Classical logic is a poor model of natural language and informal human reasoning. Paraconsistent logics are much more accurate formal models of informal human reasoning and natural language, and much closer to them in power.
One argument for this: material implication is a poor model of how implication works in natural language, as the well-known paradoxes of material implication show. With modal logic, we can introduce strict implication, which performs better, but still falls short as an accurate model of natural language implication, as shown by the paradoxes of strict implication. Relevant logics (aka relevance logics) avoid the paradoxes of strict implication as well. But relevant logics are also paraconsistent.
Using an appropriate paraconsistent logic, Berry's paradox can be turned into a formal paradox, and such a formal paradox can be viewed as a straight-forward formalisation of the informal Berry's paradox.
Attempts to formalise Berry's paradox using classical logic fail to produce a paradox, because classical logic lacks sufficient self-referential power – classical logic cannot cope with inconsistency, so classical formal systems must have their powers of self-reference neutered – compared to natural language – to prevent self-referential paradoxes – and those restrictions also prevent a formal Berry's paradox from existing. Paraconsistent logic is far better at handling inconsistency, so it does not have to restrict self-reference, and becomes much closer to natural language in power – and Berry's paradox becomes a formal paradox as well as an informal one.
So, given the above – in what sense is Berry's paradox not "real"?
One argument for this: material implication is a poor model of how implication works in natural language, as the well-known paradoxes of material implication show. With modal logic, we can introduce strict implication, which performs better, but still falls short as an accurate model of natural language implication, as shown by the paradoxes of strict implication. Relevant logics (aka relevance logics) avoid the paradoxes of strict implication as well. But relevant logics are also paraconsistent.
Using an appropriate paraconsistent logic, Berry's paradox can be turned into a formal paradox, and such a formal paradox can be viewed as a straight-forward formalisation of the informal Berry's paradox.
Attempts to formalise Berry's paradox using classical logic fail to produce a paradox, because classical logic lacks sufficient self-referential power – classical logic cannot cope with inconsistency, so classical formal systems must have their powers of self-reference neutered – compared to natural language – to prevent self-referential paradoxes – and those restrictions also prevent a formal Berry's paradox from existing. Paraconsistent logic is far better at handling inconsistency, so it does not have to restrict self-reference, and becomes much closer to natural language in power – and Berry's paradox becomes a formal paradox as well as an informal one.
So, given the above – in what sense is Berry's paradox not "real"?