Deconstruction

Stephen R.L. Clark has written a scholarly and fascinating investigation of the (f)laws of logic. He describes this examination, in postmodernist terms, as a ‘deconstruction’, but characterises it as a pre-modern enterprise. Clark's aim is ‘to show that the laws cannot be read off from experience, nor are they formal conditions for reasonable speech’ (pp. 25–6).Footnote ¹ As to the latter view, Aristotle argued, in Metaphysics, Γ 1006a10 ff., that subscription to the Law of Non-Contradiction was a sine qua non of being able to engage in sensible talk, so, in arguing for the opposite, Clark is taking on a formidable opponent. But the view that the laws of logic are read off from experience has a considerably less distinguished pedigree. However, it is not really this view that Clark attacks but rather the more interesting and plausible claim that the laws in question (Identity, Non-Contradiction, Excluded Middle and Negation) cannot fail to accord with experience. ‘Reality contains no contradictions’, says Clark, ‘but we have no more right to suppose that the world of our experience will contain no contradictions than to suppose that a long-running fictional saga contains no contradictions’ (p. 51). Clark takes his deconstruction to have a bearing on a range of topics in the philosophy of logic, including the treatment by modern logicians of vagueness, where one criterion of success is a plausible handling of the Sorites paradox. I shall argue, against Clark, that there is no mismatch between experience and the basic laws of logic, and that accommodating vagueness requires not a deconstruction of logic but a proper appreciation of its range of applicability.

Change and Difference

We know from experience that change and difference occur both in time and space. Clark gives a general argument to show that the laws of Non-Contradiction (NC) and Excluded Middle (EM) together imply that change and difference are impossible:

One obvious response to this argument is that there is nothing between the stretches L1–L2 and L2–L3, hence to attribute either A-ness or not-A-ness to x at L2 is to attribute something to nothing. Clark is sceptical of this response. He writes

Clark here displays a fundamental misunderstanding of the nature of a point. If something physical is common to two things, then it is a part of each. But a part of an extended thing must itself have extension, and that is just what points do not have, so there can be no question of there being a point common to two things. If two disks overlap (think, perhaps, of a Venn diagram printed on the page of a logic text), then there is a lens common to both, and that lens may be divided into any number of parts each common to both disks. But points are not parts. Consider the incident, recounted by Clark himself, involving a son of his who filled a glass jar with marbles (p. 37, footnote 28). Presumably Clark would not wish to say that the marbles, even when crammed into the jar, are indissolubly united as a single object. In geometry, of course, contiguous circles have a point in common, but that is because the circumference of a geometrical circle is itself extended only in length (it has no width). The circles (x − 3)² + (y − 3)² = 9 and (x + 3)² + (y − 3)² = 9 have a point (0, 3) in common with each other and with the Y-axis, but two disks separated by a barrier (however thin) have no parts in common either with the barrier or with each other.

The answer, then, to Clark's question about whether object x is A at L2 is that the question is as misconceived as ‘What is now the time at the North Pole?’. If someone — a child, say — tells you that it is now 5 o'clock at the North Pole, you may choose to take your pedagogical duties seriously and explain about time zones converging on a point, or you may, more impatiently, reply ‘No it's not’. But that latter reply is a way of dismissing the suggestion; you are not implying that it is now some time, or several times, other than 5 o'clock at the North Pole. The Law of Non-Contradiction is not threatened.

Periods, spatial or temporal, do not consist of points. Clark accepts this, but goes on to draw erroneous conclusions. He writes ‘there are no truths at points at all’. That is incorrect. There are truths about points at points. The sentence ‘That chameleon was purple at L2’ is grammatical but this does not entail that it expresses a true or a false proposition (unless by ‘false’ we mean ‘anything [e.g. ‘nonsensical’] other than true’). However ‘That chameleon is exactly three months old’ may be true at L2. Clark writes ‘our reality is ineradicably continuous, and there are therefore no abrupt changes of the kind that language and logic might lead us to suppose’. But if that chameleon was indeed exactly three months old at L2 then it was not exactly three months old at any point thereafter; an abrupt change occurred. Clark's diet of examples is unbalanced. He may be right that there is no particular moment (not even an unknown one) when night falls, but, for example, there is an abrupt transition between a vehicle being at rest and its being in motion. The suggestion, using Clark's metaphor (p. 27), that the state of rest always remains as a seed within the motion and that the state of motion always remains as a seed within the body at rest, sounds rather nice, but has no substance.

The Sorites

Clark takes his claim about an ineradicably continuous reality to undermine some attempted solutions to the Sorites paradox. He criticises Epistemicism on the grounds that it postulates a point (albeit one in principle unknowable to us) at which there is abrupt change from A-ness to not-A-ness. His criticism of Supervaluationism is more difficult to describe, for Clark's understanding of that theory seems awry. He takes the theory to advocate the replacement of vague concepts apt for the characterisation of an essentially continuous world by ‘concepts exact enough to be subject, supposedly, to the laws of logic’ (p. 30). But the supervaluationist's ‘sharp’ predicates divide a manifold, artificially, but entirely legitimately, and are introduced not as replacements for ordinary vague expressions, but as part of an apparatus supporting a non-classical semantics (in which, for example, ‘That patch of colour is red’ [said while pointing to a patch painted with a mixture of 3 parts primary red to one part primary yellow] is designated as neither true nor false). A charge, consistent with his own position, that Clark could properly make against the supervaluationist is that the proposed trifurcation into the true, the neither true nor false and the false succeeds only in exchanging one precise borderline for two.

Clark's own approach to the Sorites, which rests on his thesis of the ubiquity of continuity, does not get to the heart of the problem. In any version of the Sorites paradox, there is an ordered series of objects C₁, C₂, … . C_n, the first clearly F, the last clearly not-F, and such that an observer relying only on his or her unaided senses cannot distinguish any member of the series from its immediate neighbours in terms of their F-ness. An observer will judge C₁ to be F, then judge C₂ to be F, but the observer cannot continue indefinitely with these ‘F’-judgments for, recall, the last member of the series is judged not-F. Therefore, contra Clark, at some point along the series, there must be an abrupt judgment-switch, be that from the verdict ‘Yes, F’ on one sample to ‘Not F’ on the next, or from ‘Yes, F’ on the first to ‘Er, I'm not quite sure what to say about that one’ on the next, or some other such change of judgment between the two samples. The puzzling thing about the Sorites — and this is not addressed by Clark — is that such abrupt judgment-switches do occur, yet the difference between one sample and its neighbour is below the threshold of human perceptual discrimination.

In a Sorites experiment, a subject (let us call her ‘Ann’) has to judge of each member of a series whether or not it appears F to her (e.g. whether it looks red to her, seems to her to be wealthy, is in her judgment a heap … .). So long as the first member of the series appears to be F to Ann and each member of the series is indistinguishable to her from its neighbour in terms of its F-ness then, given premises of the form if an object in the series appears to Ann to be F, so does its neighbour, we can apparently prove logically that the last object appears to Ann to be F — which contradicts the empirical fact that it does not. That the only logical principle involved in the proof is modus ponens seems to give succour to Clark's contention that fundamental principles of logic are at odds with everyday experience.Footnote ²

The case just described bears the hallmark of paradox, but the false conclusion (that the last object in the series appears to Ann to be F) is arrived at not via the commission of some egregious though subtle error of logic but from a failure to see that at least one premise in the argument is false (and to see why it is). One can ascertain empirically (by running an experiment)Footnote ³ that there is at least one object in the series such that Ann rates it F but does not say the same of its neighbour (e.g. she might say of the latter ‘That's definitely not F’ or ‘I'd be disinclined to call that F’, or ‘I'm really unsure what to say about its colour’ etc.). This phenomenon is interesting and surprising, but is scientifically explicable. Ann is not conscious of there being any difference in F-ness between the relevant two objects, so her judgment-switch is the result of a sub-personal change that has been effected by the difference, however minute, between the two objects. Another observer might well switch from ‘Yes, F’ judgments at a slightly different point along the C-series. Compare a pair of old kitchen scalesFootnote ⁴ with a mound of ground pepper in the pan and the pointer on the face indicating 75 grams. Add one grain of pepper and the force on a spring inside the gadget increases by a tiny amount, but the pointer does not move. Add another grain; the spring may stretch a fraction, but still no movement of the pointer… until eventually the addition of a single grain sees the pointer jerk to a new position. Of course, such old scales (even of the same model) differ slightly in their performance; one may jerk from the 75 gram mark at a slightly different weight (of goods in the pan) than does another. In the case of Ann, the mechanism responsible for her jerking from an ‘appears F to me’ judgment on one object, say C₄₅₃, to a different judgment (any judgment inconsistent with ‘It appears F to me’), on the neighbouring C₄₅₄ (when the difference between the two is below the threshold of her conscious F-discrimination) is somewhat more complicated.

Complicated but not inexplicable, for the relevant neural process of sub-personal decision-making is reasonably well understood. Fascinatingly, it is functionally equivalent to the process, whimsically named ‘Banburismus’, by means of which Alan Turing and his Bletchley Park associates daily performed an initial mechanical sifting of coding hypotheses into the probabilistically promising and the probabilistically unpromising.Footnote ⁵ So, straightforwardly, the conditional ‘If C₄₅₃ appears F to Ann, then C₄₅₄ appears F to Ann’ is false, and hence the route to the false conclusion ‘C₁₀₀₀ looks F to Ann’ is blocked. There is no question here of a failure of experience to accord with logic, for Ann performs no inferences (she simply makes a series of honest judgements) and logic does not require that both C₄₅₃ and C₄₅₄ must elicit from her identical judgments as to their F-ness; it is a (surprising) scientific fact that they don't.

Do we, then, have a solution to the Sorites paradox? Not just yet. We have discussed only a Sorites experiment on an individual subject and have not considered standard versions of the paradox that make no mention of an experimental subject (i.e., which feature a vague predication ‘is F’ as opposed to ‘appears F to Ann’). We have touched upon the scientific question of why a given subject does abruptly switch judgments somewhere along a Sorites series, but have not yet broached the normative issue concerning the correct use of vague predicates.

Let us consider, then, the standard version of the colour patch Sorites, involving a series of patches, the first clearly red, the last clearly orange and each member visually indistinguishable from its immediate neighbours. Here, we are not asking whether the patches do or do not look red to a particular observer, but whether or not they are red. In this case, and, indeed, in all subject-less versions of the Sorites, Clark's observation concerning the seamlessness of transition must be acknowledged and explained. There is no point along the series of colour patches at which there is a dramatic, abrupt change from red patch to orange patch, just as there is no sudden transition from day to night.

The clue to a solution is supplied by Plato, who, in the Theaetetus, establishes that a consequence of Protagorean relativism is that a proposition ‘is more false than true, in proportion to the extent that the unbelievers outnumber the believers’ (171a). Of course, in the case of, say, mathematical propositions or scientific theories, the idea that they are not true but only partly so, and that the degree of their truth depends on the number of people who believe in them and the number who don't, is deeply suspect. But what is characteristic of judgements about vague attributes is that competent observers differ and there is no question of external arbitration to decide who is right and who is wrong. In such cases, the numbers count.

We can begin to see how the numbers count by first considering what is called ‘pure blue’. It is a hue that is phenomenally simple in that it is perceived as a blue uncontaminated by any hint of any other chromatic component. Perceived by whom? If all observers with normal colour vision agreed that a certain sample was pure blue, then we should have little hesitation in calling its colour pure blue. But, as a matter of fact, observers who are judged, by standard tests, to have normal vision, do not agree; a sample that looks pure blue to one may be perceived by another as greenish blue.Footnote ⁶ Suppose that, under excellent lighting conditions, none of a large number of competent observers judged a given sample to be pure blue. We should rightly be most disinclined to call that sample pure blue, for what is pure blue is, as already noted, what is perceived as such. Pure blue is not identified with a particular range of light frequency but is determined by how things look to competent observers. One competent observer may judge a church steeple to be leaning just slightly to the left, another, from the same vantage point, may judge it as leaning slightly to the right. Here a plumb line can arbitrate. With pure blue there is no plumb line.

What is true of pure blue is equally true of borderline red/orange, viz., it is not an objective property. Competent subjects disagree over whether a certain hue is borderline, i.e. no more red than orange and no more orange than red. Experiments done using a Sorites series of colour patches reveal not only intersubjective differences about where the supposed border lies, but intrasubjective differences too, in that a given subject is not consistent from one experimental run to the next, and subjects also tend to change their minds in the course of an experiment, revising some of their previous judgments.Footnote ⁷ Felicia Ackerman, who reports one such experiment, makes the point (following William Alston) that ‘it is the nature of these vague concepts to be dependably such as to permit such flexibility’.Footnote ⁸

To be red is to look red to competent observers. The first patch in the colour patch series has the colour of a British post box, and all competent observers agree that it is red. But, as we proceed down the series, we shall eventually come to the first patch which one or more observer judges to be not red, while the vast majority judges it to be red.Footnote ⁹ The use of the word ‘red’ is subject, of course, to normative constraints, but that does not mean that there can be absolutely no dissent in particular instances of its application. Unless we are proposing to ban all predications of the form ‘X is red’, it would be right to say of that patch about the colour of which there is just this first glimmer of dissent, that it is red. Similarly, if just a few more observers judge of the next patch in the series that it is not red, then an unbiased, colour-blind adjudicator would probably be happy to say that it is red. But, as we proceed further down the series, there will come a patch when the majority of observers dissent from calling it red and, still further down the series, consensus will re-emerge in judging all subsequent patches not red. Where there is substantial disagreement among competent observers about whether it is true that a given patch is red (or whether a given man is tall, or whether a given rock weighs a little less than 10 pounds etc.) there is not a definite answer, ‘yes’ or ‘no’, but this does not mean, as Oswald Hanfling notes, that there has to be a ‘third truth-value’.Footnote ¹⁰

A logician of the stripe Clark deprecates would wish to stipulate that the border between red and not red is exactly where the votes for ‘not red’ first become a majority. This would enable us to identify precisely which conditional in the colour patch Sorites reasoning is false. But there seems little point in doing this, when we have the richer account, briefly canvassed above, of the punctuated judgments of individuals proceeding along a Sorites series and of the consequent gradual drifts of judgment that sweep over populations of competent perceiver/speakers confronted with such series. Clark writes ‘Nothing changes from A to not-A ‘abruptly’: all such change involves its being, by degrees, less A until we judge that there is ‘too little’ of A-ness left to count' (p. 27). As we have seen, this idea cannot form the basis of a general account of vagueness or of a solution to the Sorites problem. A decelerating train does not become slow when there is ‘too little of fastness left to count’. It is slow when competent judges (each of whom may switch judgements from ‘fastish’ to ‘slowish’ at different speeds), judge it so.

The solution to the Sorites, then, has two components neither of which involves any revision or deconstruction of logic. The first is a neuroscientific account of how an individual can come to make different judgements about the F-ness of two items indistinguishable in respect of F-ness to that individual's unaided perception. The second is a treatment of subject-less versions of the Sorites (e.g., where we are talking of ‘red’ rather than ‘looks red to Ann’) as multi-subject, thereby correctly tying the normativity of vague predicates to the community of competent judges.