[FOM] Another easy solution does not work

[Todd Wilson]

On Thu, 19 Sep 2002, Sandy Hodges wrote:
> Consider these:
> 1. There is life on Mars.
> 2.  Socrates on Tuesday said "Sentence 1 is true" and he said nothing
> else that day.
> 3.  What Socrates said on Tuesday is true.
> 4.  Plato on Tuesday said "Sentence 1 is true" and he said nothing else
> that day.
> 5.  What Plato said on Tuesday is true.
> 6.  Sentence 1 is true.
> 
> By the Barwise and Etchemendy Russellian position, if 2 and 4 are true,
> then 3 and 5 are tokens of the same proposition.   But what if their
> truth status is not known?    Surely 6 is a proposition: it is not one
> proposition if there is life on Mars, and a different one if there
> isn't.    If 3 and 5 are also propositions, and not functions of the
> truth status of 2 and 4, then 3 and 5 can't be the same proposition,
> since 3 tells us something about Socrates' Tuesday activities, and 5
> does not.

If (2) and (4) are false, then it could be that the propositional
references in (3) and (5) fail.  This is not something that B/E tried
to model, although one might speculate as to how they would have
addressed the problem.  However, the essence of your example seems to
be captured in the following modification of it, which can easily be
accomodated in the B/E formal system:

1.  Claire has the 2C.
2.  Socrates believes (1).
3.  If Socrates believes (1), then (1) is true.
4.  Plato believes (1).
5.  If Plato believes (1), then (1) is true.
6.  (1) is true.

Now, (3) and (5) are different propositions, regardless of the truth
values of (2) and (4), but if the latter are true, then the former are
equivalent (but still not equal).

> In Yablo's example, the author of each proposition believed it to be
> true that someone after him in line was believing a false proposition,
> but he did not know which false proposition that was (only we know
> that).   So the propositions can't be the same, as each conveys
> information about a different collection of people.

I was under the impression that Yablo's paradox did not involve
statements of belief, but rather of ("naked") truth.  My proposed B/E
Russellian analysis of Yablo was meant to find circularity in the
propositions claimed by Yablo's infinite collection of people on
account of their structural identity, when their propositional content
was separated from the circumstances of their uttering.  One might
object that the circumstances thus elided are crucial to a proper
understanding of the paradox.  As pointed out previously by Andrew
Boucher (12 Sep 2002 20:41:02 +0200), both of the approaches proposed
by B/E can incorporate indexical elements into propositions (the
Austinian more naturally than the Russellian), and it would be
interesting to see whether any reformulations of Yablo's paradox along
these lines lead to any additional insight.  However, you are
suggesting that the paradox can be reformulated in terms of belief.
If you could propose a specific sequence of such propositions, then it
would be possible to analyze your sequence for its paradoxical
qualities under various accounts.

On Sat, 21 Sep 2002, Sandy Hodges wrote ("Re: paradox and circularity"):
> So the analogy is: sentences that refer to each other in loops (but
> without infinite chains) correspond to hypersets.    Loops do not create
> an insurmountable problem for either sets or sentences.    But infinite
> descending chains do for both.

Infinite descending chains do not present a problem for hypersets, at
least as B/E use them for modeling propositions.  The liar proposition

    f = [Fa f]

is a "finite loop", as you say, but, for example, given an infinite
set of people, {a1, a2, a3, ...}, the proposition

    [a1 Bel [a2 Bel [a3 Bel ...]]]

(i.e., that denoted by "a1 believes that a2 believes that ...") is a
legitimate proposition that poses no problems for their development.

> Alan Weir has proposed a sentence token (which I'll call W), which is so
> designed that if any token S denies W, then S will be in the set
> referred to by W, and be paradoxical.   So although we may think W is
> not true, if we say so, our sentence saying so will not be true.   W
> says (very roughly): all tokens that deny me are true.
> 
> It is this quantification over all sentences by W, that will presumably
> need to be blocked.

Quantification was explicitly avoided in the B/E approach, but a
limited form is available using infinitary conjunction and disjuction
operations.  The set of propositions thus con/disjoined, however, must
be just that -- a set -- and not a proper class.  This has the effect
of preventing quantification over all propositions, the class of which
is proper, and so would block Weir's example.  Alternatively, one
could quantify over the *set* of *expressible* propositions, but would
then find (as with Richard's Paradox) that the resulting proposition
was not expressible, and so again the paradox would be blocked.

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh