Consider a string 'self', and let 'self' denote itself -- that is, 'self' denotes 'self'. Thus, since 'self' denotes self, and denotation is a single-valued predicate, we get that 'self' is self. Alongside, we get that self denotes 'self'.
In my opinion, this conclusion seems questionable. So my question is: Can we allow sentences to denote themselves without leading to contradiction? If we can, is the above conclusion valid?
My attempt is written below. Apologies for the wall of text.
Some naive attempts at showing a contradiction failed. For example, let 'a sentence that does not denote itself' denote itself. By the above argument, 'a sentence that does not denote itself' is a sentence that does not denote itself, which is a contradiction.
Similarly, let P be the property of a sentence not denoting itself. Since 'P' denotes P, 'P' is P (that is, it does not denote itself) if and only if 'P' denotes P if and only if 'P' does not denote P, which is again a contradiction.
In both cases, we get around the contradiction by noting that that use of "is" is in the sense of predication instead of identity.
I then tried to show that the conclusion is invalid, assuming an object language and meta-language distinction, with '...' being an object language string. We further assume that the denotation relation is a relation between strings in the object language and sentences in the meta-language.
Clearly, we first need to way to mark meta-language sentences, since otherwise the following is ambiguous: A is B is C. So, let "..." be a meta-language sentence.
Using this terminology, we get that 'self' denotes "'self'" and 'self' denotes "self", and hence "'self'" is "self". In other words, the meta-language sentence consisting of the object language string self is identical to the meta-language sentence self (cf. the object-language self being identical to self). Or, self is identical to 'self' along with the additional information that it is an object language string (the '...'). This seems to me to be a better conclusion than the former.