Vagueness Challenging the Paradigm

: Sorites Paradox

In the last section we considered a cluster of problems bearing on the issue of whether the orthodox system of logic is a correct account of the fragment of reasoning it purports to represent. In this section, we continue in a philosophical vein by considering the serious problems raised for logical theory by the fact that most of the descriptive expressions of natural language are vague. By calling them vague I do not imply that their meaning or the conditions for their correct use are poorly specified or somehow deficient, but that there is a more or less imprecise borderline between the cases in which a given description applies and those in which it does not. An example or two will help to make the problem clear.

For the first example, consider a long series of coloured patches shading very gradually from red, through orange, to yellow. We can imagine the difference between each and its neighbours to be so slight as to be imperceptible to the casual glance. Indeed, there is no need to imagine: the thing itself is before your very eyes (picture). There are 256 patches in the series. Now patch number 1 is red. Moreover, of any two patches which are not directly discriminable in respect of colour, if one is red then so is the other. By repeated applications of IMPE to these premises, we are led to the conclusion that patch number 256 is red. This conclusion is false because patch 256 is yellow, not red. There must be something wrong with the argument.

The second example is that of chickens and eggs. Suppose that Darwin was right about evolution, that it happens not in large mutations but by a gradual accumulation of small ones. Any animal and its immediate offspring then will belong to the same species by any reasonable tests: they have similar structure; they are very close genetically; they are cross-fertile. Now any animal of the same species as a chicken is a chicken. So chickens can only come from chicken eggs, which can only be laid by chickens. Consequently, there have always been chickens, even 200 million years ago, before birds evolved.

The problem in each of these cases is that the relevant predicate, '... is red' or '... is a chook', is insensitive to small changes in a respect such as shade or genetic composition but is sensitive to sufficiently large changes in the same respect. Yet the large changes are made up of accumulated small ones. The logic of the argument is extremely simple, consisting of a large number of conditional premises:

If patch 1 is red then patch 2 is red
If patch 2 is red then patch 3 is red
......
If patch 255 is red then patch 256 is red

together with an initial antecedent

Patch 1 is red

from which the false conclusion follows by several uses of IMPE (a mere 255 in this case). In an alternative form of the argument, the conditional premises could all be obtained from a generalisation

For any number n, if patch n is red, patch n+1 is red

This would shorten the argument form a lot, but require appeal to another logical principle saying that if the generalisation is true then so is each of its instances.

There are only three possible ways out of the difficulty:

1. Deny that the problem is well posed. That is, hold either that no logic is applicable to vague expressions or that such logic as does apply is not coherently formalisable.
2. Maintain that the argument is invalid. That is, hold that IMPE in the form it takes here is not a valid principle but a fallacy.
3. Hold that even if the argument is legitimately set up and valid it is not damaging because one of its premises is false. That is, claim that there is a last red patch in the colour series and a first chicken in the evolutionary series.

What might look like a fourth option, to accept the conclusion that every coloured object is red, that we are all chickens, etc., really collapses back into option (a) because there are other rules of our language which dictate that the predicate '... is red' is to be witheld from many things—lemons, for instance—so that the putative fourth option involves holding the rules of our language to be inconsistent. But if they are inconsistent, then they are not (coherently) formalisable.

Some philosophers try to evade the problem by maintaining not that it is impossible to apply logic to vague discourse but that there is no need to do so. Respectable parts of language, such as science, are held to be capable of getting by without vagueness, so logicians are not required to worry about such things. My response to this is threefold. Firstly, it has not been established that much of natural discourse can be freed from vagueness. Even scientists talk about test tubes, water and the like; and this talk is vague, for how long and thin must a piece of glassware be, or how polluted may a liquid be, before it no longer counts as a test tube or as water? Real science is full of vague expressions. Moreover, even if theoretical physics were freed from vagueness, why should we believe that the same can be done for such rough and "empirical" sciences as geology or meteorology, not to mention linguistics or psychology? Secondly, the mere possibility of reformulating discourse without vagueness would not be enough. There has to be some reason why such precisified language would be preferable. In many cases a sentence like 'Fetch me a glass of water' is completely appropriate, while the alternative of ordering an exact amount of an exactly specified liquid would not serve the purpose nearly so well. Saving logicians from having to think is not a good reason to prefer a mutilated language to the real thing. Thirdly, even if there were some perfectly splendid artificial language in which all predicates were precise and on which logicians ought to concentrate, that would not solve, but merely evade, the present problem. The difficulty raised by the vagueness of words like 'red', 'chicken', 'middle-aged' and 'water' still afflicts a legitimate part of the subject matter of logical theory, and changing the subject is no solution to it.

Option (a) is unattractive. It involves holding that there is no coherent way of reasoning in most of ordinary language. This is in evident conflict with the data—with the fact that we do argue, theorise and criticise using such language, and that we apply criteria of rationality to each other's theorising etc. in a reasonably stable way. Some formal logicians may be so keen to preserve their abstract theory from confrontation with uncomfortable evidence that they are prepared thus to consign most of what people actually do when reasoning to the category of the utterly uninvestigable. Such an attitude shows a lack of respect for the ordinary. The rules in force governing assertibility in natural language may not be exactly those defined in these notes, or even much like them, but they are rules nonetheless, and can be formalised too. In any case, there are some of the familiar formal rules such as ANDE and ORI which have not been shown to lead to any problems and whose applicability in cases of vagueness is straightforward. My objection to option (a) is that it retreats too far, securing logical theory from refutation only by cutting it off from application to real life.

Option (b) is also not an easy way out. The only rule of inference needed for the problematic derivations is apparently IMPE; and surely no rule is more deeply embedded in our natural understanding of connectives. In an important paper on the subject Michael Dummett puts the point thus:

[To abandon IMPE] seems a desperate remedy, for the validity of this rule of inference seems absolutely constitutive of the meaning ... of 'if'.

Any supporter of option (b) must therefore tread a very fine line indeed, providing an alternative logic in which the deduction equivalence and IMPE are somehow retained as 'constitutive of the meaning of 'if' while the above derivations which apparently appeal only to IMPE are blocked. I believe that line can be trodden SlaneyV, though this belief puts me in a minority of approximately one. Briefly, it can be done by appealing to the distinction already drawn in the last section for the purposes of relevant logic. There is a difference between merely collecting assumptions into a set (symbolised there and here by listing them with commas between) and actually applying them to one another as when the conditional A IMP B and its antecedent A interact to produce B. This notion may again be symbolised by using semicolons in place of commas. With such a distinction in place we may keep the deduction equivalence in the appropriate form

X; A ⊢ B   iff   X ⊢ A IMP B

and continue to use both IMPE and IMPI, as motivated by this equivalence, with semicolons rather than commas on the left. This would seem to meet Dummett's point that IMPE should be preserved as constitutive of the meaning of the conditional. The rules for introducing and eliminating connectives are exactly as they were for relevant logic. That is, they are just the standard classical ones with due attention paid to the difference between commas and semicolons. The resultant "fuzzy" logic F* is superficially like relevant logic, but the two are really deeply different and their motivations are entirely independent of each other. For present purposes we do not mind augmentation of assumptions, even in semicolon contexts. So there is no objection in this section to the inference form

X   ⊢   A

X; Y   ⊢   A

On the other hand, where relevant logic was pretty casual about repetitions of assumptions, here we are concerned precisely to block arguments which work by confusing many appeals to the premises with few appeals to them. So the rule which allows repeated assumptions to be collapsed together is rejected in favour of one which allows vacuous discharge. Semantically the idea is to count not just truth and falsehood but degrees to which the truth might be stretched in order to accommodate vague propositions (e.g. that this patch is red or that that animal is a chook). Then to get  p,q  to hold it is necessary to stretch things just far enough to get each of p and q to hold. To get  p;q  to hold, however, we have to stretch far enough for p and far enough for q put together. We can think of "putting together" as something like adding together. And of course from this point of view we do not expect X;X to be the same thing as X, since x + x is not generally equal to x.

The upshot of this suggested revision to formal logic for purposes of reasoning where vagueness is an issue is not really that IMPE is rejected outright but that it gets refined. The valid sequent corresponding most closely to it is

p IMP q; p   ⊢   q

whereas the invalid sequent needed for the "slippery slope" arguments is

p IMP q, p   ⊢   q

These two are easily confused, and my present tentative suggestion (see here for details) is that by distinguishing carefully between them we may find a way to keep all the intuitions worth keeping while blocking the undesirable argument forms.

Most philosophers would disagree. They would point out the one common response which is not captured by reformulating logic as above: that "you have to draw the line somewhere". That is, the usual attempted solution to the problems of vagueness is to take option (c), holding that one of the conditional premises is false. That is, there is some number n such that it is false that if patch n is red then patch n + 1 is red. In symbols

NOT(R(n) IMP R(n + 1))

By truth functional logic, this is equivalent to

R(n) AND NOTR(n + 1)

The difficulty faced by option (c) is to give an account of how such a thing could be. Surely, after all, if you are committed to describing something as red then you are committed to describing anything indistinguishable from it as red as well. The meaning of the word 'red' is fixed only by the use we make of it, so there seems to be no way its applicability could turn on distinctions too fine to be drawn. The problem is in some ways even sharper in the case of the predicate '... looks red' rather than '... is red': how could there be two objects such that (i) no difference of colour between them can be detected just by looking, but (ii) one of them looks red and the other does not? The chickens and eggs raise a related form of the same difficulty. There are reasonably well established criteria for sameness of species, though since individuals are rarely exact clones these criteria are subject to certain degrees of imprecision. Nonetheless, any two animals one of which is the parent of the other will fall well within the area where the criteria are satisfied. Yet it is to be held that one such animal is a chicken and the other is not. Again it is far from obvious that this description of the situation is coherent.

One way of making it look more coherent is to be a little more subtle about the interpretation of vague claims. It is part of the meaning of 'red' that no precise boundary to its applicability is drawn—that there is no uniquely determined cutoff point for it—but logic demands that everything should behave as if such a boundary had been drawn. Well if we wanted to do so we could make the borderline precise by an arbitrary decision. This could be done in many ways, as there is no one place where an exact boundary is dictated, but the choice of ways to do it should not lead us into denying that it can be done. In some cases, in legal definitions of adulthood for instance, we are forced to draw sharp lines where nature provides none. We find that these artificial definitions do not correspond very well with the vague notions, and rightly so since a precise concept cannot be an accurate analysis of an imprecise one, but we also find them acceptable, and even useful, in some contexts. Now given that a vague predicate can be sharpened in many different ways, why not take the truths involving it to be just whatever comes out true on every acceptable sharpening? The falsehoods are those statements which are false on every acceptable sharpening. This leaves room for some statements like 'Patch number 120 is red' to be indeterminate because true on some sharpenings and false on others. It will therefore lack a (unique) truth value.

Some such suggestion is the most popular approach to the problems raised by vagueness. It enables us to keep ordinary logic intact while allowing for fuzziness on the semantic level by using truth tables in an extraordinary way. It does seem to capture rather well some of the intuitions we have in this area. What I see as the most pressing difficulty for it is that it results in the wrong truth values in some crucial cases. Most clearly, it implies that the statement

There is a last red patch in the colour series

is true (different patches make it true on different sharpenings). Yet this seems to me to be incorrect for the word 'red' as used in English. However, substantial philosophical positions cannot be defeated by one-line arguments, so the debate is still open.

The philosophical issues raised in this section are difficult, and they open deep issues both in logic and in the philosophy of language. I cannot pretend to have solved them in these few pages. What I have tried to do is to indicate the nature of the paradox, outline some of the difficulties facing the various potential solutions and sound cautiously hopeful about one of them. Most experts would disagree with my views both on vagueness and on the prospects for "deviant" logic. Your task is to think about the problem yourself, not learn to repeat my mistakes.

Notes and further reading

The logic F* was introduced by Ono and Komori Ono under the name DBCK. BCK is the fragment of F* in which implication is the only connective. "Affine logic" results by adding other connectives to BCK, but lacks the principle of distribution relating conjunction and disjunction. DBCK is "Distributive BCK" which supplies that inference form and its cognates while leaving the basis in BCK and some of the other fragments of affine logic undisturbed. As observed in these notes, it may be obtained very naturally as a logic in the same family as relevant logic.

IMPORTANT NOTE:   It is necessary to emphasise that F* is not the result of substituting the continuous unit interval for the two boolean values 0 and 1. It cannot be dismissed by pointing out the shortcomings of that idea.

There is a large and complex philosophical literature on vagueness. As a suitable entry point, see the Stanford Encyclopedia article by Dominic Hyde on the Sorites Paradox, and the one on Vagueness by Roy Sorensen, and follow up references from there. For reflections on the problem that are deeper than most, see Crispin Wright's collection of papers Wright.

Fuzzy logic and the associated theory of fuzzy sets Zadeh have been studied and used in computing, especially in artificial intelligence, as a basis for knowledge representation and information retrieval, and also in control. For a comprehensive account, see the book by George Klir and Bo Yuan Klir. The logics generally used in that field are a little different from F*, and tend to be less natural from a proof-theoretic perspective, so we do not consider them in any detail here.