MIT undergraduate Michael Ernst pointed out this problem to philosopher George Boolos during a course on paradox and infinity.
An expression is a finite sequence of symbols. When expression β is appended to expression α, the symbols of α are written in order and then followed immediately by those of β. So, for example, ‘b’ appended to ‘a’ is the two-symbol expression ‘ab’.
The quotation of α is the expression we get when expression α is enclosed in a pair of quotation marks. The quotation of 'Boston' is ''Boston''. The reverse of quotation is denotation: The denotation of ''Boston'' is 'Boston'.
Who could object to any of this? But now consider the nonsense string κ:
b' appended to 'a
And another string, λ:
ab
Let N(α) stand for the number of letters of the English alphabet that alphabetically precede the first letter of expression α. Then N(κ) = 1 (because that string begins with the letter b) and N(λ) = 0 (because that string begins with the letter a).
And now consider μ:
'b' appended to 'a'
μ is the quotation of κ, so the denotation of μ is κ. But look again at μ: By our own definition,
'b' appended to 'a'
is 'ab', and the denotation of μ is λ as we’ve defined it above. So now we find ourselves asserting that 0 = N(λ) = N(the denotation of μ) = N(κ) = 1.
“What has gone wrong?” Boolos writes. “Obviously, the non-identical κ and λ cannot both be the denotation of μ, and unless they are, our demonstration that 0 = 1 fails. How did we conclude that they are identical?”
When Boolos sent Ernst’s observation to Willard Van Orman Quine at Harvard, Quine wrote back, “Dear George, Thanks for Ernst’s paradox. I am delighted with it. But I find I am unable to cope with it, even when I have stopped laughing. Yours, Van.” Ernst himself suggested that perhaps we need two different kinds of quotation marks, syntactic and semantic. See Boolos’ Logic, Logic, and Logic (1998) for his own thoughts on the matter.
(Thanks, Dan.)
