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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.

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    See Self-Reference. – Mauro ALLEGRANZA Oct 7 '16 at 8:53
  • The question is different. Consider the canonical example of 'this statement is false'; it refers to itself, but does not denote (or name) itself. As an aside, if my terminology/concepts are incorrect, please let me know -- I am not a student of philosophy. – random human Oct 7 '16 at 9:28
  • To avoid conundrums, we have to find some "trick", like English for object-language and German for meta- or a clever use of quotation marks... I'll try with fonts. We have a word (expression) in the object-language that denotes an "object" in the world. Id I say "The apple is red" I'm speaking in the object-lan "using" the word "apple" to speak of the world (the apple out there). Using language to speak of language can cause problems. If I use the world "selbst" in the meta- denoting the word "self" in the object, we have that "Selbst is a four-symbols word" is true. – Mauro ALLEGRANZA Oct 7 '16 at 9:57
  • the great Raymond Smullyan wrote a long, dense, technical, and expensive treatise on this very topic, "Diagonalization and Self-Reference" -global.oup.com/academic/product/…". – user20153 Oct 7 '16 at 19:23
  • @Mauro ALLEGRANZA : your selbst example works well enough but I don't think it is an example of self-reference. – user20153 Oct 7 '16 at 19:31
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Long comment

The relation of denotation holds between a word of the language and an object of the world "out there".

We use the word (an expression) "apple" in the object-language that denotes an apple in the world. If I say "The apple is red" I'm speaking in the object-langauge and I'm using the word "apple" to speak of the world (the apple on my desk).

But we cannot reverse the relation : the apple out there does not denote the word "apple".

Thus, when speaking about language we use the meta-language to speak of the object-language: now the words of the object-language are the "objects" out there.

We have to use some "trick" to avoid mixing the two languages, like e.g. use German (selbst) for the meta- and English for the object- (self).

A usual trick is to use quotation marks :

"self" is a four-symbol word.

I think that it is not a good practice to conflate the relation of denotation with that of identity.

If we say :

"self" denotes self

it is quite clear that we cannot "reverse" the relation, while :

"self" is self

looks symmetric.

But we have to use meta-language to speak of the object-language and not the other way around.

  • hmmm. if you're using two languages then you don't get self-reference. I believe the trick is to use recursion, fixed points, etc., rather than meta/object languages. – user20153 Oct 7 '16 at 19:29
  • This neither answers nor properly deconstructs the question. In this world with interoperating languages, in what sense can a sentence denote itself, and in what sense can it not do so? I can still say 'the quoted phrase in this comment', and this system does not capture the meaning/joke in that. – jobermark Oct 10 '16 at 3:20
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If I want to say that 'the second complete sentence of Jay's answer to question #39263' denotes itself, it is simply false. Not only is it not a complete sentence, but it is part of the first of the collection of sentences it references, and not the second.

If I just slap 'Let' in front of that false statement, to shove it into the subjunctive, it is no longer false. It is just impossible instead.

If by denotation, you mean applying interpretation to a phrase, then what something denotes is not a matter for you to simply declare. It takes place in a complex negotiated environment in which phrases have denotations by social convention.

So this operation 'Let X denote itself' is simply not something that is within the powers of an individual human being to do, and all arguments proceeding from its use are baseless.

It is not impossible for things to denote themselves in this way. The phrase "'appended behind itself quoted' appended behind itself quoted" does, in fact, denote itself in this sense. Its representation does express its interpretation. But you have to find examples, you can't just declare them. (This is a variant of the famous programming problem called 'funarg'.) So the literal answer to your question, in this case, is 'Yes, they can', but the way you are going about arguing about it is confused.

If you are talking about labeling and not denotation via interpretation then it is simply not true that "'P' denotes P" implies that 'P' is P, where P is taken to be the interpretation of the meaning ascribed to the symbol as a predicate. Because we decided up front that we are not talking about denotation via interpretation, which is what evaluating the symbol P as its meaning does.

Again, I can label a sentence with anything string of symbols, sounds or other sensory triggers, including various things that can be considered that same sentence. So the answer is also 'Yes' in this case. But again the logic in this case does not work the way your arguments seem to assume.

You need some clarity on what exactly you think 'denote' means. It cannot mean both of these things, and there are not that many stable interpretations in between them.

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