[FOM] Yablo's Paradox

[Richard Heck]

There is indeed much controversy about whether Yablo's paradox involves 
self-reference. There was a good deal of debate about this matter in 
Analysis shortly after he published it there. Perhaps it is worth 
looking at one classical (in the sense of "standard") analysis of the 
paradox. (So far as I know, this kind of analysis has not appeared in 
print. If it has, I'd be happy to acknowledge that fact. The analysis, 
though, is independently mine.)

We can construct the needed sentences like this. Consider the formula:
    (x>y).~Sat(z,x).
Then by diagonalization on "z", there is a formula S(y) such that:
    Q |- S(y) <--> (x>y).~Sat(*S(y)*,x).
This yields Yablo's paradox, in effect. S(n) means: "S(y)" is not 
satisfied by any natural number greater than n. If we apply the Sat-schema:
    A(y) <--> Sat(*A(x)*,y),
we get:
    Q +Sat |- S(y) <--> (x>y).~S(y),
which is Yablo's paradox in its original form. (Obviously, one could 
pull the same trick using "true" and a substitution function, but that 
seems just to make the argument to follow messier, without any real 
gain, so let me do it this way.)

Using Sat, we can now derive a contradiction.

First we prove:
(1)     Q |- S(n) --> S(n')
   S(n) <--> (x>n).~Sat(*S(y)*,x)
   S(n') <--> (x>n').~Sat(*S(y)*,x)
But certainly:
    (x>n).~Sat(*S(y)*,x) --> (x>n').~Sat(*S(y)*,x).
QED

Now we need the Sat-schema:
(2)   Q + Sat |- (n).~S(n)
   S(n) <--> (x>n).~Sat(*S(y)*,x)
     <--> (x>n).~S(x), by the Sat schema
     --> ~S(n')
     --> ~S(n), by (1)
So ~S(n). Generalize.

On the other hand:
(3)   Q + Sat |-  (n).S(n)
   ~S(n) <--> (Ex>n).Sat(*S(y)*,x)
     <--> (Ex>n).S(x), by the Sat-schema
     --> (Ex).S(x),
which contradicts (2), so S(n). Generalize.

Obviously, (2) contradicts (3), in rather dramatic fashion.

Careful examination of these arguments makes it very hard for me to see 
where there is supposed to be self-reference. Of course, S(m) talks 
directly about the truth-values of S(n), for n>m. S(n) does not talk 
directly about the truth-values of any of the S(m), for m<=n. So the 
suggestion must be that it does so indirectly. But, so far as I can see, 
all we have are implications about the truth-values of those other 
sentences, not quantification over them. This fact can be brought out if 
we change the proof slightly, as follows. Let SatR(n) be the formula:
    (x>n)(S(x) --> Sat(*S(y)*,x)).
And let SatL*(n) be the formula:
    (x>n)(Sat(*S(y)*,x) --> S(x)).
Then examination of the proofs of (2) and (3) above show that:
(2')   Q |- S(n') & SatR(n) --> ~S(n)
(3')   Q |- SatL(n) --> S(n)
Of course, it follows from (1) and (2') that:
   Q |- SatR(n) --> ~S(n)
and so from (3') that:
   Q |- ~(SatL(n) & SatR(n)).
It is (2') that expresses the way in which S(n') has implications about 
the truth-value of S(n). But of course it only does so in so far as 
SatR(n) holds. It does not have such implications by itself and there 
is, as I said, no sense in which S(n') talks about the truth-value of 
S(n) directly. If we replace 'Sat' with 'contains an odd number of 
symbols', the analouge of (2') does not hold. (Well, it need not hold. 
Maybe it does, if you cook up the right sort of Goedel numbering. Who 
knows.) So S(n') "speaks about" the truth-value of S(n) only in the 
sense that it implies things about it in the presence of SatR(n'). But 
it does not do so on the basis of its form or, it seems to me, its 
content. If S(n) is self-referential, then, it is so only in a sense in 
which it is not if "Sat" is replaced by "contains an odd number of 
symbols". That is very much not the sense in which the Liar is 
self-referential: The sentence "I contain an odd number of symbols" is 
every bit as self-referential as the Liar. And the pair:
    (4)    (5) contains an odd number of symbols.
    (5)    (4) contains an even number of symbols.
is also every bit as self-referential as the sentences in the postcard 
paradox.

I am not terribly familiar with the work Barwise et al. did on this 
subject, but I am going to conjecture that the reason it seems, on their 
treatment, that there is self-reference here is because their 
situational notion of proposition, like the more familiar possible world 
notion, takes in facts about implication. That is inherent in such 
notions of proposition. A feature or a bug, depending upon your other views.

Richard Heck