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 instance, 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 entirely depend 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). Of course, 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', at the same time, 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.

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 can even not 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.