THE OBSCURITIES OF A CLASSICAL PARADOX
Imagine that a theorist approaches you and asks you
whether somebody whose height is 150 cm is short or tall.
Suppose you answer that such a person or, to be precise,
such a body is short. Suppose, too, that you further admit that
somebody who is only 1 mm taller than somebody who is short, is
also short. Your interrogator may then keep going and eventually
force you to conclude that somebody whose height is 200 cm or
more is still short.
'E will be ready to point
out that this is a paradox, because you would agree with
'im that somebody who is
2 meters tall is tall, and not short.
Your paradoxical partner of discourse may also employ other
examples. 'E may tell you that somebody who runs from one city
to another will remain far from that city forever, because 'e
started out far away, and being merely 1 dm closer will keep the
runner always 'equally' far away. Or, 'e may have you agree that
one grain of sand is no heap of sand, and that one grain more
will still not make a heap of what is no heap. Nevertheless, you
will be stuck at the end with a gigantic amount of grains of
sand which you have not been allowed at any moment to start
calling "a heap". It is such a 'heap' this kind of paradox
derives its classical name from: sorites.
Crucial in these paradoxes is the inductive statement that P(n) implies
P(n+1) for all n.
P(n) is, then, a propositional function reading something like "the thing
T has the predicate P if it consists of not more than n grains of sand"
or "... has a height of not more than n mm",
and so on.
To defuse the sorites paradox several solutions have been
offered. Some logicians have maintained that the statements are
false or that the general premise is somehow illegitimate and is
neither true nor false in that it does not have a truth value.
Others have given up bivalent logic altogether and have introduced
a many-valued logic to 'solve' the antinomy. Especially when
applying a 'logic of accuracy' it would seem that ordinary logic
could remain applicable to so-called 'vague' concepts like
short and far. In such logic, propositions are not simply
true or false, but it is assumed that there are also intermediate
degrees of truth, and truth-values themselves are identified
with accuracy values. On this view the accuracy of somebody's
being short would be greater to the same degree as 'e would be
shorter. The whole idea hinges very much on the assumption that
there would be only one shortness predicate and that a body has
this one predicate to a greater or lesser degree. Apart from
this assumption, it appears also to defy the fact that one can
be equally sure that two things which are not equally short are
both short. The statements in which one expresses these opinions
may be equally accurate.
Is it necessary for us to take part in all kinds of
artificial resolutions of the sorites paradox? And is it really
an antinomy we all have to suffer from? Or are we just made to
believe that there is something wrong, because we cannot prove
that everything is all right? To find this out we should not
immediately plunge away into logical or mathematical formulas,
but start at the beginning, that is, the phase which precedes
these formulas. One of the first questions is then what it
means that someone says that somebody is short, or that a city
is far away. Theoretically, these meanings must either be
related to those of opposites such as tall and close, or
be entirely dissociated from the meanings of such related expressions.
If terms like short and far are related to their
opposites, they are treated as
catenary notions; if not, then as
some sort of absolute notions. As
catenated predicates shortness
and farness, however, presuppose the existence of a particular
catenary entity, and the meaningful use of the corresponding
predicate expressions presupposes the psychological
availability of such a
catena as a frame of reference.
So there are two options open to the theorist. Firstly, the
predicate mentioned in the original premise is catenated, but
then there is somehow a catenary frame of reference. Or,
secondly, there is no catena involved whatsoever, but then the
predicate is in some sense absolute and not (necessarily)
opposed to what is ordinarily taken to be its opposite. Someone
could indeed say without making use of any particular frame of
reference that a body of 150 cm is 'short', but then it would in
no way be contradictory to conclude later that somebody of
200 cm were 'short' as well, or even both 'short' and 'tall'.
It is only contradictory if we do make use of one and the same
frame of reference in the two cases, and if the predicate
corresponding to 150 cm is a shortness predicate on this view,
whereas the one corresponding to 200 cm is not (because it is
tallness or a predicate medium tall). The schizoid lover of
paradox wants us to meaningfully say that a body of 150 cm is
short (something we can solely do by means of a catenary frame
of reference psychologically available to us) and having done so
'e forbids us to use this internalized information for a while
until we must use it again to conclude that a body of
200 cm would not be short.
The logic of the sorites lover may be sound but 'e draws on a
pre- or extra-logical inconsistence. 'E should either assume
that catenary comparison is possible all the time or that it is
not possible at any time.
In the latter case, the short and far of the premise just do
not mean what they mean in ordinary language.
And if a general, catenary comparison is possible all the time, we do not
depend entirely on a comparison with a smaller degree of shortness or
farness of the particular body or distance concerned.
We can then compare every degree of shortness or farness with what we
would ordinarily call "short" or "far" (just as the theorist asks us to do
at the beginning and at the end).
Obviously, this still leaves us with a wide, fuzzy area between
short and tall, and between far and close, but
that is not what the paradox is about.
The paradox is about things which would be 'short' and 'tall', or 'far' and
'close', simultaneously, while short and tall, or far
and close, are opposite catenated predicates which cannot be
possessed by one and the same thing at one and the same moment.
(To understand this one must clearly distinguish the predicates of a thing
from those of its component parts, as discussed in Wholes:
their whole-, part- and pseudo-attributes.
A thing may have one part which is short and another part which is long,
The supposition that the inductive premise on which the
paradox depends is false is equivalent to the supposition that
there is some n for which P(n) is true and P(n+1) false. It has
been argued that this supposition is untenable, because something
that is far at time t and not far, say, one second later, would
have to have moved a considerable distance. But negation and
opposition, and logics and pragmatics, are mixed up here.
Firstly, to say that something is 'not far' does not yet mean
that it is 'close'.
And secondly, if something has moved over an incremental distance which
cannot even be recognized as a case of coming closer, it does only not
make sense in
practise to speak
about the object's being far first and its not being far one second later.
It does not follow
tho that the object would
not have moved at all, and that it could not have crossed a more or less
theoretical borderline in the meanwhile.
And this borderline between far and not far is in practise
one between far and a
perineutral neither far nor close,
rather than between far and close, or between far and
a neutral neither far nor close. Moreover, if we took a larger unit
of measurement (something that is not a logical issue), there would
be nothing remarkable about P(n) being true and P(n+1) being
false. For example, if n is the number of half meters, S(3) is
true (somebody who is 150 cm tall is short) and S(4) false
(somebody who is 200 cm tall is short).
It has also been argued that there could be no sharp boundary
between shortness and tallness, farness and closeness, and so
on, because empirical reality would exhibit no discontinuity.
This, however, is an odd statement if we realize that it is
precisely the function of the introduction of the shortness- and
closeness-catenas to reduce the number of length predicates
(which is practically unlimited) to two or three. Saving
empirical continuity could then only mean that we would not be
allowed to speak of "short" and "far" at all, but only of the
exact distances involved. Given that this is ultimately not
possible either, one always sacrifices that 'continuity' however
small one would take one's unit of measurement, unless there is
a smallest unit corresponding to separate length predicates
which can be individually recognized or measured. But even in
that case one could not have been permitted to speak of short
or long in the first place.
We have now had enough of the dire confusion of patients
suffering from 'soritis'.
What remains very interesting nevertheless is the question of the
psychological availability of catenary frames of reference, not only
where it concerns
extremities or non-perineutral
polarities, but where it concerns all
catenated predicates, perineutral or not.
What particularly deserves our attention here is the question of what could
determine the boundaries between concatenate predicates, and the question
of how sharp they are in practise.