[FOM] Another easy solution does not work

Richard Heck 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.

Richard Heck

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