THE LESBIAN RULE 1

First published in Philosophy, XXX, July 1955, pp.195-213

THE problem with which I wish to deal in this paper is the problem of singular reasons in the humanities, whether they exist, or rather, whether they can exist: for it would seem that the word ``reason'' carried with it some idea of generality, so that the phrase ``singular reason'' was a contradiction in terms, a specification which could never be fulfilled. But humanists are always sensing the singularity of their studies: and the philosopher wondering about the nature of humane thinking either must conclude that it is really only inadequate science, its singularity being the mark of the fact that we never really have an adequate basis for any of our generalizations, an unfortunate reminder that the whole structure is built on the shakiest of inductive foundations; or, if he is unwilling to allow that humanists are just bad, or at least amateur, scientists, he is tempted to vindicate the rationality of their reasoning by inventing a special faculty of particular ratiocination which is able to perceive particular truths---the intuition of the moralists or the insight of the Collingwoodian mode.

The problem concerns the humanists alone. It is all right for the scientists; they can indulge their nostalgia for the universal to their hearts' content, because they are never concerned with particulars that are unique; in so far as they are concerned with individuals at all, the individuals are ones which differ only numerically from one another and are qualitatively the same. Scientific theory with its generous doctrines of irrelevance enables us to reduce to only a finite size the number of significant variables, and experiments can be repeated and essentially similar situations reproduced or examined, as often as we please. The great complexity and variableness of (195|196) human beings makes this impossible so far as practice goes in the study of human behaviour and human affairs; and those of us who believe that the complexity and variableness of human beings is infinite are precluded from hoping that scientific generalization ever would become practicable in the humanities or from discussing what it would in principle be like. In either case the practical problem is the same: we are confronted with a series of disciplines which claim to be rational and yet deal with persons who are unique, moral predicaments that never will recur, and historical situations to which there can be no parallels that are exact. It would appear necessary then that either the method or the subject-matter of the humanities was not what it was claimed to be, unless after all it was possible for singular reasons to exist. It will be my thesis to-night that it is possible and that the phrase ``singular reason'' is not a contradiction in terms.

The hos epi to polu nature of humane discourse, the apparent inexactitude of humane thinking and our inability to manipulate the letter so as accurately to express the spirit of the law, have worried philosophers since the time of Plato. In the ancient world it was this rather than anything else that led to the over-great use of the techne model, to the feeling that moral and political thinking must be assimilated to the medical and gubernatorial skills, because they were, like them, unformulable gems, inarticulate know-hows, rather than articulate know-thats, which, though they could be conveyed and left by intercourse and experience, could not be condensed into a series of prescriptions or communicated adequately by words alone.2 Logic in moral matters, far from being a universal ordinary, was the prerogative of the elect. Only to an aristocracy of the spirit, composed of those who by some inherited flair or natural feel or by special education, divine authorization or metaphysical insight were qualified to determine moral matters, could political power safely be assigned. Modem philosophers, while unwilling to push their speculations to a so practical conclusion, are likewise puzzled, and can find only three, equally unacceptable, escapes from their aporiai. They may play down the singularity of their reasoning and the uniqueness of the situations they describe, holding that they are not essentially singular or unique, that it is only an accident if it be the case that there are no comparable instances-there certainly could be so far as logic goes: or they may repudiate any obligations towards universality; historical connexions are discovered by inner experience or by insight, and the ultimate moral data are not abstract principles but individual decisions and concrete intuitions, which since none of them are qualitatively the same can only approximately be summed up and expressed in proverb and pithy apothegm; the hos epi to polu flavour of the language of the humanities reflects the irreducible particularity (196|197) of the underlying subject-matter and the paper logic of its reasoning never succeeds in adequately plastering over the gaps between its ultimately discrete constituents: or again, thirdly, it may be concluded that the whole of humane discourse is a regrettable lapse on the part of mankind, whose proper study is mathematics and science and whose venture into these other nebulous fields is evidence of an unhealthy aversion from the clean, clear lines of the disciplines they have deserted.

A parallel from mathematics here suggests itself. There is a comparable difficulty in the idea of the direction of a smooth curve at a point; and this is because there is an incompatibility between the generality or universality implicit in the idea of a direction and the uniqueness and particularity of a particular point. Only of a straight line, that is to say, a set of a whole lot of points each one of which satisfies a somewhat simply specified condition, can we sensibly talk of the direction: and unless we are given at least two of these points, we are unable to determine what that direction is: but with a curve this is impossible; nowhere is it a straight line, and any two points on it determine not a tangent but a chord. Sometimes, as a result, we despair of the whole idea of the direction of a non-straight line at a point; the whole concept, as the contradiction patent in the phrase ``Two coincident points'' would suggest, is a contradiction in terms; only rectilinear geometry is respectable, and conics, if admitted at all, are just constructs out of the harmonic pencils or involutions which alone really exist: at other times we conclude that curves really do have tangents, and visualize the curves as ``creeping,'' going out to join their tangents and actually coinciding with them for a space; over a small length the curve and the tangent really do touch, before separating along their respective paths; in these moods we construe all curves as being ultimately rectilinear and for that reason susceptible of the concepts applicable to rectilinear figures alone; we adopt a sort of geometrical atomism; circles, did we but know it, or, rather, had we but fine enough a discrimination to be able to see it,3 are, if not chiliagons, then myriagons or something of the sort, and in virtue of this invisible and undetectible conformity to straight-line structure may properly be subjected to straight-line treatment: at other times again we despair of the geometric method but not of the subject matter; curves really do have directions, but to draw tangents and to determine their directions cannot be accomplished by the crude line-chopping machinery of straight edges and compasses or protractors; rather, we need a trained eye and a delicate touch; curves, indeed, have directions at their various points, but ``we attain a knowledge of them,'' to parody Hume,4 ``not by a chain of argument but by an immediate feeling and a finer sense.'' These three reactions all spring from our difficulty in reconciling the covert universality with the unique particularity in our specification of ``The direction of a curve at a point'': and these three reactions exactly parallel the three reactions which the specification ``Singular reason'' evokes from philosophers who are considering the humanities, who sometimes dismiss all humanist disciplines as bunk, holding that only the natural sciences give genuine knowledge: at other times they accept the humanities, but only on the proviso that they are really natural sciences after all and will ultimately be revealed and seen as such: at other times again they accept the humanities likewise, but, taking the methods of science as too crude, postulate a trained intuition and a finer faculty for the assessment of humanist truth. The problems thus are parallel: and at a deeper level also the analogy is apt, as we shall later see.

The sceptical reaction to the demand for singular generalities, whether in the humanities or in mathematics, I shall not discuss. Scepticism is usually an unprofitable subject. The intuitionist reaction is also unacceptable: the ontological and epistemological difficulties are obvious and fatal: but rather than these, I shall press other, logical, objections. The chief motive for adopting the perceptual metaphors seems to be the logical atomism of perception: all our intellectual terms are metaphors, drawn either from sense perception or from language: the linguistic ones we cannot here adopt, because they are explicitly non- particular; the perceptual ones are on this count respectable; we can see what we are unable to describe, and the fact that we have seen ``this'', felt ``this'', or tasted ``this'', does not mean that ``this'' can therefore be adequately and completely described. But this very virtue of the perceptual metaphor makes it useless for our purpose: for whereas it is possible for one facet of our sense experience to be altered while everything else remains the same, it is not possible for our evaluative judgements to be thus divorced from their associates. This has been sufficiently argued for morals by contemporary philosophers:5 it is clearly one of the features that academic historians distrust in their more inspiring contemporaries; visions from Mistra or experiences in Cnossus cannot be relevant to the study of History or the establishment of historical causes, because the mere psychological events of a historian's autobiography could occur quite independently of the historical evidence and the records of the past. Similarly, however much the mathematician's eye discerns the direction of a curve at a point, the direction of a curve at a point is not merely that which the eye of a mathematician discerns. For not every line is a tangent to a curve at a point, and what line is so varies with the curve and with the point in some, perhaps obscure, way. The logical objection, then, to intuitionism is (198|199) that it appears that cloudy visions are merely a cover for aphasia, and inarticulateness is not by itself a sufficient guarantee of truth.

The most plausible of the three reactions is the first; which solves the dilemma by accepting the generality covertly required and denying the singularity apparent in its application. It is not true, according to this view, that there is a fundamental difference between the humanities and the sciences or that numerically different human beings never are, and never can be, qualitatively the same: it is only that human beings, and correspondingly human affairs, are more complicated; the difference between the humanities and science is one of degree rather than one of kind; it is only our dishonesty or incompetence that we do not articulate any moral principles with complete fullness and exactitude; and history, ideally, should be written in the form of, and historical causes should be exposed as, sociological generalizations, that is to say, correlations between open classes of sociologically defined situations with no unexpressed qualification, limitations or restrictions, and not subject to any closure by means of an added ``ceteris paribus'' clause.

This view I reject. Partly for personal reasons of my wider outlook, a logical protestantism that I ought here to confess, though I will not here defend, which forbids me believe that an individual human being heis hekastos with his infinite variety of particular peculiarities can ever be accounted for without remainder or completely absorbed in the folds of kath'holou terms. Partly because it does not seem to save the phenomena: it is hard to conceive what a fully articulated principle would be like; certainly there could not be more than one such, since if there were two and they were logically different, then it would be possible to imagine a situation in which they were incompatible with one another, enjoining conflicting patterns of action: and the suggestion that what a historian is really trying to say, when he asserts that Rome did not fall because Hannibal did not march on it, is some open generalisation about all cities in Rome-like situations marched upon by Hannibaline armies, is repudiated by the historians concerned: the furious refusal of historians to allow their discipline to be reconstrued as a sort of sociology, which collecting particular instances advances to omnitemporal generalizations, whence we may derive adequate explanations of the past as well as accurate predictions for time to come, should be allowed to count against this thesis at least as much as did their testimony against intuitionism. And historians to-day do distrust among their contemporaries the notebook of the sociologists no less than the mouth of the seer. The thesis is also to be rejected on more central grounds, that it is based upon a false extension of the idea of a decision procedure, and that it leads to impossible results. The thesis can be restated as the claim that value words carry (199|200) with them an undertaking to justify if required. Words then like `true', `right', `valid', `cause', `consequence', would all mean `conclusively provable', and might legitimately be used only if there is a decision procedure available, whether of observation or scientific experiment, truth table analysis, or a chain of incontestable deductive steps. In moral contexts on this hypothesis, a man could properly use the words `ought', `might', or `should', only if there were some accepted book of rules, a decalogue or a code Napoleon, by means of which he could conclusively establish his correctitude. But it is not so: the thesis fails because we have, and must have, uses for words like `right' when there are no inescapable or publicly demonstrable justifications to support it. This is shown by The Argument of the judge.

Consider a judge deciding a point of law rather than a point of fact---in Roman Law, a praetor rather than an iudex. The case is a contested and a contestable one: recourse is being had to the Courts to determine and not merely to enforce the law: that is, it is not a clear case. Such cases occur. And our judges who decide them are for the most part good judges. They do not decide the cases in accordance with some bad rule- say that of deciding for the party which bribes them most-or they would be bad judges. Nor do they show their impartiality by deciding cases by the toss of a coin in court; or they would still be bad although now impartial judges. But they do not decide the case according to some good rule: else the parties would have been able to see what the decision was going to be and would have settled out of court. So good judges decide their cases neither according to any rigid rule, good or bad, nor randomly, that is accordingly to a no-rule. There is thus not an exhaustive disjunction between being in accordance with some definite rule and being completely unruly, between the conclusively justified and quite unjustified. judicial decisions are not accordance with inescapable reasons, but nonetheless are capable of being reasonable and right: not kata ton orthon logon but still meta tou orthou logou6.

It might have been objected that although we normally do use value words to express conclusions which are not conclusively justified, common usage here is wrong: we ought, it will be urged, to restrict words such as ``right'' to conclusions which are conclusively and demonstrably proven: the activities of judges are not in point; that judges ratiocinate does not prove that they are rational; and in fact they are not: what goes on in their heads is of interest to the psychologist alone, certainly not to the logician, witness to this that when they are unable validly to confute the arguments of unsuccessful counsel they have the grace to use some psychological word to express (200|201) their tentativeness ``I feel that these considerations are outweighed. . . `` they say, and this shows that their decisions are a matter not of reason but of feeling and of sense. Such an argument is powerful and attractive; but it cannot hold for considerations of continuity and in order to avoid an infinite regress. For the only reason for upsetting normal usage was that there were unclear cases where it was impossible to draw a rigid line, and difficult to know which way to decide them. This is not sufficient warrant for holding that such cases are describable only in psychological terms, for clear cases also exist in which the judge merely subsumes the details of the case under the relevant rule and gives judgement accordingly. Here at least the judges' activities must be described as rational. If judges' activities in dubious cases be not rational, then there must be a sharp and clear distinction between clear and unclear cases: for the distinction between rational and irrational, capable-of-logical assessment and not-capable-of-logical- assessment, is sharp and clear, a black-and-white distinction. But there can be no sharp and clear distinction between clear and unclear cases. Suppose there were: then we could make our legal system absolutely fair by deciding clear cases according to their obvious merits, as at present but substituting for the psychological caprice of judges the evident impartiality of the coin. But then there would be legal disputes exactly comparable to our present ones, in order to determine the new question whether a given case came wholly under some rule---yielding 100% chance of winning- --or whether it was only dubiously covered---with only a 50% chance of winning. And this problem is no different from the one it was introduced to eliminate. There is a steady gradation of cases from the certainly one way through the probably that way and the pretty evenly balanced to the probably the other way and the certainly the other way. If it is difficult to decide which shades of grey are to count as white and which as black, it will not make the problem easier to replace this question by the two questions ``Where does grey shade into white?'' and ``Where into black?''.

It is no reproach then to a judge that his reasoning is not inescapable and his arguments only tentative. The arguments he uses can still be assessed from a logical point of view, though by no logical criterion, and the law he enunciates can properly be evaluated as good (or bad, as the case may be) law. Although such judgements in law, as well as those of morals and the humanities, cannot be completely defended, that is no reason why they should not be attacked. The fineness of the discrimination does not confer incorrigibility or immunity from criticism even though in law, as opposed to morals and academic disciplines, social utility may. It is not an implicit claim to infallibility that marks the use of evaluative terms but a perpetual possibility of proving wrong. ``How'', it may be asked, ``is it possible to (201|202) prove these nicely balanced judgements wrong, when it is said not to be possible to prove them right?'' In giving an answer to this question we shall also be answering our major problem how to reconcile the particularity with the universality of moral judgement and humane reasoning.

I shall now introduce the concept of a dialogue, a conversation between two persons in which one of them is seeking to justify or to defend some judgement he has made. This is the central concept of this paper and will pervade the rest of it: but if any find the idea of a dialogue objectionable, it can be eliminated in favour of more orthodox concepts of quantification. I am not bound to use it, though I think it is suggestive: too much philosophy has been done on the assumption that language is to be monologously construed.

Consider two conversations that might follow the utterance of a judgement, whether of morality or of history or of any other humane discipline. The first dialogue exemplifies Kant's canon ``Act only on that maxim which you can at the same time will that it should become a universal law.'' Two professors, the Professor of Ethiopian Language and Culture and the Professor of Comparative Education, are talking over the port. The Professor of Ethiopian Language and Culture leads off:---

``You ought to live within 4.5 miles of Carfax, you know.''

``Why?'' says the Professor of Comparative Education.

``All professors should.''

``All professors! What about you? Bibury is more than 4.5 miles from Carfax you know.''

``Ah, when I said `All professors,' I was speaking broadly: I meant `All professors who profess soft option subjects have to live within 4.5 miles from Carfax.' Ethiopian language and culture as you are aware is an autonomous discipline with standards all its own; so the ruling does not apply to me.''

``I see, but what about the Professor of Experimental Theology? you would call that a soft option subject I suppose, yet he lives at Stanton-in-the-Vale.''

``I ought to have been more precise still. It is only those professors holding soft option chairs who are not obliged to indulge in experiments who have to live within the radius.''

``Well, then, what about Dr. Lee's Professor of Natural Philosophy, residing in Whitehall. That is a soft subject if ever there was one and he gave up experiments years ago. How are you going to deny that he ought to reside within the radius?''

``Oh, but I am not. `Ought' does not imply `does,' you know.''

``Well, then, how about your friend the Professor of Psychokinetical Friction? He is a non-experimental soft optional professor if anybody is.'' (202|203)

``I am sorry. I see I still was not exact enough; what I ought to have said was . . .'' and so on.

In this dialogue the structure is clear: every singular judgement, such as ``You ought to live within 4.5 miles of Carfax'' must be subsumed under some universal principle, some open rule; the onus is upon the proponent of the singular judgement to set forth this rule in a satisfactory and acceptable way, and it is open to his adversary, the disputant, to show that the rule as he formulates it does not cover exactly the cases required. The second dialogue which I now shall exhibit differs in that no subsumption under a universal principle is attempted and that the onus of proof is differently disposed. It might go thus: the Professor of Ethiopian Language and Culture starting:

``You ought to live within 4.5 miles of Carfax.''

``Why?'' retorts the Professor of Comparative Education. ``What about you?''

``Ah yes, but my case is different. You profess a soft option subject, whereas I don't.''

``Well, what about the Professor of Experimental Theology?''

``Ah, but his case is different too; he does experiments.''

``Well, what about . . .'' and so on.

Now to compare and contrast these two conversations: the contrast lies, as I indicated, in the placing of the onus Probandi and all that follows from that; but of this I defer my consideration until after I have examined the broad similarities.

Both dialogues might end in the capitulation of either party: not that this surrender must be one forced by logic, but it could be the case that in either dispute one of the parties might succumb to the pressure of sweet reasonableness and come round to his opponent's viewpoint. The disputant, that was, the Professor of Comparative Education, in either conversation may be convinced and may resolve to act accordingly, selling his house at Bicester and migrating to Boar's Hill; or the proponent, that was, the Professor of Ethiopian Language and Culture, might come to see that he was wrong in his original statement and withdraw his original value judgement: ``Yes, I'm sorry,'' he might say, ``you are quite within your rights to reside outside the radius, I take back all I said.'' Alternatively, if the dialogues did not end in agreement, they might end in a fundamental disagreement: the parties to the dispute might find that they each held views internally consistent but ones that were incompatible with one another. The disputant has been carrying out a series of spot tests of the proponent's position, trying it against our general, or at least his own, moral sense; and he might find that they everywhere diverge: this would have been the case if Polemarchus had claimed that it was always obligatory to return people's possessions, even (203|204) when the people are insane and the possessions are lethal weapons; in that case Polemarchus' position, though strange and perhaps repugnant, would not have been inconsistent. In these three respects, as well as in the requirements of relevancy and the ban on special pleading, the dialogues are similar.

The two dialogues differ in the onus of proof. In the first dialogue the onus is on the first speaker---the proponent; in the second upon the second speaker---the disputant. The onus of proof carries with it the possibility of proof, I mean logical proof, which I am distinguishing from the pressure of sweet reasonableness I mentioned above which can operate on both parties indifferently in both dialogues: the second dialogue is at a logical end if the second speaker, the disputant, can find a case which the proponent is proposing to treat differently from the case under consideration, but cannot show that it is significantly different; if that happens, the proponent's claim collapses on formal grounds: likewise in the first dialogue the first speaker, the proponent, can reckon to go on heos epi ti hikanon elthoi,7 until he succeeds in finding an acceptable principle to cover his case, and if he does so succeed then he has succeeded, the dialogue is at an end, for he has won. It is this difference in decisive ending that determines the different course of points attempted and points made in the two dialogues, and also, as will subsequently appear, the fact that the first dialogue is in terms of principles and subsumptions while the second is concerned with distinctions and differentiating attributes. In the course of the first dialogue the interchanges are articulated into a series of ``throws'' in which the proponent is seeking for a hold in which he will be able decisively to throw the disputant and the disputant is on the defensive trying to break any hold and avoid being thrown, whereas in the second dialogue the roles are reversed, the disputant takes the initiative and it is the proponent who has to ward off his attacks. But though it would thus seem that the first dialogue favours the proponent and the second the disputant, by a variant of Professor Popper's argument the balance of advantage is seen to lie the other way: the possibility of proof is a liability to prove; the initiative is rather a logical obligation than a logical privilege; in the first dialogue in each interchange it is the proponent who has to try and make good his claim, has to make the first move, and it is his move which is vulnerable to counter- attack: it is the principle which he is required to formulate to which the disputant may take exception: whereas in the second dialogue the separate interchanges, which are articulated in the other order, have the disputant trying to make his case and the proponent permitted to be merely on the defensive. It is an onus probandi we are assigning. If there is a draw, that is to say, if neither side is convinced and no fundamental divergence is revealed, (204|205) then according to the scheme of the first dialogue this would have to be construed as a defeat for the proponent, according to that of the second as an unspectacular win.

Although the dialogues are different the difference has not been clearly appreciated; and there has been much confusion between them when an explication of the requirement of moral consistency has been offered. It is easy to see how this confusion can arise. It seems quite acceptable to say, as Mr. Hare 7 says, that when we use a ``You ought . . .'' sentence, we are ``implying (in a loose sense) that there is some general principle that we are invoking''. But the word `some' here is ambiguous. If it will translate into Latin as quidam, if we are meaning some particular and definite principle which, though we can not cite it at the moment is the sort of one we could cite and should be able to cite after a few minutes' consideration, then it is the first dialogue which represents the correct way of justifying a value judgement. If, however, the word `some' means merely `not none', nonnihil in Latin, if all we are doing is to deny that the ``E'' proposition holds, to wit that we are acting on no principle, or unprincipledly as one might put it, then the second dialogue is the one that correctly exhibits our logical commitments. What has happened so often in ethics is that philosophers, sensing the requirement of consistency which the use of value words enjoins on us, have formulated that requirement in the shape of some law of universality which, as it stands, cannot hold. They have made an unconscious and illegitimate transition from the second dialogue to the first, and have been led by this confusion to look for an impossible rigour in ethical rules.

The moralist should not do, as Mr. Hare has sometimes urged,8 and seek ever to screw up the formulation of his principles, making them ever ``more rigorous'' until the unsatisfactory hos epi to polu feature is eliminated and they really do become major premises from which it is possible to deduce all and only those conclusions which it is desired to justify. Rather he is required only to remember, with Mr. Nowell Smith,9 that ``he is abusing language if he says it is a matter of moral principle with him to pay his debts and he pays Jones while refusing to pay Smith without being able to give any reason for the discrepancy.'' Discrepancies must be accounted for; which can be done only by showing that there was some relevant difference between the two cases. So that if there had been no relevant difference adducible then they ought to have been treated similarly. Instead of the canon which regulates the first dialogue ``In such and such circumstances one ought always to act thus and thus'' we have (205|206) the more accommodating one ``one ought always to act similarly in circumstances which are sufficiently similar or, as I might put it, which sufficiently closely resemble one another.'' In the two latter formulations ``sufficiently'' is the crucial word; for it is this that shifts the onus of proof and relieves the proponent of the obligation to specify antecedently the criteria for similarity and then to submit to attack, and gives him instead the right to wait for attacks and ward them off with criteria of similarity that he need not have already, antecedently, specified. Instead of all the obligations of antecedence which go with the burden of proof, the proponent is given all the subsequent privileges in the right to the last word.

The full force of this alteration will become clearer if we consider another pair of dialogues.

The parallel problem which we found in mathematics, that is to say the problem of determining the tangent to a curve at a point, can be solved by the method of taking limits. And I shall now argue that Weierstrass' formal explication of the process of taking a limit can be cast into a dialectical form, and that the crucial point at issue is where the onus probandi is to lie, that is to say, whether the dialogue shall be of the first or of the second form. The normal way of explaining a limit is by means of the epsilon notation and quantifiers. We say that f(x) tends to a limit f(xo) at xo if (for All delta) (there Exists an epsilon) such that if x is within epsilon of xo then f(x) is within delta of f(xo). We reject the similar explication which differs only in having the quantifiers reversed; that would give us the normally impossible requirement that (there Exists an epsilon) such that (for All delta) if x is within epsilon of xo then f(x) is within delta of f(xo). The point is that when I announce that f(x) approaches a limit f(xo) as x approaches xo, I do not mean that I can specify a degree of closeness of x to xo which will make f(x) approach f(xo) as near as you please; that is to say I do not claim that I can specify a degree of closeness of x to xo such that you cannot find objection to it. I make instead the weaker claim that if you first specify your requirement of how close f(x) is to be to f(xo), I will then find a degree of closeness of x to xo such that your requirement is satisfied. Weierstrass' definition of a limit is thus regarded as a dispute which Weierstrass will always win; but only on condition that in every interchange he has the last word.

This suggests that if only we were able to construe the connexions of humanist reasoning as functions we might be able to apply the procedure of mathematical analysis and exhibit the rules of reasoning as part of the theory of functions. This I shall now try to do. I shall be trying, that is, to demonstrate that the esprit de finesse is geometrical after all. But for various reasons it is expedient to change the mathematical model: hitherto we have been contrasting the smooth curve with the straight line---differentiability as against rectilinearity; now the contrast will be between continuous and constant functions. (206|207) The difference is not great. Both contrasts depend essentially on the Weierstrass process of taking a limit.

Let us construe as a biunique function the relation between a moral situation and right response to it; the relation of historical cause and effect and the relation between the facts of a case and the correct legal consequences of those facts can be likewise construed as biunique functions. To every moral situation there must be, then, one and only one correct response, and every response would be appropriate only in one moral situation. In order to accomplish such a construction we shall have to translate out of our ordinary terminology, because we ordinarily describe the responses to many different moral situations as being ``the same''. We normally describe situations as ``the same'' if they are similar in most important respects, and this is usually quite acceptable: but, especially with human situations, no two ever are exactly the same; individuals are never identical; between any two historical contexts there are always significant differences. Let us therefore go against normal usage and construe no two situations as being the same as one another; at the least they will differ in some unimportant respect of place or time. Every situation then is to be different; the interesting question still remaining whether they in important ways resemble one another. With this one piece of redescription accomplished, the correlation between moral situation and response can be construed as a biunique function. To every possible moral situation there will be one and only one correct response, and every response will be appropriate only in one situation; for although there will be many other similar responses, they will be different as regards time or place or something, and so will not be the response.

We may therefore legitimately construe the relation between a moral situation and its correct response as a biunique function. We are picturing the various moralities as being each one a function such that given any possible moral situation, which we take to be the argument of the function, there is determined one and only one correct response, which is the value of the given function. Likewise feeding into the function of ``Historical consequence'' the various possible historical states of affairs as ``arguments'' or ``causes'' we shall get out respective consequential ones as ``values'' or ``effects''; or again feeding in the facts of the case to a judicial officer of some legal system should evoke from him the appropriate judicial decision.

If we construe these relations as functions we then see that the formal requirements imposed by the dialectical scheme of the first and the second conversations are to be taken as restrictions upon the functions which are formally acceptable for these purposes. The second conversation, the conversation we accept, places upon all correlations which claim to be moral correlations the restriction that (207|208) if the responses to two situations are not similar, then the situations must be shown not to have been similar either: that is to say, whatever standard of similarity of response is given, it must be possible to find a standard of similarity of situation such that unless the responses according to their, the antecedently chosen, standard are similar, the situations according to their, subsequently chosen, standard are not similar, i.e. such that if the situations are similar, then the responses are similar too; which is to say, however stringent a standard of closeness of resemblance is chosen for the responses it is possible to find a standard of closeness of resemblance for the original situations such that if any situation is sufficiently close by the latter standard its response will also be sufficiently close by the former standard. And this is none other than the topological specification for a function's being continuous; for a function is continuous at a point if, however close we require its values to be to its value at that point, we can find a neighbourhood of this point in which this is everywhere so. If, however, the form of the first dialogue is preferred, we are placing upon the function the more stringent restriction that it be not merely continuous but constant. For when the order of each interchange is reversed, we are saying that the proponent shall be required to lay down some standard of closeness of resemblance which, though it may be as strict as he likes, must be such that, once he has laid it down, the disputant will be unable to find any difference whatsoever between the proponent's treatment of situations within the standard he, the proponent, has set himself. But it will be impossible for the disputant, who has the last word, to find any difference only on condition that there is no difference to find. The responses, therefore, must be the same within that degree of closeness of resemblance. This is what we should expect. For the satisfactory ending of the first dialogue was that the proponent should lay down exact criteria determining all and only those cases similar to his initial case: that is to say, he was to specify the relevant variables, and the variation of any other variable whatsoever was to leave the response unaltered, that is to say, constant. This is the aim in the sciences: there should be an exact and simple formulation which should be universally and invariably applicable-no qualification is necessary; every one that can be stated can be stated antecedently-only the specified factors are relevant, all other circumstances are irrelevant, and all situations which fulfil the specification are, from the scientist's point of view, qualitatively identical. But in the humanities this will not work, for as Plato says,10 Adunaton eu echein pros ta medepote hapla to dia pantos gignomenon haploun

The requirements of the first dialogue are too strict. It is the second (208|209) dialogue alone that will give us what we require, consistency, without exacting a price we cannot pay, constancy. Perhaps it was to this desideratum of consistency without constancy that Hegel was referring when he said: ``A living bond of the virtues, a living unity, is quite different from the unity of the concept; it does not set up a determinate virtue for determinate circumstances, but appears, even in the most variegated mixture of relations, untorn and unitary. Its external shape may be modified in infinite ways; it will never have the same shape twice. Its expression will never be able to afford a rule, since it never has the force of a universal opposed to a particular.''11

Certainly in history the shift from the first dialogue to the second manages to eliminate all the crudities of the Comte-ian approach while preserving the tenuous thread of rationality which we seek in the historical discipline.

Although the historians vehemently resist the attempt to force statements about the possibility of Rome's having fallen into the straight-jacket of ``Whenever Rome-like situation A, then fall-like situation B'' they are willing to accept the liability of demonstrating how apparently similar situations which were not followed by similar results were in fact different in some significant circumstance, which would explain the dissimilarity of consequence. Though they vehemently assert that no two historical situations are ever exactly alike, they do not in their saner moments deny that similar situations often do recur; situations, that is, which are alike in many, though not all, relevant respects, and where, if in each case a similar succession ensued, this was only to be expected, though if no further similarity occurred, this also could be explained. Indeed that so much is, simplicity at least, conceded can be seen if we consider the Strawsonian implications of many of the historian's descriptive terms: if there were no similarities between the development of the various Greek City States, how could we talk of the Age of Tyrants? Similar situations often arise: what is meant by the claim that no two situations are exactly alike is that analogies must never be treated according to the scheme of the first dialogue, with its rigid formulation of some rule or other which must always hold good, but according to the flexible method of the second dialogue, where there is no demand that criteria be antecedently laid down and no knockdown falsification by counter-example. Parallels in history can often be drawn but must never be pressed. The requirement of consistency, which is a formal requirement of all value judgements in the humanities, is then, I contend, to be (209|210) compared with the concept of continuity in topology and contrasted with that of constancy which characterizes the sciences rather than the humanities. Both in the humanities and with continuous functions in mathematics, it has to be possible, given a situation or given an argument, to find that any other situation or any other argument sufficiently closely resembling or sufficiently close to the given one has a similar response or a similar value to that of the given situation or argument, provided always that the standard of closeness of resemblance or closeness for the response or the value of the function is to be given first. In this respect the analogues are analogous, and this is what I mean when I claim that the esprit de finesse is geometrical after all.

In three other respects the analogy is not complete. For one thing, in the theory of functions we are usually given the function by the correlating formula which specifies it---e.g. y = x3--- whereas with evaluative judgements we do not, because we cannot, produce a formulation of the correlation between possible situations and our responses to them, and so it is up to the disputant to probe and discover what our response is in spot cases. In the mathematical analogue to view the limit process as a dispute may be illuminating, but actually to have a dispute would be unnecessary because, given what the function is, that is, a means of calculating its value for any argument we please, we are able to see at once in advance how the dispute must end; the dialogue is unnecessary because the conclusion is inevitable: in the moral case there really is a dispute, and must be; for we have no other means than that of asking the proponent of determining what his response to particular situations will be; to view the dispute as part of a limit process may be illuminating, but to claim that it actually is one would be misleading, since we have no means of assuring ourselves that however long it went on the proponent would win.

In this respect, then, the difference is deep and ineluctable: two further distinctions can be partially obliterated. These are the problem of closeness and relevance and the problem of special pleading.

In the mathematical model there is no problem of closeness. In the metric spaces that we naturally visualize it is intuitively given, and we can give formal accounts in terms of intervals and their size or measure: and even with the non-metric spaces, there are approaches which start with the idea of each point's having to it a ``neighbourhood'' attached. In the humanities, however, the concept of closeness of resemblance provides a great problem, in fact the whole problem, since if we could agree upon a criterion of closeness, our disputes could for the rest be settled by formal means. But though we see clearly that some differences---alterations of time and space---increase by very little if anything the distance between two cases, whereas (210|211) other differences-the addition of the word ``not'' to an utterance can cause a great separation, we have neither any rigid criteria of relevance nor any way of conflating differences of different sorts on to one scale. The absence of criteria of relevance is irremediable, but it is feasible to circumvent the problem of measuring closeness. Not all approaches to the concept of continuity begin with the idea of a ``neighbourhood''; and I shall adopt an approach which begins, instead, from the concept of an ``open class''. A function is defined as continuous at xo if, given any open class containing f(xo) as a member, there exists some open class which contains xo as a member, and such that for every one of its members x f(x) is a member of the given open class. Or, using some symbols, ``A function is said to be continuous12 ... at an element xo if for every open set V such that f(x) is a member of V, there exists an open set U such that xo is a member of U and such that the condition x is a member of U implies f(x) is a member of V.''1 Or, using more symbols, f is continuous at xo, =df. (for All V)(there Exists a U)(for All x):.f(xo) is a member of V. xo U:-->:D : x is a member of U.-->.f(x) is a member of V.{not quite correct: when symbols available in HTML, correct from handout}

That is to say, however stringent are the criteria for a response's being similar to another given one, there will always exist some open class of situations all of which will evoke responses which win, according to those criteria, be similar; or, to contrapose it into a more useful form, however stringent are the criteria for a response's being similar to another given one, if some situation evokes a response which according to those criteria is dissimilar, then there exists an open class of which this situation is not, and the original situation was, a member; or, moving from classes to qualities, if the responses according to some criteria are dissimilar, then there is some differentiating quality possessed by the one but not by the other situation ---i.e. the situations are different. This was exactly the point at issue in the second dialogue, the question of whether the proponent was being consistent or not, turning on whether, whenever the response was acknowledged to be dissimilar, the situation could be reasonably claimed to be different or not.

The difficulty of the concept of closeness of resemblance is thus cleared up and the problem of closeness and relevance is thus far solved. Here no further progress can be made, f or the criteria of relevance are not similarly susceptible of exposition. This is because questions of relevance are material questions while consistency is only a formal requirement. Thus while some of the objections the proponent may interpose in order to save his case are obviously irrelevant---``But he has got red hair''---other objections of this sort are neither trivial nor obviously irrelevant. It is a debate of great moment whether the counter ``But he has got a black skin'' is never, sometimes, or invariably, relevant. It may be felt that the absence of (211|212) criteria of relevance destroys the beauty of the test for consistency since in the absence of such criteria a person can always fulfil the formal requirements for the correct use of evaluative terms by making use of irrelevant distinctions to save his case: formal criteria, it 1 felt, are by themselves too thin: they can be bent to fit any shape This is true, and a fully fledged morality needs in addition at least; sense of relevance, if not explicit criteria. But a formal criterion by itself is none the less useful. For one thing, it may in practice be easier to obtain agreement on questions of relevance than upon questions actually of morality. For another, formal requirements, even though without material content when strictly considered, may be effectively substantial requirements; the formal requirement of consistency is like the formal maxims of the lawyer, Suum cuique, and of the scientist, Every event has a cause, which, though strictly speaking without content, nonetheless are possessed of some substantial force.

The third problem which distinguishes morals from mathematics is the problem of special pleading. This falls into two parts, both of which are already covered. The first is the case when the proponent begins to use proper names or token-reflexives to secure his case. If the Professor of Ethiopian Language and Culture were permitted to argue that ``But my case is different because I am me'', clearly the dialogue would be futile. But we consider this sort of answer already ruled out by the requirement in the definition of consistency that the class containing the original situation as a member be an open class. And however unexplained the idea of an open class may be, it can at least be shown that classes consisting of a finite number of named individuals are closed classes. I am permitted, so far as the ban on special pleading goes, to claim that your case is different from mine because you are not a Fellow of Merton and I am; but I cannot claim that because you are not one of ``Garrod, Lambert, Levens . . .'' and I am, therefore our cases may be treated differently. The second type of special pleading is that where though no stonewall answer of the ``But I am me and you are not'' form is actually articulated, this principle is in effect adopted and differentiating circumstances are produced upon this principle ad infinitum. It always is possible for a person to point out some difference between a pair of similar instances, however much alike they are-identity of indiscernibles and so a class can be refined down by the addition of ever further qualifications to as near as you please to a unit class which can (logically can) have only one member; and it is possible to regard such a unit class as the limit of such a process. If a person were conducting his defence on the principle that he would allow as similar only those cases which resembled the given one in. every conceivable (instead of in every significant) circumstance, then he would be guilty of special (212|213) pleading. Clearly, the case is already covered by the ban on irrelevance, since it must needs utilize the non-significant but conceivable differences of circumstance to save itself. It is elegant, however, in order to help outlaw this type of special pleading to add this consideration: that although no term was set to the dialogue, it must not be interminable: the dialogues are infinite in the sense that they may be as long as we please, but not in the sense that they are so constructed that they could have no ending. And a dialogue which is to exploit to the full the Identity of Indiscernibles is so constructed, and the offence of special pleading is for this additional reason indictable. Clearly such a charge cannot be brought conclusively home on any finite length of dialogue. But if the proponent confessed the principle upon which he was acting, if he gave us the rule whereby his answers were to be determined, which guaranteed that he would never cease from making distinctions, then he would be conclusively discredited. We cannot formally condemn an arguer for special pleading on his record of argumentation, however long and black it be: but if to the question ``Are there any circumstances at all which you would ever allow as sufficiently similar?'' he were to answer ``No'' then we could convict him of Special Pleading. Both types of Special Pleading are in any case ruled out by previous restrictions, and for this, the most common type and most common cause of irrelevancy, we have this additional safeguard and check.

It has been the aim of this paper not to produce a calculus of morals after Locke's prescription13 but to exhibit one of the features of moral argument which is formal, but which explains how the requirements of universality and particularity can be reconciled. This, which was impossible in ordinary ``flat'' discourse, can be achieved if we are allowed to go out into the dialogue form; or to put it another way, if we are allowed to use that curious quantificational scheme based upon a core of two universal quantifications separated by a negation, which, as Mr. Hampshire14 has noticed, possesses the peculiarity of having no substantial content. Whichever way we have it, we have had to go outside the plane of simple discourse. And there, by a simple reversal in the natural dialectic, we discover that the most stringent requirements of logic are not to straight-jacket the mind along unnatural lines, tou gar aoristou aoristos kai ho kanwn estin hwsper kai tes Lesbias oikodomias ho molubdinos kanwn,15 finitude cannot measure what cannot be confined, and limitations of consistency are to be construed not as a rigid regulus but as a Lesbian rule.


1. A Paper read to the Oxford Philosophical Society on Friday, 12th, November, 1954.

2. Republic IV, 425 b7-427 a7.

3. Plutarch, Quaest.Plat. v.2.3.

4. Enquiry Concerning Morals, Sect. I. ed. Selby- Bigge Section I34.

5. R. M. Hare: Language of Morals: pp. I29-133.

6. Aristotle: Nicomachean Ethics, Bk. VI:I3:5, 1144b.26-7 ``Not according to the right rule but still with right reason.''

6. Phaedo, 101 e 1, amended, and slightly out of context.

7. R. M. Hare: Language of Morals, p. 156.

8. 2 R. M. Hare: Language of Morals, p. 52.

9. P. H. Nowell-Smith: Ethics, Harmondsworth (Penguin), 1954, p. 309.

10. Politicus 294 c7-8 ``What is altogether simple cannot fit well things that are never simple.''

11. 1 Hegel: The Spirit of Christianity, Section 3 p. 295. tr. Hegel's Early Theological Writings, p. 246. I am indebted to my colleague, Mr. W. H. Walsh, Fellow of Merton, for bringing this passage to my notice.

12. General Topology: by Waclaw Sierpinksi. tr. Toronto I934 (1st edition only), p. 16, Section 10.

13. Essay on the Human Understanding. Bk. IV: ch. 3: Section 18.

14. S. N. Hampshire: Analysis 1950. Multiply General Propositions.

15. Aristotle: Nicomachean Ethics. Bk. V: IO: 7, II37 b20-31.

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