UPDATE II: I've just heard that this reply will be appearing in Analysis after Caret and Cotnoir's piece (which I believe is scheduled for the July 2008 issue).
UPDATE: I've now drafted a more thorough reply.
Caret and Cotnoir's reply to my Analysis paper on Beall's 5-valued approach to the Liar is now online. Their main contention is that it is not a requirement on Beall's model that designation should be expressible in the model language. They make lots of helpful clarifications, but I am not convinced by the main thrust of the paper, which I'll just say a little bit about here.
"Jenkins has provided no argument for the requirement that 'untruth' be expressible; but the requirement appears to be based on an unwarranted assumption, namely that 'true' in English is a classical notion ... it comes as no surprise that classically bivalent notions will yield inconsistency."
I disagree with Caret and Cotnoir that because designation in Beall's model is a 'model-dependent, instrumental notion', it is no problem for the model if its language cannot, on pain of Liar-like paradox, include any predicate expressing it. I don't see how helpful a model can be, with regard to resolving the Liar, if there are any semantic facts about the way the model works that cannot, on pain of Liar-like paradox, be expressed in its language. It's certainly not that I'm assuming that 'true' in English is classical. It's just that it seems to me that the problem the Liar presents is the apparent impossibility of our language's being capable of expressing certain kinds of claim about its own semantics which it seems to be able to express. Models which embrace the inexpressibility of key semantic claims don't satisfy me as ways towards a resolution of that problem.
I'm planning to write up a proper reply soon.