[FOM] Yablo's Paradox

Volker Halbach Volker.Halbach at uni-konstanz.de
Mon Sep 16 08:26:25 EDT 2002

There is a whole family of paradoxes similar in structure to Yablo's 
paradox. Recently Hannes Leitgeb wrote a paper on this (Studia Logica 68, 
2001, 69-87) on paradoxes of this kind and how they are related. Usually 
these paradoxes don't yield an inconsistency but only an 
omega-inconsistency or they even can only be shown to lack a standard model.

Related paradoxes are: Vann McGee's paradox on omega-inconsistent theories 
(JPL 1985) , Albert Visser's theorem on infinitely descending hierarchies 
(Handbook of Phil Logic), an additional theorem by Hannes in the above 
mentioned paper.

Jeff, Hannes' paper contains also a version of Yablo's paradox along the 
lines you suggest:
>On one approach, which doesn't use diagonalization, you add Yn's as
>primitive sentence letters, and add biconditionals as axioms such that Yn
><=> "all sentences after Yn are not true". It led to an omega-inconsistency,
>which seemed kind of interesting. But I can't find my write-up of this.

In order to exhibit the relation of illfoundedness and these paradoxes and 
in order to generalize them, Hannes, Philip Welch an I worked out a formal 
possible semantics for truth as a *predicate* (as opposed to a sentential 
operator). We try to assign to the elements (worlds) of a frame (just a 
binary relation on the worlds) models of the form (N,S) for the language of 
arithmetic plus a unary truth symbol T.; N is the standard model of 
arithmetic, S the extension of the truth predicate. Consider the frame (in 
the sense of modal logic) with the natural numbers as elements where every 
number can see exactly its successor. This frame is converse illfounded. 
Can we assign to every number (=world) n a model (N,S_n) such that S_n is 
the set of true sentences of the model (N,S_{n+1})? The above mentioned 
results can be used to refute the existence of such an assignment of models 
to elements of the frame. Apropos converse illfoundedness: We can use 
Loeb's theorem as well in order to refute the existence of such an 
assignment (although the accessibility relation is not transitive, but we 
can define its transitive closure in the language).  The liar paradox 
becomes in this setting the theorem that there is no assignment for the 
frame with a single world that sees itself. Of course this frame is 
converse illfounded as well. In some sense we succeeded that in showing 
that a large number of paradoxes  correspond to the fact that we cannot 
find such assignments of models to certain converse illfounded frames. 
Hannes even proved a completeness theorem to the effect that a theory of 
truth (with certain restrictions) is inconsistent in omega-logic iff there 
is no possible worlds semantics for it in order sense.  Thus all paradoxes 
seems to flow from Loeb's paradox.

If the frame is converse wellfounded, in contrast, it is obvious how to 
define such an assigments of models (N,S) to the elements of the frame.
There are, however, converse illfounded frames that admit such an 
assignment of models, but they are much more complex than the simple 
infinitely ascending chain of worlds - pace Loeb's theorem.

Volker Halbach
Universitaet Konstanz
Fachbereich Philosophie
Postfach D21
78457 Konstanz
Office phone: 07531 88 3524
Fax: 07531 88 4121
Home phone: 07732 970863

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