In an effort to re-start discussion, what follows is a quote from the current draft of my dissertation proposal, illustrating why I reject Tarski/Azzouni-type solutions to the semantic paradoxes.
"To see why approaches that concede the ‘inconsistency’ of natural language but hope to create a consistent language, or isolate a consistent fragment of natural language, miss the mark, imagine the following scenario. Take a world in which Shröedinger’s Cat was not a thought experiment but a real experiment. Scientists discovered, or thought they had discovered, a cat that really was in simultaneous, superimposed states of being alive and being dead. Moreover, this discovery was taken to not only revolutionize physics but also logic. The cat was alive and not alive, so we had a true contradiction.
Imagine, then, that clever logicians responded by creating an artificial language in which the aliveness predicate could never be applied to cats, or carefully formulated rules prevented it and its negation from ever being applied to the same cat, or perhaps even where talk of cats was eliminated entirely. Perhaps some clever new quantifier would allow us to say most of what we want to say about cats in non-paradoxical contexts--that certain saucers of milk were depleted at certain times, or that certain creatures purred as they were petted--without actually referring to cats. These logicians then declared that although languages with sufficient expressive resources to discuss cats are inconsistent, their special artificial language was now consistent. A sensible response might be that the problem has not been solved but avoided.
"Now, there may be good solutions in this world. For example, a sensible monaletheist could point out that superposition and conjunction are very different things, that mathematically superposition is represented by linear combination, and that this sort of linear combination might not in fact make true the conjunction of the a statement describing one of these states and its negation. Surely, however, the beginning of wisdom would be the recognition that a language without the expressive resources to discuss cats would simply be a bad medium in which to discuss this thorny problem."