The following remark is inspired by Williamson's "Everything" piece. Here is a liar paradox that uses no direct reference (as in "This sentence is false"), and indeed where the only funny business going on in it is a quantification over all sentences:
No actually tokened written sentence is true if it both ends with a decimal number which is the MD5 checksum of all of that sentence minus its last sequence of non-space symbols and if the MD5 checksum of all of that sentence minus its last sequence of non-space symbols is 187835884982830523138282294681725949791.The paradox relies on the extremely likely claim (probability about 1−2−128, I suppose) that nobody ever tokens a different written sentence satisfying the condition after the "if". Take my word for it that the sentence above does satisfy the condition.
Note that "that sentence" is not directly referential—it is, rather, a bound variable, bound by the quantification over sentences.
What should we do? Well, I think we can should either reject quantification over sentences, or reject something like compositionality. Neither is an appealing prospect, though I've got other reasons to be suspicious of compositionality and its relatives.
If one says that one should reject quantification over sentences, but allow quantification over sentence tokens, then I'll offer the following variant:
No actually written sentence token is true if it both ends with a decimal number which is the MD5 checksum of all of that sentence token minus its last sequence of non-space symbols and if the MD5 checksum of all of that sentence token minus its last sequence of non-space symbols is 127533944667835603647534200477710876898.
This yields interesting arguments. If one allows compositionality, then one should reject quantification over all sentences or all sentence tokens. I think this forces one to be an irrealist about sentences and sentence tokens. Or one can just disallow compositionality, and thus deny that the items in block quotes are bona fide sentences, expressive of propositions.