> Regarding the examples you raised: the liar paradox can be solved
There are various proposed solutions, but they all have drawbacks. You have to compare the drawbacks of the different options. (From my limited memory, your proposed solution is not one of the standard solutions proposed in the literature.)
> As for Russell's paradox, as far as I know this has been solved in ZF theory
ZF theory pays a price – the restriction of the axiom of comprehension. The point is you don't solve Russell's paradox for free, every solution has its price, every solution involves giving up some component of naïve set theory; ZF chooses to partially give up the axiom of comprehension; inconsistent set theory (part of inconsistent mathematics [1]) chooses to partially give up the axiom of non-contradiction instead. If we have to give something up, how do we decide which part to give up? You think that giving up the axiom of comprehension is a smaller price to pay than giving up the axiom of non-contradiction – but is that a subjective value judgement? Or, can it be objectively justified? (And if so, how?)
A good argument for paraconsistent logic is that relevant implication is a more accurate model of natural language than material or strict implication, and relevant implication is paraconsistent. (That said, dialetheism goes beyond mere paraconsistency.)
I'd suggest (if you have the time/inclination) reading Graham Priest's book In Contradiction. It explains the arguments for all this far better than I can from memory. (I read that book 15 years ago.)
I was referring to Arthur Prior's solution to the paradox.
As for dialetheism being a better model of natural language, I'm familiar with the claim but I think its main argument fails to grasp that contradictions in natural language do not happen "in the same sense and at the same time". For instance, the sentence "This sentence is false" is true and false in natural language, but in succession and not at the same time. Moreover, dialetheism does not seem to be able to explain hierarchies and cause-effect relationships, which are key constructs of language.
I think that when it comes to logical systems such as mathematical theories, the point is not necessarily to come up with a theory that is true (that would ultimately be impossible due to Agrippa's trilemma), but to have a theory that has predictive power and is self-consistent, and this is what ZF is. Self-consistency seems to be the minimum requirement for any theory, lest it falls into triviality.
The little I know about paraconsistent logic is that it suspends consistency in a few select cases. But generally, consistency still holds, otherwise, as you said, the system would become trivial. I guess my question was: would paraconsistent logic claim to be true? Or, at least, identical to itself? If yes, then if would itself be subject to consistency and identity. And if it chooses to temporarily suspend those, then in that time it can no longer claim to be true. But if in general it complies with consistency (lest it becomes trivial), then it must reject the subsets that are not true, otherwise it could no longer be self-consistent.
I guess my point (and I could be wrong) is that paraconsistent logic is entirely non-consistent, unless it chooses to redefine what consistency means, in which case, we're back to triviality.
Thanks for the book suggestion, I'll definitely read it and hopefully find some answers.
There are various proposed solutions, but they all have drawbacks. You have to compare the drawbacks of the different options. (From my limited memory, your proposed solution is not one of the standard solutions proposed in the literature.)
> As for Russell's paradox, as far as I know this has been solved in ZF theory
ZF theory pays a price – the restriction of the axiom of comprehension. The point is you don't solve Russell's paradox for free, every solution has its price, every solution involves giving up some component of naïve set theory; ZF chooses to partially give up the axiom of comprehension; inconsistent set theory (part of inconsistent mathematics [1]) chooses to partially give up the axiom of non-contradiction instead. If we have to give something up, how do we decide which part to give up? You think that giving up the axiom of comprehension is a smaller price to pay than giving up the axiom of non-contradiction – but is that a subjective value judgement? Or, can it be objectively justified? (And if so, how?)
A good argument for paraconsistent logic is that relevant implication is a more accurate model of natural language than material or strict implication, and relevant implication is paraconsistent. (That said, dialetheism goes beyond mere paraconsistency.)
I'd suggest (if you have the time/inclination) reading Graham Priest's book In Contradiction. It explains the arguments for all this far better than I can from memory. (I read that book 15 years ago.)
[1] https://plato.stanford.edu/entries/mathematics-inconsistent/