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- Shalom & Welcome! - 
The initial transition from direct experience to language is, clearly, effected by means of ostensive definition.
"That!"1 provides the nearest possible approach to denotation without connotation - the minimum of egocentricity.
1. Russell in Chapter 7 of his Inquiry into Meaning and Truth contrasts "This is" with "That was". This correctly interprets the temporal separation between sensation and memory in private experience. However, communication always involves spatial separation, so "that" is correct here. It can, of course, be replaced by an indicative gesture.
We will proceed to examine the necessary characteristics of "thatness", i.e. what is necessary to provide the assurance of perceptual acceptance.
These involve:
Condition (b) implies that an æsthetic factor is operative, since interest and importance are concerned with actual or potential improvement or deterioration in the quality of experience.
Condition (c) implies that change itself - the continuing flux which 'shows' itself in conscious experience - can never be ostensively defined, although such definition is possible for a state of change.
In any case, it is not possible to furnish a verbal or symbolic representation descriptive of change, since the whole virtue of word and symbol, for communication purposes, is that they can be assumed to denote an invariant!
The occurrence of change can be indicated, as for example in a mathematical equation, 2 + 3 = 5 shows such a passage from state to state: what is first thought of as a parity and a trio is later considered as a quintet. It is the "=", not the "+" which signals the change; this shows clearly that it is not an attempt to describe the particular changes in question, since the same sign is used in all mathematical equations.
The continuing identity of the symbolically undiscerned is a necessary presupposition for the logical use of word and symbol. The undiscerned is not, however, the indiscernible. Discernment only occurs where there is felt to be a need for it - the principle of Occam's Razor is generally applicable here.
Thus a pillar box and a peony will both usually be described, correctly, as "red", even though in some contexts their colours may need to be discerned as "scarlet" and "crimson" respectively.
"Indiscernibles" is in fact nonsense, since the use of the plural shows that discernment has in fact occurred. all"universals" are singular.
Since it is impossible to symbolise the actual process of change which is inherent in all conscious experience, it follows that there can never be complete correspondence between direct experience and our attempts to express it symbolically.
The "correspondence" invoked by the supporters of the correspondence theory of truth is not, in fact, between experience and symbolic expression at all. What seems to have been generally overlooked is that symbolic expression has to be retranslated into direct experience before it becomes "meaningful".
A word or proposition is only meaningful inasmuch as it stimulates a memory and/or an expectation. These constitute its meaning, which must be sought, not within the language itself, but in direct experience.
A word is in fact the symbolic representation of a Humeian 'idea'. The equivalent of Hume's principle "no idea without an antecedent impression" is "no meaning without an antecedent ostensive definition". The conditions necessary for ostensive definition are those which make sure that the "impression" does actually generate an "idea".
In contexts where an expectation is involved, the word becomes a sign, but not otherwise. The written word "fire" is not a sign; the exclamation "Fire!" is.
Language, and symbolic representation in general, is in fact an attempt to communicate direct experience by symbolising its persistent elements relying on the interpreter to reinsert the element of change during the re-translation process in such a way as approximately to recreate the experience symbolised.
Under what circumstances can we then say that a statement is true?
Tarski has established that for logical and semantic purposes the statement 'snow is white' is true if, and only if, snow is in fact white. But when is snow white?
We must assume that "snow" and "white" have been ostensively defined. If the memory or expectation produced by the statement "snow is white" is not discordant with the feelings produced by the direct experience of seeing snow, or with the memories of former occasions when we have seen snow, we believe that snow is, in fact, white. This belief is provisional, since future experiences of seeing snow may produce feelings which clash with the expectations aroused by the statement, but we tend to retain it unless it is refuted in this way.
Belief is in fact a feeling of harmony between the memories and expectations evoked by the statement and those produced, either by other memories and experiences, or by other statements. In the first case we have correspondence, in the second, coherence.
Correspondence and coherence alike are æsthetic assessments of feeling on the part of the interpreter. In cases where the verbal and behavioural reactions of all the interpreters of a symbolic statement fail to reveal any sign of internal discord (i.e. doubt), the statement is said to be generally true. But the truth is not inherent in the statement, but in the unanimous belief in the merit of what is, presumably, its uniform interpretation, when this is combined with memories of experiences or other statements which are believed to be true.
It is, I suppose, just possible that a general belief in the truth of a statement might arise in spite of differences in its experiential interpretation, but in view of the mass of experiential material with which every interpretation has to harmonise, this is extremely unlikely. We are pretty safe in assuming that a statement which is generally accepted as true is always being interpreted in identical fashion.
What are the conditions necessary for such uniform interpretation? Clearly, they are the same as for ostensive definition. The more effective the definition, the simpler will be the task of retranslation. Uniformity of interpretation can be fairly relied on when the entity denoted by a word is (a) clear and distinct, (b) of general importance and (c) persists relatively unchanged.
The paradigm which best satisfies these conditions is the standard solid. Its fulfilment of (a) and (c) is obvious; solid objects are also of obvious importance from early infancy as manipulation (favourable) and as obstacles to motion (generally unfavourable). They also provide us with perceptual assurance, since they are distinguished by the quasi-simultaneous provision of at least three types of sense - visual, tactile and kinæsthetic. By kinæsthetic I mean the sense of motion and orientation of which we are usually unconscious because it is always with us - it is controlled by the semi-circular canals of the ear.
The primary properties of standard solids - shape, size, position and inertial qualities - are denotable with the maximum likelihood of uniform interpretation. It would also appear that smoothness (visuo-tactile) and symmetry (purely visual) are of direct æsthetic interest and importance to all human beings, except perhaps in a few very primitive societies. We all reckon to have an idea of what is meant by a 'straight line' or a 'circle', even if we are unable to define it or even to exemplify it exactly.
This is the proper field of mathematics and symbolic logic, where the entities discussed and symbolised though frequently undefined, preserve their identities unchanged, and can also be considered to preserve their mutual independence. 2 + 3 = 5 is only true if these conditions are observed; the 'numerical similarity' or one-one correspondence, used by Russell in his definition of 'number' - also requires these conditions for the members of his 'classes' if the number of any class is ever to be considered constant.
Mathematics will clearly be applicable to the basic data of physical science, which consist of pointer-readings, marks on photographic plates or other persistent states of standard solids. Even time is measured by a quasi-stationary pointer reading (actually stationary if a stop-watch is used) obtained by counting repetitions of a motion which is considered to be rhythmic.2
2 The constancy of a rhythm must in the last resort be considered intuitive - there is no final court of appeal as to the rhythmic character of a repetitive phenomenon.
Communication in the whole of this field is pragmatically speaking, outstandingly successful. Since general scientific principles are mainly used for forward planning, their truth is mostly assessed by the correspondence of expectation with direct experience.
Improvements in symbolic expression, however, can sometimes produce an enormous increase in the degree of coherence between the symbolic formulations of natural 'laws' which have hitherto been thought of as unconnected. Newton's association of the fall of the apple with planetary motion, and Clerk Maxwell's electromagnetic interpretation of light are obvious cases in point. Such instances result in a euphoric intensification of intellectual harmony, and the inventor is apt to feel that he has made contact with 'absolute' truth. The euphoria induced by the Greek's geometrical achievements is the obvious source of Socratic idealism, and even in Kant's day Newtonian physics and Euclidean geometry were still considered 'absolutely' true.
The truth of statements about past events is mainly assessed on the basis of coherence, but as a rule some of the related statements will give rise to expectations of finding standard solids, which add an element of correspondence. Such finds constitute the only 'solid' evidence for or against the statements.
The 'truths' of fiction, which are intended to stimulate memory but not expectation, are 'true to life' in the sense of coherence alone. If by some mischance they do stimulate expectations, the result may be an action for libel!
Inventions which, like fictions, result from new arrangements of symbolic materials, arouse expectations of experiences which could result from the corresponding arrangement of material entities denoted by the symbols. The 'truth' of the new formula is therefore judged entirely by correspondence. This is the proper field of the 'instrumentalist' theory of truth.
As soon as we leave the field of standard solids, we can no longer rely on the unanimous interpretation of our words and symbols. Consider the famous saying of Heraclitus "You cannot step twice into the same river, for fresh waters are ever flowing in upon you". At first sight this appears to be obviously true, but it cease to be so if we define the river in terms of whatever water lies between the quasi-permanent solid banks - as is done, for example, on a map. The word 'river' has several connotations, so that its constant interpretation can no longer be relied upon. No statement dealing with anything other than the primary properties of standard solids, and their states of change, can lay any claim to general truth. Nevertheless, behavioural tests may sometimes show that the vaguest type of statement can succeed in communicating the feelings of its author, at least to a limited number of interpreters.
An extreme case of this kind is provided by music, a symbolic form of communication without any indicative content whatsoever. A composer imagines certain sound-sequences, which for him symbolise certain changes of mood and feeling. He writes down symbols representing persistent elements of the imagined sound-sequences for the purpose of communication. The performer retranslates them into sound, and it can happen that the composer, listening to the performance, will go so far as to admit that the reinterpretation symbolises his feelings better than the sound that he originally imagined!
In such cases we speak of a great sympathy, i.e. likeness of feeling, between the composer and the interpreter. It is, of course, unusual for communication to attain such a pitch of efficiency in this type of field. A composer may reckon himself lucky if he finds more than a handful of interpreters who can produce a thoroughly satisfactory performance of his work.
At the other end of the scale, where what is communicated is memories of standard solids and their properties, everyone can be relied upon to effect the same interpretation. Between these two extremes there is a vast field where identity of interpretation is uncertain; but on the whole communication is efficient at elementary levels, so that failure is probably to be attributed to accidental differences of connotation rather than to failure of sympathy.
'Analytic' Truths
I have so far written as though there were no 'absolute' truths. I believe this must be correct in all cases which involve the double translation from experience to language and back again.
But there are absolute truths, which have provided the examples which we attempt to imitate in our construction of the quasi-truths of correspondence.
These are purely internal to language, and apply to its indicative but not to its poetic use. They result from the way in which we multiply our concepts in accordance with the æsthetic and practical3 necessities which arise in dealing with direct experience.
3. All "practical" necessities are ultimately æsthetic in origin.
Analytic truths arise whenever a wide, vague concept is dissected into narrower, more precise concepts for æsthetic or functional reasons. A positive analytic relation then becomes established between the word denoting the parent concept and that denoting each of the sub-concepts, and a negative one between the words denoting the sub-concepts themselves.
The recognition, but not necessarily the naming, of the parent concept must temporally precede the naming of the sub-concepts.
These relations remain absolutely true as long as the denotation of the word for the parent-concept is not curtailed; as this sometimes occurs, they cannot be regarded as "eternal" truths.
As soon as linguistic communication begins, verbal propositions become available to supplement or replace pre-verbal sign-situations, enabling us to juxtapose memories of signs and their expected interpretants.
Now the primary object of indicative language is to avoid 'no'-feeling. They also indirectly confirm the positive analytic relation between the words for concepts and those for sub-concepts derived from them, since they would condemn any attempt to negate it.
The clearest example of this principle is provided by colour-concepts. In this case there is no possibility of confusing classification by appearance with classification by function, since appearance and function are identical.
A pillar box and a peony can both correctly be described as 'red', even after the description 'scarlet' has been applied to the pillar-box, and 'crimson' to the peony.
The sub-concepts have been dissected by contrast out of the main concept for functional reasons - here obviously æsthetic.
The resulting analytic truths are as follows:- "If anything can be described as scarlet (or crimson) it can be described as 'red' ", and "If anything can be described as scarlet it cannot be described as crimson (or vice versâ".
Colour contrasts are functionally interesting long before æsthetic notice is taken of the colours themselves, since they define the boundaries of the images which are the signs for the universally important standard solids. It so happens that in nature certain contrasting shades predominate, which come well within the range of the elementary colour-words 'white', 'grey', 'black', 'brown', 'red', 'yellow', 'green' and 'blue'.4 These can therefore be easily dissected from the basic concept 'colour' and can be ostensively defined with confidence. Marginal shades, which occur less frequently, will initially be ignored.
4. This particular list may not apply in regions where climate and vegetation differ from ours.
We must note that the word 'colour' will probably be learned later than those denoting its sub-concepts; but the wider concept must be familiar before the sub-concepts can be ostensively defined. The answer to the ostensive question "what is that?" will certainly not be a colour-word unless attention is specifically drawn to that particular aspect. The main concept must have been recognised before the sub-concepts can be named, but is initially of subordinate importance, since the main interest lies in the contrasts between the sub-concepts.
The statement 'If anything can be described as red it cannot simultaneously be described as green' has not generally been considered to be an analytic truth, since the names of positive attributes have generally been considered to originate independently from the recognition of similarity in their individual exemplications. In such a case there could be no contradiction between them.
Yet we all feel that the statement is necessarily true, and know that the attempt to imagine a "red-green sense datum" is going to result in a "no"-feeling. This becomes readily intelligible if the original definition of the words "red" and "green" was accomplished by the method of contrast.*
* Editor's note: It is widely agreed both that many theories of perception are false and that the language employed by epistemologists is frequently fantastic, but there is also a danger that Solomon's more than fully justified onslaught on a confused and often sterile tradition may blind us to any number of valuable insights its originators were struggling valiantly to express. Readers unfamiliar with the 'green'-'red' question are initially referred to Friedrich Waissmann's The Principles of Linguistic Philosophy (Macmillan, 1965); I hope to explore this area somewhat further at a later date.
On the other hand "a red-orange sense datum" does not produce a strong feeling of negation, since we can imagine an intermediate shade which could not be described by either word with complete assurance. We can pass gradually from red to orange without traversing a field denoted by any other colour-word. Indeed, before the fruit became generally familiar, it is probable that some shades of orange were described as 'red' and others as 'yellow', just as the vernacular name for purple fluorspar is 'bluejohn'. Before the word 'orange' came into general use the negation beween 'red' and 'yellow' must have been weak, which is no longer the case.
In many cases the sub-division of a broad concept is possible either on a basis of appearance or of function. It is invariably functional differences which have given rise to new names for the sub-concepts.
Take, for example, the parent-concept denoted by 'horse'. Here we have two types of functional sub-division, neither of which curtails the denotation of the original word. On the one hand we have 'stallion', 'gelding', 'mare', 'colt' and 'filly'; on the other hand, 'carthorse', 'racehorse', 'hack', etc. 'Pony' might be considered an exception to our rule, since it simply denotes a small horse, but smallness, in this instance, involves quite important modifications of function.
Each sub-concept asserts the compresence of the parent-concept 'horse' with an additional function, e.g. in the case of 'stallion', active male function. The contradictions between 'colt' and 'stallion' and between 'filly' and 'mare' are weak; the others are strong; should a hermaphrodite horse ever make an appearance, we would certainly find a new name for it!
Words which merely denote appearance remain as separate adjectives. 'White horse' denotes the compresence of white appearance and horse function. When applied to the horse-images on the chalk hills, it necessitates the presence of horse-appearance alone. If applied to the foam of breakers, its use is poetic, and if used as a name, as for whisky, 'White Horse' only necessitates the presence of the word 'Horse' with a capital H.
Generally speaking, it is only functional differences that are important enough to give rise to new words; the principle of Occam's Razor appears to be firmly implanted in human nature.
Indeed, in a case where appearance may be the principal function, the denotation of a functionally defined word may be extended to cover objects of similar appearance, as in the case of 'artificial flowers'. In this case there ceases to be an anlytic relation between 'artificial flowers' and 'flowers', since the original denotation of the latter has been curtailed, removing the botanical function and leaving merely the appearance (cf. 'White Horse', supra).
New words are seldom absolutely necessary, but they are obviously convenient when they arise from a functional difference which can only be clumsily expressed, and which is of widespread importance. They tend to multply with each increase in human discrimination - as already exemplified in the case of colour-words. Cross-classification, the sub-division of a major concept in respect of two or more functions, may render them indispensable as abbreviated code-words for the more important sub-concepts resulting from this process.
The end-point of this procedure is reached with the Proper Name, which is officially given only to persistent objects which uniquely fulfil a variety of functions important to many people.
The denotation of such an object by a combination of function-words would be impractically long; for example, a person's name would have to be replaced by a greatly amplified account of the kind to be found in Who's Who, which is a very brief summary of a few of the most important contexts in which the person named has been functionally unique up to the time of compilation.
The above account of concept-formation tacitly assumes that our earliest concepts are extremely wide and vague. The atomic 'sense'data' required by epistemologies such as Russell's may eventually form part of our acquaintance, but only as the result of a lengthy process of dissection out of an experience which begins, in infancy, as an integrated chaos, more akin to the deliverance of hearing or even, as Whitehead suggests, of visceral sensations than to that of vision or touch.
William James called it a 'blooming, buzzing confusion', but this is a misnomer, describing how it would probably appear to an adult. To the child, it is simply inchoate until it has been differentiated.
Concept-formation begins as soon as the infant begins to interpret certain repetitive elements of conscious experience as signs indicative of future possibilities - usually an improvement in the state of affairs. Concept-differentiation can then proceed by conjecture-refutation, verbally expressible by a syllogism in modus tollens, viz. "The appearance A which I (conjecturally) took as a sign for the imminence of function a (generally 'nice') has been followed by an interpretant which was decidedly not- a (generally 'nasty'); I must therefore either reject A altogether as a sign for a, or find out how the genuine sign differs from A".
If the significant difference is discovered, the outcome will be the dissection of a sub-concept from A. The nasty 'not-a' may also be conceptualised as a possible interpretant of the original A.
Thus every new concept originates in a negation.5 This is why we are so addicted to syllogistic logic, which is valid whenever the word-relations are genuinely analytic.
5. This generalisation does not remain valid after verbal habits are well-established.
We constantly extend it to pseudo-analytic propositions, but this extension, although often useful, is not valid.
A major premise of categorical form, such as "all men are mortal" asserts the existence of an analytic relation between the concept words "man" and mortal"; this is nonsense, since the concepts originated independently of one another.
In the conditional form "If all men are mortal", it means "let's pretend that the relation is analytic". Such a pretence is very useful since it enables us to apply syllogistic reasoning to the quasi-analytic connections provided by empirical generalisations to which there is no known exception.
For example, the names in the Telephone Directory form a synthetic "class"; the names existed long before the "class" was created. It is arranged so that it can confidently be divided by progressive differences into sub-classes and eventually into individual items, in accordance with a conventional (alphabetical) order which makes it possible to find the entry required.
There is always the possibility of a slip in compilation - the relation between the pseudo-parent class, sub-classes and single entries is entirely pseudo-analytic - but it suits our propensity to analytical methods of thought, which are those which served to effect our concept-function in early childhood, so that we willingly accept the pretence that it is genuinely analytic.
Much error, along with much highly undesirable dogmatism, results from our frequently faulty diagnosis of quasi-analytic relations as genuinely so. Logical inference is the model we like to use for the kind of implication involved in empirical investigation, with synthetic concepts as raw material; but the two should never be identified.
The above account of analytic process owes much to Prof. Sir K.R. Popper. It is simply an extension to learning at its most primitive level of the conjecture-refutation theory which he has so successfully applied to the methodology of science.6 Every sign-system constitutes a conjecture, which is refuted as soon as the coincidence of expectation and experience results in an unpleasant 'no'-feeling.
6. In The Logic of Scientific Discovery and Conjectures and Refutations.
Like him, I have followed Hume in rejecting induction as a method of reasoning; but memory is undoubtedly strengthened by repetion, so that habitual memories are influential in framing conjectural sign-systems which may help to solve the infant's problems.
Any similarity which stimulates the memory is likely to be seized upon for this purpose as better than nothing at all!
The problems themselves are all of æsthetic origin.
This interpretation removes the need to ascribe any supra-empirical element to logic; all we need to accept is the universal occurrence of two propensities in every child. The first is the propensity to develop likes and dislikes. The second is the tendency to form sign-systems - to interpret certain elements of present experience as a guide to future possibilities - with a view to satisfying its likes and avoiding its dislikes.
I find it difficult to believe that Locke himself would have challenged these reservations to his tabula rasa theory of mind. Their acceptance is implicit in every mother's treatment of her child; and even of every animal-trainer's treatment of his subjects. We retain these propensities throughout our lives, and I can see no objection to regarding them as empirically given.
I hope I have succeeded in showing that they are sufficient to account for the whole of the a priori element in human thought processes. Couple them with the propensity to communicate, and the development of our verbal and symbolic methods of representation is readily explained.
Belief, as Hume claimed, is in fact a part of the sensitive side of our natures. There is indeed no need to postulate the existence of a separate cognitive side - this is simply the name we apply to our attempts at communication by way of word and symbol, and its functioning can be satisfactorily explained in terms of our natural and apparently inescapable propensities.
Conclusion
In the final chapter of his Inquiry into Meaning and Truth, Russell divided philosophies into three main groups:
If my diagnosis of the relationship of language to experience is correct, we can see quite clearly how much justification there is for each of these points of view.
The first group is correct in so far as the properties of the world can be expressed in terms of the unchanging.
The second group is correct in so far as "knowledge" is supposed to be acquaintance with 'absolute' truth, which is internal to word- and symbol-systems.
The third group is correct in that pure change, an inherent element in all conscious experience, cannot be symbolically described, though we can use language to point to its occurrence.
Wittgenstein, however, wrongly identified the "ineffable" with the "mystical" - a word which is only applied to certain exceptional types of experience.
Bergson saw quite rightly that the intellect, attempting to express direct experience in terms of unchanging symbols, was bound to remove the vital element of change, but failed to notice that this ingredient is always reinserted during their re-interpretation; after all, the physicist who plots his isolated, static observations as 'points' on graph-paper joins them to make a smooth, continuous curve, and the cinema-goer insists on making a continuum out of what he knows to be a discontinuous succession of 'stills'.
This failure led him to regard the operations of the intellect as fundamentally opposed to those of intuition. His extrapolation of this supposed dialectical opposition to world-process as a whole made his philosopy an easy target for ridicule, and obscured the value of his original insight.
The relationship between words and experience is in fact closely comparable to that between dessiccated coffee and the actual beverage. Just as coffee powder is an intermediary between the original brew and its 'instant' equivalent, so language is an intermediary between the experience expressed and its experiential reinterpretation.
Enjoyment - that is, value - is obtainable from the original drink and from its reconstituted substitute, but not from the powder. The latter is easily transportable and will keep almost indefinitely, but will only provide a reproduction of the more elemental qualities of the original brew; all the aromatic subtleties vanish, along with the water, during the dessiccation process.
In similar fashion, indicative language preserves and helps to communicate the cruder, more persistent features of direct experience, minus the fluid element of change. The subtleties of experience actually fare rather better in reproduction than does the aroma of coffee, since some interpreters have sufficient expertise - i.e. sympathy with the author of the language - to reinsert some of them, whereas the 'interpreter' of coffee-powder is not likely to reinsert anything but water.
The symbolising intellect can now be seen to be the imperfect servant of intuition, helping to preserve and communicate the more significant parts of direct, intuitive experience. But if the words and symbols used are expected to be amenable to logical treatment, their use must be confined to the material field, where their unanimous interpretation can be relied upon.
The poetic use of language - the attempt to communicate subtleties of feeling by directly evoking similar feelings in the listener or reader - is more akin to the technique of music than to that of indicative language. Nevertheless, the overwhelming consensus of informed opinion as to what constitute the major achievements in the arts, music, literature and higher mathematics, suggests that symbolic communication in these fields must be quite successful. The concept of "truth" is hardly applicable here - even in higher mathematics, where æsthetic considerations play quite a large part in assessing the merits of a "proof".
The feelings communicated at this level are almost universally considered more valuable than those associated with the verification of propositions symbolising factual "knowledge", with the exception, already mentioned, of those which accompany the discovery of an enormous extension of coherence.
Truth is, for the most part, only a low-class representative of Beauty; it is of great value on account of its generality, but the very property which renders it so useful practically confines its application to human cognition at the animal level, where uniform interpretation of word and symbol can be relied* upon even from the most primitive, unsubtle members of the human species.
* Editor's note: So reads the manuscript; the typescript, however, has "relief" which the author has not corrected.
(a) The Arrow
Zeno's version: "At every instant, the arrow must be somewhere. Therefore it can never be in transit from one point to another, therefore motion is unreal". Russell's amendment, preserving the "reality" of motion: "Motion consists of the fact that the arrow is first at one point and then at another". This is coupled with the proviso, derived from the work of Dedekind and Cantor, that there is never a 'next' point.
Bergson's "refutation": "The moving arrow never is anywhere". This is not altogether convincing if motion is relative, since it can be at rest in terms of one set of coordinates and in motion in terms of others.
The genuine refutation is provided by the circumstance that only the persistent can be expressed in word or symbol. The arrow, whether at rest or in motion, can never be said to 'be' at an instant.
To 'be' at a point is meaningful, since states of relative rest in space are part of everyday experience. A state of rest in time, however, forms no part of our experience and is quite unimaginable.
(b) The Achilles
Zeno's version: "Achilles, who runs twice as fast as the tortoise, gives the latter a start of a mile. When Achilles has run a mile, the tortoise will be half a mile ahead; when he has run another half mile, the tortoise will be one-quarter of a mile ahead … and so ad infinitum"; the conclusion being that Achilles can never overtake the tortoise.
In the above form, this is not a paradox at all.
If both Achilles and the tortoise halve their speeds after Achilles has run one mile, and again after he has run the next half-mile, after the subsequent quarter-mile and so on ad infinitum, the original conditions will still obtain, and Achilles in fact will never overtake the tortoise.
If, however, we specify that their respective speeds remain uniform - which is generally taken for granted - we can restate the puzzle, giving the start in terms of time - say, one minute. We then obtain the following:-
"When Achilles has run for one minute, the tortoise will be half a minute ahead; when he has run for another half a minute, the tortoise will be a quarter of a minute ahead …" and so on.
We can now see that the puzzle depends on assuming the possibility that two minutes will never elapse. This is, of course, quite ridiculous. Achilles and the tortoise may have some control over their velocity in space, but none over their 'velocity in time', which is quite independent of anything that they can do!
Zeno's rejection of the reality of time is seen to be the rejection of anything that cannot be expressed in word or symbol. The Parmenidean school, followed by the Platonic Socrates, were quite understandably inebriated by the expressive triumpths of the Greek language and of Greek geometry. In our own day Russell, though he made more concessions to commonsense, was inclined to regard the symbolic descriptions of the physicists as more 'real' than direct experience.
(c) The Liar
"The statement I am now making is false" - or words to that effect. We have seen that the falsehood of a statement is defined by its evocation of "no"-feelings when its experiential interpretation is combined with our other beliefs and/or experience.
Now "The statement I am now making" is not a statement. When it is used in practice as part of a sentence, as for example in "The statement I am now making is true", it is always the name of a statement; the "now" always covers a period extending beyond the actual utterance.
Considered in isolation, it is clear that our original sentence says nothing at all. The only memories it can evoke are exactly the same as those evoked by the words "False statement", which is certainly meaningful, but cannot be true or false.
There is a simple rule for determining whether a self-referring expression is well-formed, and possesses genuine content which would make it possible to assign to it a truth-value. It can be stated as follows:-
This would appear to be obvious commonsense; it can nevertheless be shown that the classical paradoxes of logic, such as the 'Liar' and 'Class of classes which are not members of themselves' have only persisted because the implications of this rule have been entirely disregarded.
A paradox, being self-contradictory, has to be negative in form. What I propose to demonstrate is that its positive counterpart, which can itself give rise to no paradoxes, is either ill-formed or vacuous when used for purposes of self-reference.
Let us first consider the case of the man who says 'I am lying', 'This statement is false', or some combination of sentences which adds up to the same thing.
The positive counterparts here are 'I am telling the truth' and 'This statement is true', both of which are in everyday use.
But truth is an external relation between a sentence and something else, and not an inherent property of a sentence. 'I am telling the truth' is therefore ill-formed, unless the context provides it with an external reference - which in actual practice it always does. It has the same linguistic status as 'I cannot', 'you seem', 'he believes', which are incomplete and without real content unless their context supplies them with implied objects or complements.
The status of 'This statement is true' is similar. In this case the position is demonstrated effectively by the use of Tarski's truth formula, which is a simple and clear statement of the alio-relative nature of truth.
We obtain -
Now 'this statement', being a phrase and not a sentence, can be neither true nor false. This circumstance alone is strong prima facie evidence in favour of the view that no truth-value can be obtained for 'this statement is true'.
Since however, the assertion is in common use, we can modify Tarski's formula slightly so that it does express the meaning of the expression as used in actual practice, by making it clear that 'this statement' refers to its definiendum, which is a sentence, and can therefore be true or false.
We thus obtain:-
This is satisfactory. Substitute 'snow is white' for X, and we obtain:-
In the case of self-reference, this becomes clearly circular:
It is now obvious that no truth-value can be obtained for 'this statement is true', taken by itself, and that it must therefore be condemned as vacuous.
It is interesting to note that 'This sentence is meaningful' yields a completely different result, although apparently similar in form. Applying the Tarski formula, we obtain:-
Here no substitution is needed, although 'meaningful' is an alio-relation, because there is already an implied alio-relation between 'this sentence' and its definiendum; a relation which can be described as 'meaningful'.
This produces an apparent paradox in the case of 'this sentence is meaningless', but only because 'meaningless' can apply both to 'this sentence' and to 'this sentence is meaningless'. But if the epithetic 'meaningless' is applied to 'this sentence', then the latter cannot stand for 'this sentence is meaningless', since this is a sentence, so that the substitution would render 'this sentence' meaningful.
"This sentence is meaningless" is therefore false if considered as self-referring.
Russell's notorious paradox 'the class of all classes which are not members of themselves' originates in much the same way as the 'Liar'.
The fact is that 'a class which is a member of itself' is complete nonsense in the English language, since 'membership' is always understood to be an external relation as far as the member is concerned. No valid exemplification of such nonsense is possible.
The only attempt I have come across is Prof. Ayer's 'Class of things which can be counted'. This is clearly unsatisfactory, since counting is a process to which no end can be imagined, and so is the creation, or discovery, of countable 'things'. The resulting 'class' can therefore never be completed, and cannot be counted.
If we attempt to remedy this defect by imposing a deadline on the counting, then the class will not be completed until the deadline is reached, and consequently its own counting will have to take place beyond the deadline, so that it will not be a member of itself.
It should be an easy matter to find similar objections to any attempt at exemplifying 'classes which are members of themselves'.
In conclusion, I hope that the simple rule stated at the beginning of this essay will be adequate to dispose of much vacuous nonsense, including most of the classical paradoxes, and that it will serve to distinguish paradoxes from pseudo-paradoxes. High-powered logical methods should not be needed for the solution of problems which are fundamentally silly.
What is the purpose of making informatory statements?
It is, clearly, quite different from that which underlies the primtive use of language, which we first encounter in infancy. Here it is used to communicate commands, requests and warnings, with the expectation of immediate or proximate responses; it is for this purpose that it is uttered. A child quickly learns to interpret it as an expression of the utterer's desires, and its response is usually intended to elicit signs of their satisfaction, since children remain relatively helpless and dependent on their parents for a long time after they learn to speak, and therefore seek to please them, if only from motives of fear.
Now the purpose of uttering informatory statements can likewise only be some desire on the utterer's part, but, unlike that which issues in a command, it is satisfied by his conjecture of a delayed response on the part of his hearers. He conjectures that they will find them useful in situations of doubt. They may wish to repeat them in answer to verbal questions, but it is more likely that the "questions" will be non-verbal in type, and inherent in their experiential situation. Such "questions" are all variations on the same theme, namely "What elements in current experience can I regard as signs indicative of the type of effort likely to achieve the most favourable sequel?"
A sympathetic utterer anticipates the questions with which his hearers are likely to be confronted, and supplies them, in advance, with answers which he himself has found satisfactory. A malevolent utterer may on occasion express an answer which he has found unsatisfactory, in order deliberately to harm his hearers; a careless utterer may also mislead them if he expresses a "satisfactory" answer in such an inaccurate fashion that his hearers misinterpret his utterance, and perhaps even fail to recognise the sign-situation that he has attempted to describe.
It is clear that the ethical merit of truthfulness is derivative from the ethical virtue of the sympathy attributed to the utterers of information. This is why the ethical demerit of deliberate falsehood is generally rated higher than that of mere inaccuracy, even when the latter is disastrous in its consequences.
For example, we can be sure that if anybody had intentionally ordered the charge of the Light Brigade, his name would have been held up to public execration as long as any record of the incident survived. As it is, the "blunderer" remains practically anonymous.
Disregarding the case of deliberate malevolence, it is clear that the basic "meaning" of an informatory statement depends on the nature of the question to which the utterer has provided an anticipatory answer. The commonest form of question is "What is x?", where x stands for some unfamiliar structural element in current experience; the answer, in subject-predicate form, to which it can usually be reduced, is "x is y". The subject is the current focus of importance, and the answer, y, expresses predicates, with which the utterer supposes that his hearers will be familiar, and which will be most useful to them in the conjectural sign-situation.
With this point in mind, it is instructive to examine some of the examples used by Russell in his formulation of the "Theory of Descriptions".
I will begin with "Scott is the author of Waverley". Russell's analysis of this runs "One man, and only one man, is the author of Waverley, and that man is Scott".
This interpretation, which is, indeed, a likely one, makes it clear that such a statement is intended to answer the possible question "Who is the author of Waverley?", not, "Who is Scott?". Its failure to fulfil the normal function of subject-predicate statements is due to its incorrect form; it should run "The author of Waverley is Scott". Here the name "Scott" is a very useful predicate of "the author of" a good many books, notably the "Scott novels".
The question "Who is Scott?" is unlikely to be asked nowadays, but the parallel question "Who is Homer?" is quite a possible one, and the answer "Homer is the author of the Iliad and the Odyssey" is very adequate, since it embodies practically the entire reference of the name.
It is clear that in both cases the name refers, not to the person to whom it may or may not have been attached, but to his official biography. This is why it is in order to use the present tense when framing the question. The surface to which the name "Scott" is believed to have been attached on the occasion of its first appearance, and which it retained as long as it continued to appear and behave "humanly", and, presumably, to be identifiable by the quasi-invariant structure of its finger-prints, has disappeared long ago.
The purport of the question "Who is x?" is very ambiguous, since it may be a desire to ascertain the nature of certain quasi-invariant visible signs for "x", or, alternatively, to found out how he is likely to function. Both purposes are quite common. We often want to "put a face to a name"; passport photographs and "Wanted" notices on police-stations fulfil this function. The other purpose is, perhaps, more frequent; for example, the answer to the question "Who is Lord Butler?" would probably be "Lord Butler is Master of Trinity College, Cambridge". No mention is made of his appearance, and provided that the question is in the present tense, no reference to his former career is required; furthermore, this answer is in simple subject-predicate form.
Such an answer is only possible because one of the functions which he at present fulfils is of outstanding importance; he has to be unique in respect of his position as well as his finger-prints, and the answer simply expresses the spatio-temporal coincidence of the two kinds of "quasi-invariance".
Whenever we interpret a word-sound as a name, we clearly assume that it expresses some thought in the utterer's mind, and this thought, which I call the "significant" of the word,1 must be considered to "exist", so that in this sense "Scott exists" is true, but otiose. But if the word "Scott" is presumed to have an invariant reference which is "public", and more important than the mere name itself - in other words, to refer to something "real", then "Scott exists" is false; any such reference that it may have possessed has disappeared long ago.
1. Cf. Peirce's "interpretant".
It is clear that "proper names" can never be used in framing logical arguments, since their reference is far too ambiguous. They cannot even serve as "arguments" of a propositional function, since these have to be "constants".
Let us now consider the case of descriptions. For my example I take another of Russell's stalking-horses, namely "The present King of France is bald". This is clearly a partial answer to the question "What is the appearance of the head of the present King of France?"
Unless the utterer considers that somebody is likely to ask this question - which is improbable - he has not uttered it with the intention of conveying information. Nevertheless, there is always a remote possibility that it may encourage some hearer to be on the look-out for the sign-situation that it expresses; moreover, it implicitly embodies a further proposition, to wit, "A man is at present fulfilling the function 'King of France' ". If the hearer is an autograph-hunter, or even a "lion-hunter", he may go in search of the individual concerned, naturally confining his attention to the bald-headed. He may suffer grievous disappointment, correctly condemn the statement as "false", and possibly go so far as to denounce its utterer as a "liar".
But is it really sensible to take such "falsehoods" seriously? If so, we will have to bar all forms of verbal expression which are liable to give rise to the slightest chance that any hearer will misinterpret the utterer's intentions, which form an essential element in its "meaning". All figures of speech, all metaphors must, in the interests of "truth", be eliminated from language, in case they are mistaken for "information". No proper names may be used, since the ambiguity of their references makes it impossible to ensure that they will not be misinterpreted, and the same veto must be applied to the use of all "object-words", which we use simultaneously to denote the visual sign for an object and to refer to its functional interpretant.2 It will very soon become apparent that there is hardly anything that we can say "clearly and distinctly"; are we on that account to condemn ourselves to silence?
2. As, for example, when we say "I see a chair".
The sensible alternative is to abandon the ridiculous over-valuation of "Truth" which has characterised most philosophical writing, and to realise that its value is restricted to a small, though important, part of the field of linguistic communication. It is only important inasmuch as false statements are liable to lead to unpleasant consequences for hearers who use them as aids to the choice of effort appropriate to a doubtful situation.
This applies also to statements descriptive of "past events", which are doubtless attempts to express actual experiences with which they have some structural correspondence. Their "truth" is only important in so far as they tend to corroborate or to refute such of our beliefs about the nature of present processes as appear to be relatively invariant, and which we use as guides to help us to choose courses of action which are likely to be favourable.
The "past facts" themselves are intrinsically of negligible importance; they can express nothing more than the mutually corroborative elements in the memories of eye-witnesses. These may have expressed the memory-structures in speech or writing, thereby to some extent circumventing the fading process which characterises memory in general.
Of course, a historian will try to combine such statements into a Gestalt which can be enjoyed from the æsthetic point of view. He is conventionally expected to construct it entirely from statements which he conjectures that the utterers have intended to be interpreted as "information", but if it is to be enjoyable, he is usually obliged to supplement it with a certain amount of "corroborative detail, tending to give verisimilitude to an otherwise bald and unconvincing narrative".3
3. Pooh-Bah, in W.S. Gilbert's The Mikado.
Except possibly in the context of such a Gestalt, what does it matter whether Columbus crossed the Atlantic in 1485 or 1486? The only possible importance is, as Russell himself suggested, that one of them, but not the other, is the expected answer to an examination question. If the "truth" of a statement consists of a structural correspondence between an event and its verbal description, it is obvious that statements about past events can never be "verified".
Experience of events may have prompted their utterance, but our estimate of their "truth" is entirely unconcerned with the events themselves. Any historian's account of such events is inevitably coloured by his notions of the nature of present processes, which he conjecturally extrapolates into the "past". Indeed, when it comes to prehistory and geology, this methodological procedure is explicitly recommended; Lyell's "principle of uniformity" specifically prescribes the policy of "backward induction" as the best method of conjecturing the nature of past "events" in the Earth's history.
The "past" and the "future" are both conjectural, our notions of their nature depending in both cases on the assumption of the continuing operations of certain relatively invariant elements of process which we experience "here-now".
When the "past events" are judged to be very recent, our conjectures as to their nature have a particularly strong influence on our notions of the nature of present process, and therefore on our judgment on what it is best to do next. This is why strong measures have to be taken to attempt to prevent the giving of false evidence in a court of law.
But witnesses who attempt to tell the "whole truth" - and this often happens - tend to obscure the main issue by the wealth of "corroborative detail" with which they embroider the salient facts, and which, for them, is equally "true". All that is really required is that their answers to the questions put to them shall not be materially false.
There is, indeed, no reason to suppose that the value of truth is more than coextensive with the disvalue of falsehood. The latter is confined to excessively intense experiences of refutation, i.e., of discordance between expectation and actuality. This is always a form of what we call "shock"; it is clear that the degree of sensitivity to "shock" varies very greatly between individual human beings. We can, however, confidently assume that they all dislike violent impacts, drastic changes of temperature, very loud bangs, and, in general, all experiences above a certain level of abruptness and intensity. These common "prejudices" are the basis of what Kant called the "Synthetic a priori" in human thinking.
The sensitivity of human beings normally increases during the course of their development, so that while in infancy "shock" is practically confined to visceral and tactile experiences (including taste), its incidence tends to spread to ugly sights and sounds, and eventually even to minor inconsistencies in the Gestalte created by historians, mathematicians and scientists.
Now the Gestalte created by the Greek mathematicians, especially Euclid, appeared, to the most sensitive minds of their time, to be completely free from such inconsistencies, as well as being very beautiful. Since that time it has been the persistent aim of all logicians, and of most philosophers, to extend this degree of consistency to the linguistic expression of Gestalte in all fields of knowledge.
Such an aim is completely ridiculous. Language-structure is firmly anchored to the expression of distinctivenesses which we can discern when our sensitivity is at the stage of development normal to a two-year-old, and cannot be used for the adequate expression or general communication of any subtleties which require a higher sense of discrimination.
In practice, attempts to communicate subtleties above the level of "hard fact" invariably demand the invention of new language-structures as well as new terms, and the resulting expressions can only be interpreted by persons who are sensitive enough to attach importance to the distinctive modes of experience that their utterers have attempted to express. In cases such as certain specialised branches of higher mathematics, only a handful of people understand the "language" employed, and it would be impossible to translate it into the medium of speech, which does not dispose of subtleties of structure adequate to express the "significants" which the utterer attempts to communicate. This is simply because the tempo of succession of our acts of aural awareness is of a slower order than that of our scanning of the visual field.
Such language as is amenable to logical treatment can never express any more than the rigid skeleton of thought. Fortunately, human sympathy seems adequate, in many cases, to allow its interpreters to reclothe it in flesh which does not differ too much from that which has disappeared during the process of expression. Linguistic communication at higher levels of subtlety is therefore valuable, but its efficiency - which at "basic" level is the purpose of "truth" - cannot hope to approach the level of that achieved within the field of "hard fact", so that the concept of "truth" is altogether inapplicable within the "higher" field.
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