1. Suppose that we run a Sorites-type experiment in which the subject is a frog in a pan. We know that frogs survive perfectly happily in water at 20oC. If the frog survives at this temperature, then surely, we reason, it will happily tolerate 20.001oC ….. and so on, for each increment of one thousandth of a degree Centigrade. After 80,000 such steps of reasoning, we would reach the false conclusion that the frog is going to survive in boiling water and will find it tolerable. But, typically, we don’t reach that conclusion, for at some stage in the reasoning (for most people, it is at around 50oC) our judgments flip-flop as a result of the neurological processes that were mentioned in the last lecture. There is some temperature at any greater than which we reckon the frog’s chances of finding it tolerable as low, at any rate, if the frog stays immersed for any substantial period of time.
2. In the old public bathhouses, if the bath water was getting cold, all the bather needed do, when having a bath in cubicle N, was to shout `More hot water in Number N’ and an attendant on the outside would open the hot valve, releasing a flow of very hot water into the bath. When the bath water got close to uncomfortably hot, one shouted `Enough, thanks’. The height of luxury. However, if the valve malfunctioned and the inflow of very hot water was not cut off, then one would hop out of the bath pretty damned fast. I take it that this would happen even if one had no inkling that the valve had malfunctioned – at some point, we should find the water intolerably hot. (It is a little known fact that, although intolerance levels vary considerably from one individual to the next, broadly speaking, the intolerance level for blue-eyed people is different for that of brown-eyed people.).
3. Communication with frogs is a problem, but we can assume that a frog votes with its feet, so to speak, and says its `Enough, thanks’ by jumping out of the heating pan of water when it finds the water intolerably hot. Since the frog is more in touch with its feelings than we are, one would expect some discrepancy between the frog’s judgments and ours about when to vote `enough’. And indeed, there is such a discrepancy, but it is much greater than you might think. The surprising thing is that, in the gradual heating situation, the frog’s judgment never flip-flops; it never jumps out of the pan, but just boils to death. Put your live frog into a pan of nice, lukewarm water, and then gradually heat it until boiling, by which time the frog is dead.
4. It is, as we can see, a purely empirical matter when, if at all, a Sorites subject, human or bestial, makes a judgment-switch. Obviously, the frog’s brain is very different from ours in terms of the A- and B-type processes. Identifying such processes and understanding how, for different species of animal, they operate in tandem are fascinating projects for neuroscience. But, if the `no philosophical solution’ solution is correct, there remain no philosophical mysteries about paradoxes in the Sorites family, and the assumption that a solution requires an extension or alteration of classical logic or semantics is mistaken. The reason for this goes deep -- right down to the very nature of logic. Frege declared that logic is a normative science, and that has certainly been the dominant conception before and since. People tend to reason badly; it is not the rôle of logic to reflect how people reason, but to set the standards for how to reason correctly. Although this is true, the connection between how we do reason and how we ought to is more subtle than Frege thought. Frege held that logical and mathematical relations were the immutable denizens of a Platonic realm, divorced both from the empirical world and from the minds of individuals. However, the validity of an inference is a function of the meanings of the logical terms that feature in it, and the meanings of such terms is not to be found in a mind-independent realm, but is simply the correct use of those terms. And the correct use is the community-wide (or almost community-wide) use; there is no other standard. In formal semantics, the meaning of a predicate is determined by fiat, there generally being a proper subset of objects in a domain to members of which the application of that predicate is assigned the value `true'. The meaning of a predicate in natural language, by contrast, is its correct use -- how it is correctly applied by speakers. Now, as we have seen, although there may be broad agreement in how, in a large class of central cases, we apply a vague predicate, there are cases in the perimeter area where not only do people differ in their judgments about whether a particular predicate applies but also one individual may make inconsistent judgments from one occasion to the next; and, most often, it would be wrong to declare any of these judgments wrong. So a `semantics' for such a predicate would be personal and dialetheic. And would have no logical value (taking `logic' to be a normative science) for it would be only a record, for each speaker, of that speaker's usage.
5. This becomes especially clear if we consider Sorites experiments with frog subjects. We have considered the gradually boiling frog – we can call this the analog frog version of the Sorites. The digital frog version is different in that here there are many pans of water, ranging between 20o C and 100oC in increments of .001oC. The frog is thrust into a pan and its reaction (either staying put or jumping out) recorded. Then it is allowed to dry and return to its original state before being thrust into a different pan, and so on until it has been dipped in each -- the order of the dippings does not matter, just so long as each pan is sampled. Precautions must, of course, be taken to avoid making the frog resentful or cantankerous, so that, during the course of the experiment, the only factor relevant to its behaviour is the heat of the water. For any given frog there will be temperatures q and q+dq such that he finds the first tolerable, the second not, even though dq is too small a difference for a frog to be able to detect. Clearly, the frog behaves in a way that requires some fathoming but that is a project for the biological sciences. And this has been my theme. The frog is a pure Sorites subject, not a reasoner or a theoretician. Like us, it flips in its reaction to one condition and another when the difference between those conditions is invisibly tiny. Yet these animals have no language, so any proposed logic or semantics of vague predicates is quite beside the point in a proper general explanation of the Sorites phenomenon.
6. Crispin Wright has argued that there are different types of Sorites paradox that call for different types of solution. What I have offered above is supposed to apply to all versions of the paradox, but are there problematic cases? Take Wang’s paradox: 1 is a small number, so 2 is too, and so is 3 ….. and so is 1000,000! It is not clear that the points I have made about the effects of observation apply here. Timothy Williamson makes what I think is the correct point – that, sure, we can discriminate between two numbers, but what we are being asked to do is to compare the smallness of one with the smallness of the next. And that, it seems to me is a matter of observation, because it is what is measured numerically that is being assessed for smallness, and usually that's something concrete and observable. In other words, the question `Is 1,000,000 a large number?’ doesn’t make much sense. If someone has only 1,000,000 brain cells, then they’re in trouble. And, for a mathematician working on paired primes (numbers n, n+2 both of which are prime), 1,000,000 is really small. On the other hand, 5 is a large number if that is the number of miles you have to walk back home late at night, having missed the last bus.
7. Let us now consider one last type of solution, the interest-relative theory due to Delia Graff. See her `Shifting sands: An Interest-Relative Theory of Vagueness’, Philosophical Topics 28 (2000), pp.45-81. You can also inspect (but not download) this article at http://instruct1.cit.cornell.edu/research/graff/papers/shifting.pdf
8.
As we saw in paragraph 6 above, the standards that we
use in making judgments vary from one context to another. For example a certain number may be large in
one context, but not in another. Graff’s
solution, as she acknowledges, is similar (but not identical) to solutions that
say that vague expressions are context-dependent. The differences beween the two are
nicely explained in Jason Stanley, `Context, interest-relativity and the
sorites’, Analysis 63 (2003), pp.269-280 – and
9. One crucial element of Graff’s view is her point that we can use a vague predicate with different standards on different occasions. This difference is sometimes accounted for by the fact that, on those occasions, we are using different implicit comparison classes, but, for her, this is not the crucial kind of case (for we can easily `strengthen’ a Sorites paradox by relatavising the vague predicate to a comparison class). Her example of the use of different standards is when we say to a particular man `You are not bald’ (when we are trying to find a lookalike to act as a stunt double for the completely hairless Yul Brynner), but we say to that same man `You are bald’ when we are trying to find a lookalike for Mikhail Gorbachev. On both occasions, the comparison class is the same, but our interests are different on the two occasions. Of course, one needs to know from Graff what it is to use a predicate according to a certain standard, and she offers a number of constraints, one of which is: `Whatever standard is in use for a vague expression, anything that is saliently similar, in the relevant respect, to something that meets the standard itself meets the standard’.
10. The notion of salience is important for Graff’s theory. In the colour patch Sorites, at least one statement of the form `If Patch n is red, then so is Patch n+1’ must be false yet, says Graff `the very act of our evaluation raises the similarity of the pair to salience, which has the effect of rendering true the very instance we are considering. We cannot find the boundary of the extension of a vague predicate in a sorites series for that predicate because the boundary can never be where we are looking. It shifts around’.
11. Notice the difference between Graff’s theory and Epistemicism. According to the Epistemicist, in a sorites series, there is a definite (fixed) boundary between the samples that are F and those that are non-F (but we cannot know where that boundary is). For Graff, there is a boundary, but it moves around – it is always somewhere different from where we are looking. Note the similarity between Graff’s position and the `No philosophical solution’ solution. In the latter, no fixed boundary exists between the Fs and the non-Fs – this can be tested empirically. The difference between the two accounts is that the `No philosophical solution’ solution acknowledges that, although we may be looking at two adjacent patches that are visually indiscriminable, we may nevertheless judge one to be F and the other to be not F because our sensory mechanism may be affected differently by the different patches, even though it is not a difference that we can notice (be aware of).
12.
Reading: R. Keefe and P. Smith, Vagueness,
pp.1-57. This is a `survey’
article, the introduction to the book which contains many of the important
articles on the Sorites. Another useful
collection of articles is D.Graff and T. Williamson. Other survey pieces
include Stephen Read, Thinking about Logic, Chapter 7; R.M. Sainsbury
and T. Williamson, `Sorites’ in B. Hale and C. Wright (eds), A Companion to
the Philosophy of Language, Chapter 18.
The articles I have mentioned today – Graff (2000) and Stanley (2003) –
are difficult but they repay the effort of careful reading.