Use and Mention
Many logical fallacies arise due to the confusion of use and mention,
even though the topic is pretty simple. It's a good distinction to master. It shows up
in programming, too, and by the way, plays a central role in Gödel's Incompleteness
Theorem due to the fact that it allows the construction of a particular kind of self-referential
statement.
What's the Difference?
Go through the following sentences, making sure you understand them. (Some sentences
were borrowed from Smullyan's Satan, Cantor, and Infinity.)
- Ice is frozen water.
Ice has three letters. FAIL
- "Ice" has three letters.
- This sentence has five words.
- "This sentence" has two words.
- Alice is reading Hofstadter's book.
- Alice is reading "Hofstadter's book".
- It takes longer to read Hofstadter's book than to read "Hofstadter's book".
The uses are unquoted and the mentions are quoted.
Here's another example, inspired by a question asked by Jimmy Kimmel. One of the following questions is a question about branding, and the other one is useful to show people that they might not be as informed as they thought they might be. Which is which?
- Which do you like better, Obamacare or the Affordable Care Act?
- Which do you like better, "Obamacare" or "The Affordable Care Act"?
Quining
Now let's define the quine of a phrase as the phrase followed by its quotation,
i.e., a use followed by its mention. So the quine of the word "woot" is:
Look at these two sentences. The second is... self-referential!
- Alice is reading "Alice is reading".
- Alice is reading the quine of "Alice is reading the quine of".
Wow. We got self-reference without the use of an indexical (such as
the word "this"). That is, the following sentences are self-referential,
but only because they have an indexical:
- (#1) Alice is reading this sentence.
- (#2) Alice is reading sentence #2.
- (#3) This sentence is false.
Sometimes indexicals are self-referential, but sometimes they are not.
- She said "he is an idiot" but said this sentence was only partially true.
See how the indexical doesn't refer to the whole sentence but rather to a part of it? The
quining trick is stronger in a way; there is no ambiguity.
A famous quine:
- Is false the quine of "Is false the quine of".
And another:
You can also do quining by preceding with a quotation, as is done in Gödel,
Escher, Bach. That makes some quines sound a little better in English. From this book: (heh,
I said "this")
- "yields falsehood when preceded by its quotation" yields falsehood when preceded by
its quotation.
- "yields falsehood when quined" yields falsehood when quined.
- "yields a tortoise's love song when quined" yields a tortoise's love song when quined.
Oh, I got one:
- "is not a theorem when quined" is not a theorem when quined.
Woah! If we can just find a way to represent the notion of quining within TNT, we can
create a statement expressing its own non-theoremhood, and that would show TNT to be
either inconsistent or incomplete! Wait.... I think someone beat me to that idea.... What
was his name again?
Use-Mention in Programming
Here is a Python script:
print(3 + 4)
print("3 + 4")
The first line prints a line with one character: 7 and the second
prints the five characters
3
+
4. See the use and mention here?