[FOM] Another easy solution does not work
heck at fas.harvard.edu
Wed Sep 11 00:12:16 EDT 2002
The suggestion that we should simply "forbid liars" as a way of solving
the liar paradox has a long history. One can think of Tarski's response
as an extreme version of this strategy. Hodges's examples would, of
course, be regarded as out of bounds by Tarski. But, as Kripke pointed
out some time ago, that's a problem for Tarski: One can easily construct
very natural examples in which self-reference of various sorts is
essential to one's intuitive understanding of what is said.
Here's Kripke's now famous example. Suppose Dean says:
(1) Everything Nixon says about Watergate is false.
And Nixon says:
(2) Everything Dean says about Watergate is true.
It may well be the intention of each of them to include such utterances
in the scope of their remarks. If Dean had said that everything Haldeman
said about Watergate was true, perhaps Nixon would indeed want to
endorse that claim. We have, in most cases, no trouble assigning
truth-values to these utterances. If Nixon once said that the Watergate
break-in was a bad idea, in retrospect, then that was true, and (1) is
false, whatever else Dean may have said. Hence (2) also would be false,
in the imagined circumstance. But if, as a matter of empirical fact, all
of Nixon's /other/ Watergate-utterances have been false while all of
Dean's other Watergate-utterances have been true, then (2) is true iff
(1) is true iff (2) is false, and we have a paradox. (If you think about
it, under these assumptions, (1) and (2) are, in effect, a version of
the postcard paradox.) Tarski's reply, that (1) and (2) are ill-formed
or whatever, simply doesn't seem plausible in this case.
As Kripke also points out, one can get more plausible sorts of examples
using quantifiers like "most".
Once you get the idea of how to construct these examples, it is a fairly
trivial matter to construct cases in which a sentence (or pair of
sentences) is paradoxical if and only if some arbitrary mathematical
sentence A is true. If you take A to be, say, CH, then you get a case in
which the question whether a sentence is paradoxical is equivalent to CH.
Another example worth noting here is one due to Steve Yablo that
purports to be, and I think is, an example of a semantic paradox in
which no self-reference is involved. Here's the example. Consider an
infinitely long line of people. Each person in the line is to utter one
sentence. As it happens, each person in the line says: Everything the
people behind me will say will be false. No one's remark includes his
own remark in its intended scope. And yet, we get a paradox. I'll leave
the informal argument as an exercise, as well as the formalization of
the example, which is actually a nice exercise in diagonalization.
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