisfying metasystem consistency. In a sense, these forms of paraconsistency manually join a subset of non-self-referential contradictions into a partially or weakly consistent sub-first-order object logic. Zizzi's creates a complex relationship between a proposition and its negation which is analogous to classical AND but explicitly accounts for the complex degrees and decomposition of the propositions. She identifies it with the process of superposition. Her Lq models a qubit as a weak contradiction; it is unitary.

I conjecture removing the axioms of identity and using neither a consistent, nor an inconsistent metasystem will allow the formalization of the turnstile for unrestricted domains. The trick is in what happens to trivial notions when you don't have non-contradiction, the excluded middle, and identity axioms/theorems to for the absolute emptiness of a set. I've been trying to wrap my brain around a way to formalize a paraconsistent system which is independent of and essentially undecidable from consistent formal systems and consistent metasystems.

My intuition is that a non-dual relationship is paraconsistently defined--analogous to an automorphism--between an empty set and universal sets. In (Empty XOR not Empty) logics the Empty set is singular and absolutely empty; the negation of the Empty set can't possibly be universal sets because universal sets entail Russell's paradox, but I think their undefinable because there can't exist an absolute separation relationship of the two concepts. They are not absolutely true or false, so negations of either can not be absolute in general. Think of it as an uncertainty relationship between the truth values of the Emptiness and non-Emptiness of a set. The more certain you are it is Empty the less certain you can be about its non-Emptiness.

The notions of Reductio ad Absurdum and Proof by Contradiction need to be revised in such a system. As well as vacuous truth and empty functions. You still use the techniques of consistency, but consistency is something recovered in certain cases of conditions of the paraconsistent metasystem. Proof by contradiction splits into two parts: proof by contradiction tolerance and proof by contradiction intolerance. Proof by contradiction tolerance is new and to be used for describing recursive paraconsistent arguments. Proof by contradiction intolerance is a relabeling of classical proof by contradiction; it applies for restricted or conditional domains.

Under consistency, combonatorics, computing, Boolean algebra, the set of recursive functions, and others are equivalent systems. Cantor and cardinality proofs tell us that. The Chomsky hierarchy gives us further insight into how the formal languages, grammars, systems, and Turing machines all interrelate. The difference is in where we go from there. Calculus, Higher order logics, Diophantine polynomials, recursively enumerable languages, the set of all Turing machines, and uncountable sets spin off into infinite infinities. The Turing point is proof by contradiction. We can trust Gödel's proof in so far as we accept his assumptions or the premises of his argument. If you reject the consistency of his consistency assumption for instance, you reject the conclusions of his proofs in general. Truth preservation does that. For Gödel and his contemporaries, there was only one kind of contradiction and it could not be tolerated or trusted. Arbitrary contradictions led to explosion, so it was argued that contradictions in the conclusions of an argument necessitates the erasure of the premises; and there is effectively assumed to be exclusively contradictions or no contradictions.

Let us doubt Gödel, Tarski, Russell, and the great thinkers of Western history. Take a radically skeptical stances regarding the need for consistency and the intolerance of contradictions. Arguments to the effect, "There exists at most one of either p or not p. We need only consider consistent systems or not consistent systems. All other options can be reduced to these extremes." Can be dismissed because we can write a Gödel's sentence which gives us another option, and we can show the necessary independence of contradictions from Gödelian systems. We can point directly to examples of paraconsistent logic as proof of neither consistent nor inconsistent alternatives. So we refuse to accept the assumption of consistency and demand again: prove or disprove it.

Now consistency is a hypothesis that we need to theorize a way to develop it from paraconsistent formal systems. But we need a paraconsistent number theory. We need paraconsistent computing. We need paraconsistent algebras and set theories. And in at least restricted domains, some of it needs to reduce somehow back to the familiar consistent results.

What changes when you can partially define a curve joining what would be a singularity in classical mathematics? When 0/0 can be given a paraconsistently definite meaning? How does diagonalization work with complex-valued membership and non-membership? What are the properties of recursive contradictory expressions?