Statistics Using Neural Networks: Chance Effects - American Chemical

VOL. LXXXI

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[October, 1972

MIND A QUARTERLY REVIEW OF

I.—LOGICAL POSSIBILITY BY GEORGE SEDDON

Is it logically possible that a bar of iron should float on water, or that there should be carnivorous rabbits on Mars? If the category of logical possibility is set up by exclusion, so that any state of affairs that is not logically impossible must be regarded as logically possible, and ' logically impossible ' is defined in terms of self-contradictions of the form ' p and not-p ', then either we must show that there are contradictions here or the answer will be that it is logically possible that a bar of iron should float on water. That statement is not obviously self-contradictory. Yet we all know that it is not possible for a bar of iron to float on water, or that there be carnivorous rabbits on Mars, and if we insist that it is nonetheless logically possible, we invite the comment that we are using the word ' possible ' in a very odd way, and that we will need good reasons for such a striking departure from ordinary usage. I doubt that good reasons are forthcoming. I believe that ' logical possibility' has become a debilitating term because it blurs the distinction between science and pseudoscience, one variant of science fiction. I am no foe of science fiction, but see no reason why what can be imagined or what can be conceived should be confused with what is possible. Although I am not sure that I can do so myself, many claim that they can imagine a bar of iron floating, and we can certainly talk about it (I do not want to say that such talk is meaningless: but I do want to say that it is impossible—short of a miracle, which we will come to later). An important part of education is the acquiring of a sense of the possible; it comes slowly, with difficulty, and it must be nourished by a feeling for 16

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inter-relation, the knowledge that in nature one can never do only one thing. Talk of ' logical possibility ' may weaken this sense, I think, although at no time in our history has it been so important to distinguish between what we can and cannot do, and to foresee the consequences of what we do. The negative term, ' logically impossible ' is also confusing. I can recognize a contradiction of the form ' p and not-p ', but no one goes around saying " p and not-p ". My local newspaper had the headline: "Floating crane sinks", and we all knew quite well that it was still a floating crane, even on the bottom of the harbour. We knew not to reduce to the form ' p and notp '. A Hungarian friend of mine once remarked that there are no husbands in Australia, only unmarried and married bachelors. This we recognize as a paradox rather than contradiction: we certainly do not reply: " That is logically impossible." 'Logical impossibility' has nothing to do with what we ordinarily understand by ' possible ' and ' impossible '; it has to do with logic. Problems arise so soon, as we ask whether the category of the ' logically impossible ' is supposed to exclude statements or states of affairs. What does it exclude them fromi Clearly, from ' the possible '. But that does refer to states of affairs. So ' logically impossible' must also refer to a state of affairs. But there are, and can be, no logically impossible states of affairs. These problems stem from a prior assumption, which we need not make, that some statements in a natural language can be known to be true, and others fake, solely by inspecting the meaning of the constituent terms. But when apparent contradictions arise in a discussion, we never say: " That's a logical impossibility" (although we might say: "You are contradicting yourself"). Suppose the unlikely case that we hear someone say that a man gave birth to a child (or: " That man is a mother "). We reply that' he ' must in fact be a woman, because men do not bear children, and our linguistic distinctions have been devised expressly to reflect this reality. I can think of only two circumstances in which a statement such as ' that man is a mother ' might be made. One is the case of someone who is learning the language, and to him we would say, not " That's a logical impossibility", but " Don't you mean ' father'? " i.e. we would say that he did not understand the words. The other is the case where we might say " That man has been both a mother and a father to his child " meaning that he has played both parental roles in its upbringing, and not that he fathered the child upon himself. The question of ' logical impossibility ' here could not arise unless we know that the

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latter sense is not intended, or something like it: we must be sure that the man is saying " p and not-p "; that is, we must be sure that he really has failed to understand the words. And if we are sure, why say more than that? What further clarification is achieved by talk of ' logical impossibility '? Now for ' logical possibility '. What should we say about the statement that it is logically possible that a bar of iron float on water? (' Bar ' is intended to exclude the needle, supported by surface tension, and the new ' Queen', Q4, floated on Zurich capital.) One move is to argue that although the statement that ' A bar of iron floats on water' is not explicitly self-contradictory, it is implicitly so, for we are saying that a mineral with a specific gravity of less than one (i.e. itfloats),has a specific gravity in the range 7-3-7-8 (i.e. it is iron), and this is a contradiction, and is therefore logically impossible. This move retains the category of logical impossibility and removes the absurdity of supposing that a bar of iron might float. But the debate now shifts to whether or not a s.g. of 7-3/7-8 is to be regarded as a defining characteristic of iron. There is no doubt that the mineralogist regards it as one; when an unknown mineralogical sample comes up for assay, one of the first tests performed is to determine its s.g., and if this falls outside the range of Fe, then the possibility that it be a fairly pure sample of metallic iron is totally excluded. However, the philosopher might argue that it is merely a contingent fact that Fe has a s.g. about 7; this may be true of all the iron encountered to date, but new forms might be discovered in the future with an s.g. less than 1. " Defining characteristics ", says the philosopher, " have not been constant in science. Look, for instance, at wood. In the eighteenth century it might confidently have been asserted that wood floats (if it doesn't float, it's not wood). But this assumed that all wood has an s.g. less than 1, which proved not to be the case. Jarrah (Eucalyptus marginata) from the Swan River Colony in Western Australia and several of the ironwoods from eastern Australia, are considerably heavier than water. ' This makes it impossible to handle logs by floating, and has on occasion caused trouble in overseas ports, logs dumped from the ship into the water to be floated to their final destination finally sinking to the bottom ' (McLuckie and McKee, 1955, p. 599). So how can we say that iron must have a specific gravity greater than 7? " The philosopher may offer another illustration nearer home, discovery of isotopes: the term was introduced to apply to so called ' atomic species' having the same atomic number but different nuclear structure, as indicated by different atomic

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weight or different type of radioactivity. The physical and chemical properties of hydrogen, for example, were known with assurance, but our account of them was subsequently modified to accommodate deuterium, with a mass approximately double that of ordinary hydrogen, and with quite distinctive physical and chemical properties. For example, the heat of vaporization of D20, or heavy water, is about 260 cal. per gram greater than that of B^O. The point of the example, says the philosopher, is this: " We might, before the discovery of deuterium, have claimed a heat of vaporization of 540 cal. per gram as a defining characteristic of water, and argued from this that any subsequent statement that water has a heat of vaporization of 540 cal. per gram would be analytic (because we have already implied its heat of vaporization of 540 cal. per gram in calling it ' water '). We might then have argued that to deny that water has a heat of vaporization of 540 cal. per gram would be self-contradictory; therefore it is logically impossible for water to have a heat of vaporization of, say, 800 cal. per gram. Yet we would have been wrong. It turns out that heavy water has such a heat of vaporization after all, and we have had to change our defining characteristics." But two simple points emerge from this discussion. One is that we have been discussing a language shift. The physical properties of H2O have not changed. To the student who suggests that it is, after all, possible that water have a heat of vaporization of 800 caL per gram we say, " Oh, you mean heavy water ". The ordinary language word ' water ' may be used by a physical chemist until such problems arise, but as soon as they do, it is abandoned: we must speak of Hj,0 and D2O. If the student persists, saying " No, I just mean ' water' ", we say that he does not understand the way these words are used in physical chemistry, like the man who did not know how to use the word ' father '. Questions of ' logical possibility ' would not normally be raised here at alL The second point, however, is that if a question does arise about the possibility that water have a heat of vaporization other than what we now think it to be, because perhaps of the discovery of further isotopes of hydrogen (tritium?) or oxygen, we are certainly not talking about logical possibility, we are talking about the possibility of a language shift plus some sort of empirical possibility (' empirical' is at least a conventional philosophical term here, although scientists in fact talk about ' physical' and ' theoretical' possibilities, for reasons that I shall turn to presently). A similar story can be told about the bar of iron. As any

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first-year chemistry student knows, iron is tetramorphous, and the four crystal forms differ slightly in physical properties. However, below temperatures of 760°C, iron is found at equilibrium only in the a-form (' ferrite '). Ferrite is a soft, tough, grey-white metal with a cubic body-centred crystal lattice. Let us then discuss ferrite. It has an atomic number of 26 and four stable isotopes; 54, 56, 57 and 58, the atomic weight of a typical sample being 55-84, with a s.g. at 20°C of 7-87. It does not, therefore, float in water. The four isotopes are separable, and we might have a bar of each of them; the four bars would differ very slightly in s.g., but none of them would float in water either. (Note that the range of s.g. 7-3/7-8 given earlier in this discussion is now imprecise: it allows for minor impurities in what, at a crude level of analysis, we were still prepared to call iron.) Specific gravity is dependent on density, being the ratio of the mass of a given volume of a substance to the mass of an equal volume of water at a specified temperature and one atmosphere pressure. But density is not an independent property either: the density of a pure substance with a fixed chemical composition and crystallizing in a specific structure (in this case, body-centred cubic) is constant at a stated temperature and pressure. Quartz, for instance, which is practically invariable in composition, has a density of 2-65 g/cm8 at ordinary temperature and pressure. The other polymorphs of SiO2, which crystallize in different structures, have different densities, that of cristobalite being 2-32 g/cm3 and of tridymite 2-26 g/om3. For a given substance, if the dimensions of the unit cell have been measured and the number and kinds of atoms in the unit cell are known, it is possible to calculate the density. In point of fact this is standard procedure. Densities determined from x-ray measurements provide a most useful independent check on densities as traditionally measured (e.g. by direct weight measurement, and volume determination via Archimedes principle); and conversely, densities so measured can be used to check the correctness of a proposed structure or chemical formula. Any change in the density of an element would therefore, not be accompanied by, but in fact be either a change in the size of the unit cell or in the number or kind of atoms of which it is built up. If the latter, a change in atomic species is involved, by definition —and if the former, the physical properties which are determined by the lattice structure (e.g. crystal habit, cleavage, optical properties and so on) would also change in such a way that we could no longer be talking about ferrite. None of this is in any way novel or surprising: it is thoroughly well known that the

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1

What the engineere are more likely to Bay is that the hridge is feasible. I t would usually be called ' technically possible ' only if this were being


physical properties of a given mineral are interdependent. Locke said that there is no connection between the colour of gold and its malleability, but Locke was wrong. A glance at the Periodic Table shows that iron is one of the transition metals, all with related physical properties, and among them, density, which increases steadily from scandium (3-1 g/cm3) to copper (8-91 g/cm3); iron has as its immediate neighbours, manganese (7-21 g/cm3) below, and cobalt (8-70 g/om3) above, and this alone indicates the possible density range for iron, which must fall between these two values. If by some oddity iron had not yet been discovered as a naturally occurring element, all of its major physical properties, including its density and specific gravity, could have been successfully predicted from its position in the Periodic Table, as was the case with hafnium. But here the weary logical possibilist may break in impatiently and say: " Yes, we all know that iron does not in fact float in water, but you persist in talking about factual consequences and empirical possibility, which is not the point at issue. It is still true that the specific gravity of iron is a contingent fact: it is not self-contradictory to conceive that it might have been different." My problem, of course, is to conceive what this' might' might possibly mean when it clearly has no meaning in the linguistic context in which terms such as ' specific gravity' are applicable. Moreover, I would contest the claims that I have been talking about ' empirical' possibility, or rather, I would suggest that this, too, is an inadequate term for the purposes of this discussion. We are, by philosophical convention, offered three categories, technical, empirical and logical possibility, of which the first is a proper subset of the second, and the second of the third. There is no serious difficulty about understanding what is meant by ' technical possibility ' (within the range of present technology), although the difficulties commonly encountered in estimating it are worth brief investigation. We ask a firm of engineers to determine whether of not it is technically possible to build a single-span bridge from Point A to Point B, across the Swan River. They may be able at once to point to a bridge built under circumstances which are similar in all relevant respects, in which case we are talking of actualities rather than possibilities, provided their professional judgment of the circumstances is sound. Failing this, they do some calculations, on the basis of which they will either say that the bridge is technically possible,1 or demonstrate from engineering theory that it

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contrasted with other aspects of the total situation, e.g. technically possible, but economically ruinous, or politically dubious, or aesthetically undesirable.


is not possible, or ask for a second opinion. Let us suppose that they choose the first and go ahead and build the bridge. Either it stands, in which case they were right in claiming that it was technically possible, or it falls down. If it falls down, there is an enquiry, and a firm of consultant engineers with an international reputation is called in. They give evidence: there are again three conclusions available. Either the bridge was technically possible, but something (specifiable) went wrong. Or they will show that the bridge is after all a theoretical impossibility. Or they will say that they do not know whether such a bridge is technically possible or not, i.e. engineering theory is not sufficiently developed to make a firm prediction from the available evidence. Several points emerge from the disoussion of this example. One is that possibility claims are, in general, predictive—a claim is being made about the future. Next, possibility claims can only be assessed relative to the evidence. As the word is most commonly used—some exceptions will be noted—possibility is a form of probability. (It is true that we sometimes say " that is possible, but not probable " but by this we mean only that we are assigning a low probability.) Now the meaning of the word ' probable ' is incomplete without specification of the evidence. We can no more say (other than elliptically) " that is probable " than we can say " that is equal " or " that is relative ". Probability claims are predictions based on evidence. A prediction is not a guess: if a man claims to make a prediction, but can give no evidence for it, and denies that he has any, we say that he is guessing: he doesn't know whether the event is probable or not. The only honest answer to many questions in science is " we don't know ". But it is trivial to claim that when we do not know whether something is possible or not, we cannot exclude its being possible, and therefore it is possible. In this situation, neither can we exclude its being impossible, and we should speak of ignorance rather than possibilities. This, in my view, is how we should talk about extrasensory perception: we don't know whether it is possible or not, because there is not enough evidence, or theory, to form a reasonable basis for estimating probabilities. It is possible only in the trivial sense that we do not know it to be impossible—but then neither do we know it to be possible. I am now in a position to make a final point about the bar of

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(credo quia impossibile est).

Have I then committed, myself to the view that there is only empirical and technical possibility? Not quite. In discussing the bridge we passed from technical to theoretical possibility. The hitter category needs discussion, and perhaps one should explain why scientists rarely discuss empirical possibilities. However, I have also noted above that there are some legitimate uses of the word ' possible ' that do not fall within the present framework of discussion, and perhaps we should give an example, for contrast. I ask a friend whether he is going skiing over the week-end. He says: " I doubt it, I have a lot of work on hand. Still, I guess it's a possibility." This is a common use of the word, but a specialized one, because the ' possibility' referred to has little to do with physical possibility—we know our friend has skis, can ski, and often does so during the week-end. We also know that if the snow conditions deteriorate—if there is a quick thaw, for instance—the whole conversation is cancelled. We take all that for granted. " It's a possibility " here translates : " I haven't made up my mind: I haven't definitely decided against it." In this and similar cases, we do not ask for evidence, but in talking of technical, empirical, and theoretical possibility, we do, and to these I return. The term ' empirical' has somewhat derogatory overtones in science. For instance, hydraulic engineers make use of a streamflow velocity equation, known as the Manning equation. Here is a discussion of the Manning equation from a recent book on hydraulic theory: " It is truly surprising that engineering practice has depended to such an extent on a formula as empirical as this one, derived nearly a century ago " (Leopold, Wolman and Miller, 1964, p. 158).


iron. Suppose that I have said all that I have about the specific gravity of iron and its place in the Periodic Table. My persistent questioner then asks: " But what would you say, if you had done all the tests, including a specific gravity test, and got all the right answers, and then put the bar in water, and it floats? " My answer, of course, is that I would say it was a miracle, and I now define ' miracle ' as an event of zero probability. This should dismay neither those who believe in miracles, nor those who reject them, since their probability could only be estimated from the evidence of observed regularities, whereas miracles by definition run contrary to these regularities. When we are discussing possibilities we are not discussing the possibility of miracles. I would like to think with Sir Thomas Browne that miracles are impossible, and believe in them because of that

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Discussing problems of sediment transport, it is later (p. 129) argued that " The number of empirical equations itself testifies to the continuing need of a more fundamental knowledge of the basic processes governing sediment movement. Because of the variables involved, it appears likely that major advances will be made primarily through advances in theory and critical experiments rather than by amassing volumes of additional data." Comments such as this are commonplace in any branch of science which is just emerging from the pre-theoretical phase. For example, tidal prediction was so scandalously empirical in the early nineteenth century that the British Association for the Advancement of Science, founded in 1831, devoted 10 per cent of its budget in the first decade of its life to tidal research. And rightly so: the empirically derived tidal tables, like the Manning equation, have limited explanatory power, and also limited predictive power in the sense that the range of their application cannot be fully grasped without a theoretical basis. But crystallography and mineralogy have not been empirical in this sense for many years, and it would be profoundly misleading to say that the specific gravity of iron is a mere empirical fact. The specific gravity of iron, the place of iron in the Periodic Table, and its physical, chemical and atomic character can be discussed only in the context of a highly integrated and coherent physical theory, and this is why one wants to say that it is theoretically impossible that the bar should float, and not merely empirically impossible. Much more is at stake than a series of observations on the behaviour of iron bars in water. It is true that there appear to be degrees of theoretical impossibility—some things seem to be more impossible than others —but this need not puzzle us. Our confidence in the assertion of theoretical impossibility will depend on two things; first, the completeness and precision of the demonstration of impossibility from theory; and second, our confidence in the theory itself. Science does not deal in certainties, and no assertion of theoretical impossibility can claim to be incorrigible. But the only residue of possibility here is of the trivial kind already discussed. What scientists can do is to claim that a given event has zero probability relative to the evidence, or is incompatible with a given theory. It used to be argued, for instance, that it is impossible to exceed the velocity of light (e) within the special theory of relativity, for within that theory a body accelerated from a speed less than c to one greater than c would acquire energy and momentum expressed by an imaginary number—and this does not seem to have any physical meaning. It is curious that Pap,

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who spells out this example in detail as an example of logical impossibility (1962, p. 64), prefaces his discussion thus: " Is it meaningless to suppose that there occur, or may occur, motions exceeding the velocity of light? It seems to be a perfectly meaningful supposition, for even if it be true that the velocity of light is the upper limit for all physically possible velocities, this is a physical fact; in other words, what we suppose to be the case is at most empirically impossible, not logically impossible, hence it must be allowed to be a significant supposition." He then goes on to show that the supposition is nonetheless ' meaningless' within the context of the special theory of relativity. But why suppose that it is possible when no context is specified, no theory indicated or evidence cited, without which the probability of the supposition cannot be assessed at all? This is to tali no language (other than philosophy!) Perhaps Pap's question was intended to mean: " Does this supposition have meaning within the context of subatomic physics? " This question has been revived by physicists (Torsten Alvager and Michael N. Kreisler, 1968: The Physical Review, vol. 171, no. 5, pp. 1357-1361; Gterald Feinberg, 1967: The Physical Review, vol. 159, pp. 10891105; 1970: Scientific American, voL 222, no. 2, pp. 68-77). But they have not asked the question as Pap does. First, Feinberg argues that the now classical objections hold only in the case of bodies that are accelerated through the speed of light. But " we now know that.. . subatomic particles can easily be created or destroyed, and that in their mutual interactions their energies and other properties change discontinuously . . . therefore one can imagine the creation of particles already travelling faster than light, and so avoid the need for accelerating them through the ' light barrier' with the attendant expenditure of infinite energy " (Feinberg, 1970, p. 69). This might suggest that Feinberg is arguing, in the philosopher's way, that velocities greater than c are after all logically possible within relativistic theory. But of course he is not doing that, as we see from what follows. Having shown that such particles are not ruled out by theory, he goes looking for them. He has not found any yet, and there may not be any: " The . . . possibility is . . . that nature has not filled the niche that is allowed by the theory of relativity " (1970, p. 77). But this is not accepted as a ' brute fact'; that is, such particles could exist, but just happen (' contingently') not to. If they do not exist, then this immediately calls for further explanation, although " we may not understand why it should be so until we reach a much deeper understanding of the nature of elementary particles than now

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Are rabbits possible on Mars? Clearly not. For one thing, they would be frozen for most of the year (this is not meant, however, to raise the possibility that there are frozen rabbits on Mars). Are carnivorous rabbits possible, anywhere? No, this is a theoretical absurdity. Rabbits belong in the order Lagomorpha, and the lagomorphs have a place in the infra-class of eutherian mammals which is admittedly not as tightly structured as that of iron in the Periodic Table; nevertheless certain characteristics are grouped in a theoretically necessary way. Rabbits have the dentition of a herbivore—they could not eat meat as they have no canines to tear it with. They have a


exists " (1970, p. 77). Thus in no context is it adequate to speak of the velocity of light's being an upper limit as a " physical fact", as Pap does, and then to consider the logical possibilities beyond this "fact". What about the carnivorous rabbits on Mars? Is this a possibility of any sort? I say that it is not, that it is theoretically impossible, and if we admit that it is not quite so impossible as the bar of iron's floating, this is not because of a residue of possibility, but because of a residue of ignorance. We could, of course, again turn the trick of showing that it is logically impossible, in that in saying that they are carnivorous it can be shown that they are not rabbits, because ' herbivorous' is a defining characteristic of' rabbit'. But this does not satisfy me; there is more to it than that. So let us anatomize. Is life of any sort possible on Mars? This is a matter on which scientists disagree, but their disagreement can be explored from a prior agreement as to what—at least roughly—is to count as evidence. For instance, the white polar caps, which wax and wane with the seasons, are thought to be areas covered with dry ice to a thickness probably not exceeding an inch. The seasonal changes in the polar caps are accompanied by changes in the dark surface markings on Mars: with the shrinking of each cap the region around the cap darkens: the dark areas become more distinct, and some of them take on a greenish hue. It is just possible that these changes are vegetational changes in lowly plants like our hardy mosses and lichens. But when we say that it is just possible, we mean that there is some positive evidence, not merely that we do not yet know it to be impossible. There is also much negative evidence: the mean temperature for the entire planet is about 60° below zero F, compared with 60° above zero for the earth. Free oxygen is generally estimated as much less than 1 per cent and water vapour only a tenth of 1 per cent of our supply.

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Perhaps we are most tempted to think that there are possibilities that survive demonstrations of theoretical impossibility in cases like the following: I take my wife to the doctor. He says she has cancer, and that it is impossible that she live for more than six months. I think: " Well, I won't buy the coffin


herbivorous digestive system, and could not digest meat. Rabbits have the Rabelaisian habit of coprophagy: the stomach and intestinal system are adapted for the ingestion of large quantities of vegetation quickly, resulting in the production and excretion of capsules (soft pellets) rich in bacteria, and of high nutritive and digestive value, which are swallowed whole (reingested) at leisure in the safety of the underground burrow. A carnivorous rabbit would get a nasty shock in the burrow. It follows from the Principle of Natural Selection that any biologically successful species is broadly adapted to its environmental niche. Rabbits are prey rather than predators, but they are well adapted to this role. Their very high birth rate, for instance, would be disastrous in a predator, which would rapidly outstrip food supplies. Even the familiar white scut and the long ears are functional: the scut is a social alarm signal which protects the group; it would be inappropriate in a stealthy predator. The eye socket is large to support the large crepuscular eye. The powerful feet are adapted to bounding escape flight, and to digging and shovelling earth during excavation. These characteristics are not, admittedly, logically entailed by each other, or by some essence-of-rabbit. But they cannot be treated entirely independently within biological theory either. To make a rabbit a successful carnivore we should have to restructure him so thoroughly that no one—leave alone the zoologists—would want to call him a rabbit any more. We would also, incidentally, have to rewrite the evolutionary history of the lagomorphs, which is already fairly well known. This demonstration of theoretical impossibility may be less convincing than that of exceeding the velocity of light within the context of the special theory of relativity used to seem. If so, it is because the rabbit demonstration cannot be given in mathematical form, and perhaps because the theory of Natural Selection may seem to some less fully articulated or less well founded than the special theory of relativity. But there is no room here for logical possibility. The only way in which we could come to call a carnivorous animal a rabbit is that we could, after all, come one day to use the word ' rabbit' to refer, for example, to the animal we now call a tiger. But this has no bearing on the category ' logical possibility'.

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Yet before I conclude I must make an admission: I am conscious of seeming to beg a question throughout this paper. When I say that the density of iron must fall between the values for manganese and cobalt: when I say that rabbits must be herbivorous: what is the force of this 'must'? This is the question that I cannot quite answer. Are these necessary truths? Only if we claim that with the help of definitions, the denial of these propositions can be shown to be self-contradictory.


yet, anyway." Or the same example in a slightly different form: I have a friend who tells me that he took his wife to the doctor five years ago, was told that she had an incurable disease and could not possibly live more than six months. " Then I took her to all the specialists and the clinics, no expense spared, and they all said the same thing. Not a chance." But there she sits on the sofa opposite you, beaming testimony to the triumph of motherhood and femininity over the laws of medical science. What do we say here? If we are prudent, we do not say anything, but we doubt that the doctors did say just what was attributed to them. Good doctors are not much given to unconditional prediction. But suppose they did: then they were wrong. A claim that a given event is theoretically impossible from a medical point of view may be falsified either because the demonstration is defective: the doctors got the theory wrong, or applied it incorrectly: or because the theory itself is defective or inadequate. Medical science is highly pragmatic. Any doctor admits that there is much that we do not know about health or sickness, and he usually prefaces his conclusion with ' so far as we know', and he means it. One runs beyond the boundaries of medical science into sheer ignorance fairly quickly. But there is still no room here for logical possibility. What there may seem to be room for is a weak but non-trivial form of a theoretical possibility of the form: we cannot be quite sure that it is theoretically impossible because the theory is not good enough—but we cannot say for sure that it is impossible, so I guess you can say that there is some possibility here. The doctor says: "All the evidence is against it, but you never know." But he does not really mean that all the evidence is against it, but only the specific and immediately relevant evidence. He reads this against background evidence (called ' professional experience ') of the form: " Doctors make mistakes; patients we would not give a nickel for sometimes just mysteriously do get better." So his concession is based on evidence, and the possibility here is a weak form of empirical possibility, assigning a low probability to an event he thinks unlikely.

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University of Western Australia REPBBENOES

MoLuokie, J . and McKee, H. S. (1955). Australian and New Zealand Botany. Associated General Publications, Sydney. 768 pp. Leopold, Luna B., Wolman, Gordon W. and Miller, John P. (1964). Fluvial Processes in Qeomorphology. W. H. Freeman & Co., San Franoiseo. 522 pp. Pap, Arthur (1962). An Introduction to the Philosophy of Science. The Free Press of Gleneoe, New York. 444 pp. Alvager, Torsten and Kreisler, Michael N. (1968). ' Quest for faster than light particles '. The Physical Review, 171 (5): 1357-1361. Feinberg, Gerald (1967). ' Possibility of faster-than-light particles'. The Physical Review, 159 (5): 1089-1105. Feinberg, Gerald (1970). ' Particles that go faster than light'. Scientific American, 222 (2): 68-77. Waismann, Friedrich (1951). 'Analytic-Synthetic, I I I ' . Analysis, 11 (3): 49-51.


Yet I claim that there is more at stake than this. So I must be talking about the world, about contingent truths, about statesof-affairs that might have been different. But a rabbit cannot be a carnivore; and so we go round again. Waismann (1951: 'Analytic-Synthetic III', Analysis, 11 (3): 49-51) discusses a comparable dilemma, of which he says: " The reason for this wavering between opposite poles is that this is a case in which the philosophical antithesis ' contingent-necessary' loses its edge." Yet his examples (" time is measurable " , " space has three dimensions "), are much more rarified than mine. Iron sinks in water, but it could have been otherwise. Perhaps: but how much else are we allowed to know ? It couldn't if it is still to be one of the transition metals with a place in the Periodic Table. But the Periodic Table could have been otherwise: it was not inscribed on the underside of the Mosaic tablets. Perhaps it could be otherwise: and perhaps there could be a world in which organisms are not functionally adapted to their environment. WeU, perhaps: you never know: anything can happen. Nature is bountiful, and full of surprises, and there's plenty more where that came from. Think cornucopian.

Statistics Using Neural Networks: Chance Effects - American Chemical

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