As far as I know, dialetheism starts from an ontological assertion (there are true contradictions), while this idea of dual constructive mathematics is more of an epistemic framework, so I would say: if they are related, they are only weakly so.
Actually, I think constructive mathematics makes a strong case for dialetheism, or at least for not caring much about logical inconsistency. The problem is that the “constructive mathematics” equivalent to full general recursion is an inconsistent logic. Thus, logical inconsistencies are a lot like non-terminating programs in a Turing-complete programming language. (This can be formally stated in proof theory and type theory; it follows from the properties of progress and preservation, and from the fact that there is no construction of Falsum.) Of course one can limit oneself to only talking about finite, terminating proofs, but since the halting problem is unsolvable, all such limitations are “hacks” which will exclude some interesting constructions.
From a Platonic point of view, it is implausible that the “real” mathematical universe implements one of these ugly hacks; it’s much more plausible that “paradoxical” objects actually exist, but we are limited in talking about them by the need to exclude non-terminating proofs (such as “the set of all sets that don’t contain themselves is a member of itself, assuming that it isn’t a member of itself; but in fact it isn’t, assuming that it is; but in fact it is, assuming that it isn’t …” and so on).
Dialethists requires paraconsistent logic, as you have to be able to reason in the presence of contradictions, but paraconsitent logic can be used to model other things than truth. For example, constructive logic is often given the semantics of showing what statements can be proven, rather than what statements are true. There are similar interpretations for paraconsistent logic.
OTOH, if you think that paraconsistent logic is the correct logic for truth, then you probably do have to be a dialethist.
Does your new concept have anything to do with dialetheism?
As far as I know, dialetheism starts from an ontological assertion (there are true contradictions), while this idea of dual constructive mathematics is more of an epistemic framework, so I would say: if they are related, they are only weakly so.
Actually, I think constructive mathematics makes a strong case for dialetheism, or at least for not caring much about logical inconsistency. The problem is that the “constructive mathematics” equivalent to full general recursion is an inconsistent logic. Thus, logical inconsistencies are a lot like non-terminating programs in a Turing-complete programming language. (This can be formally stated in proof theory and type theory; it follows from the properties of progress and preservation, and from the fact that there is no construction of Falsum.) Of course one can limit oneself to only talking about finite, terminating proofs, but since the halting problem is unsolvable, all such limitations are “hacks” which will exclude some interesting constructions.
From a Platonic point of view, it is implausible that the “real” mathematical universe implements one of these ugly hacks; it’s much more plausible that “paradoxical” objects actually exist, but we are limited in talking about them by the need to exclude non-terminating proofs (such as “the set of all sets that don’t contain themselves is a member of itself, assuming that it isn’t a member of itself; but in fact it isn’t, assuming that it is; but in fact it is, assuming that it isn’t …” and so on).
Dialethists requires paraconsistent logic, as you have to be able to reason in the presence of contradictions, but paraconsitent logic can be used to model other things than truth. For example, constructive logic is often given the semantics of showing what statements can be proven, rather than what statements are true. There are similar interpretations for paraconsistent logic.
OTOH, if you think that paraconsistent logic is the correct logic for truth, then you probably do have to be a dialethist.