[FOM] Fuzzy or Extensible? On ill-defined Concepts
laureanoluna at yahoo.es
Tue Nov 20 13:34:50 EST 2007
Let D be:
'the least indefinable natural number'
D defines no number.
It is often argued that 'definable' is not
well-defined and that it is so because in the absence
of reference to any particular and well-defined
language it is not clear what counts as a definition;
'define' and its derivatives are, like many other
terms in natural language, vague or fuzzy: the
concepts/predicates associated to words like 'heap',
'tall', 'blue', etc. are all fuzzy, in the sense that
their extensions are fuzzy sets.
Therefore, it is often argued that the ill-definedness
involved in definability paradoxes, such as Berry's
and Richard's, is just fuzziness.
I tend to dissent.
I think that fuzziness is an estimative phenomenon. It
stems from our impossibility of producing clear-cut
conceptual classifications for our sense data. For
example, our estimative capacity hesitates when
confronted with questions such as:
'how many grains make up a heap of sand?'
'what color are the eyes of Charlize Theron?'
I suggest that such estimative hesitations are not
essential to the definability paradoxes. Perhaps even
if we could free our language from all estimative
doubts concerning what counts as a definition of a
natural number, D would still define no number.
Let's now assume that 'define' has to be typed into a
hierarchy of logical levels. If this is so, then D is
ill-defined just because 'indefinable' in D bears no
level mark. Take instead DD:
'the least n-indefinable natural number'
DD is of type higher than n and appears to be
prevented by no paradox from defining a natural.
We can say that a concept/predicate P is well-defined
iff for all x
Px v -Px
Now there are at last two ways in which a
concept/predicate can be ill-defined: fuzziness and
While fuzziness seems to be an estimative phenomenon
that ultimately depends upon our incapacity to impose
clear-cut conceptual divisions on our sense data,
'typedness' is a phenomenon of a quite different kind,
it's a logical phenomenon.
Let me introduce now a notion of concept extensibility
closely related to the idea of Shapiro and Wright in
'All Things Indefinitely Extensible' (in 'Absolute
Generality' OUP, 2006). The existence of a hierarchy
of levels of definition can be interpreted as the
'extensibility' of the concept 'definition of a
natural number' with respect to the concept 'set', in
the sense that for any set of (n-)definitions of
naturals there is a (m-)definition (with m higher than
n) of a natural not in the set.
The consequence seems to be that there is no set of
all definitions of naturals. If such a set existed,
since each definition defines just one number, it
seems that there would also be a map from those
definitions to the defined numbers (if you feel
inclined to object that definitions are not set
theoretic objects, take them as urelements, Im
obviously not reasoning inside the universe of pure
sets); if so, by Replacement, the set of all definable
natural numbers would also exist; and then there would
also exist the least natural number not in that set.
The concept 'set' is itself extensible with respect to
the concept 'set', in the sense that for every set S
of sets there is a set not in S (e.g. the diagonal set
of S). The consequence is that there is no set of all
sets. Properties can fail to define sets not only
because of fuzziness but also because they are so
typed as to be extensible with respect to the concept
Consider the set of all strings of letters of the
English alphabet. We cannot separate from it the set
of all English definitions of naturals, and I guess we
could not do it even after having purified English
from each pertinent fuzziness.
So, I suggest that fuzziness might be inessential to
definability paradoxes, and that 'English definition
of a natural number' is perhaps not essentially worse
defined than 'set'. Both could be just extensible
And if this is so, the impossibility for a collection
to be a set could depend on logical conditions other
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