Some Reflections on the Theory of Systems

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Some Reflections on the Theory of Systems

Nelson Goodman Philosophy and Phenomenological Research, Vol. 9, No. 3, "Second Inter-American Congress of Philosophy".

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Some Reflections on the Theory of Systems Nelson Goodman Philosophy and Phenomenological Research, Vol. 9, No. 3, "Second Inter-American Congress of Philosophy". (Mar., 1949), pp. 620-626. Stable URL: Philosophy and Phenomenological Research is currently published by International Phenomenological Society.

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SOME REFLECTIONS ON T H E THEORY OF SYSTEMS When I first heard that the general topic of the present session would be "The Philosophical Bearings of Modern Logic," I wondered in exactly what sense the term "bearings" should be taken. Bearings are used in machinery in order to reduce friction and noise; but those of you who have attended these meetings in recent years will agree that the joining of logic and philosophy has accomplished neither purpose. In the course of my remarks, I shall indicate some possible reasons why the application of logic to philosophy tends to make the wheels spin less freely, rather than more so. By calling my talk "Some Reflections on the Theory of Systems," I have rather put myself on the defensive, because the term "system" is in bad repute in philosophy. I t calls to mind the magnificent structures of an earlier day, that were built with more regard to their own grandeur than to the comfort of the facts that had to live in the'm. In reaction, recent philosophy has tended to emphasize what might be called "spot analysis": the attempt to deal with each separate problem on its own terms as it arises. One danger in this is that we may find ourselves running around in vicious circles-or, what is worse, that we shall run in vicious circles without finding it out. But quite apart from that, we may be overlooking something important in our eagerness to avoid any charge of trimming facts to fit systems. Although a philosophical system does not create fact, any more than a system of coordinates creates motion, still it begins to look more and more as if some facts are relative to systems, just as motion is. When we investigate the nature of analytic statements, or try to set up an adequate definition of confirmation, or deal with any of a number of related problems, we repeatedly come upon difficulties that seem to be insoluble except relatively to systems. For instance, an analytic statement is presumably one that follo~vsfrom the definition of a term. But while it may be quite clear what the definition of a term is in a given system, the notion of the definition of a term in any absolute sense is as elusive as the notion of an absolutely fixed point. To take another example, when we try to set up explicit standards for confirmation, me are harassed by counterintuitive results that follo~vwhenever properties of certain troublesome kinds are in question. One remedy, suggested by Professor Carnap, is to restrict ourselves to simple properties and certain specifiable compounds of these. But again it is difficult to see how simplicity of the sort intended can mean anything except with respect to a given system. A reverence for fact, then, does not call for a renunciation of system. But a t the same time, no system is indispensable; none is more true than

some alternative system. Yet the fact that there may be a number of systems, each as good as any other, does not mean that there is no point in constructing any particular system. The customer who feels that an automobile of one color is as good as an automobile of any other color, and ~ v h orecognizes that any automobile he gets will be of some color, does not thereby acquire something to drive. Good philosophical systems are perhaps even a little rarer these days than good automobiles. In both connections, openmindedness is advisable; but in both, the first need is to get something that will run. For the attempt to construct a philosophical system, modern logic provides us with both a model and a tool. What I want to do here is suggest a few problems, and fewer solutions, that result from the effort to use the tool and profit by the model. The systems I am concerned with are primarily systems of definitions, and one of the first general theoretical problems me face is the question: by what criteria shall we judge whether a given ,definition is acceptable? A familiar subject of controversy is the adequacy of the criterion of extensional identity, by which a definition is judged correct if the definiendum and the definiens denote just the same things. I t is often contended that such a standard is too lax, that the definiendum and definiens must not only denote the same things-as might, happen in some cases "purely by accidentn-but must have the same meaning, the same intension. On the contrary, I believe it can be shown that the criterion of extensional identity, far from being too loose, is too tight; that it would exclude many definitions we want to admit.

Suppose we are concerned with this diagram, in which there are four lines, a, b, c, and d, and four points of intersection, K, L, M, N . Let us assume that the lines are primitive elements of our system, that they comprise the field of our primitive relation, which we need not specify for our

present purposes. Now we might define the class of the points Ii, L, iM, N as the class of classes of two intersectinglines; in particular, defining Ii as the class of the lines a and c (or as the corresponding whole). A definition of this sort is quite common and often very useful. To exclude it on the ground of a belief that point K "really is" something different from the class of the lines a and c would be to misunderstand the purpose of the definition; and would exclude all the many useful definitions reached, for example, by such a means as Whitehead's extensive abstraction. Moreover, if the proposed definition is allowed, this can hardly be on the ground of a belief that point K "really is" the class of the lines a and c. For suppose there are also in the diagram the lines e, f, and g. Then we might define K as the class of the lines a and c, or the class of a and f, or the class of a and c and f . Anyone who regards any of these defintions as admissible will hardly deny that the others are equally admissible. Yet the system recognizes that the class of the lines a and c is not identical with the class of a and f nor with the class of a and c and f. Sincg extensional identity of definiendum and definiens can thus obtain in a t most one of these admissible definitions, the admissibility of a definition cannot depend upon such extensional identity. Other versions of the extensional criterion fare no better. The test of a definition is often supposed to be whether replacement of the definiendum by the definiens will always leave the truth-value of a sentence unchanged. This would exclude all the definitions we have been discussing. For consider the following sentence:

s. " K is identical with the class of the two lines a and c." To replace "K" by its definiens in each of the three definitions proposed in turn will result in one true sentence and two false ones. Obviously not all of these can preserve the original truth-value of S-whatever we take it to be; yet all are admissible definitions. What we must do is not merely rephrase the criterion of extensional identity, but drop it in favor of some weaker criterion: for example, some kind of systematic isomorphism. Such definitions as satisfy the criterion of extensional identity will satisfy this criterion also, just as definitions which satisfy a criterion of intensional identity presumably also satisfy the weaker criterion of extensional identity; but some definitions which satisfy neither of these stricter criteria will now be admitted. And adoption of a definition will no longer carry with it any commitment as to the extensional or intensional identity or non-identity of the definiendum and definiens. Besides settling upon what constitutes a satisfactory definition, we have to examine the factors governing the choice of the basic vocabulary in

terms of which all the definitions of a given system are to be stated. In addition to the logical apparatus, we shall need one or more extralogical primitives. There is no one best extralogical basis, and the reasons for choosing a given basis at any time will often be very complex. But whatever our other demands may be, we shall always want our basis to be as economical as possible consistently with those demands. I t is rather the fashion to decry economy as trivial. We often hear it said that apart from the purely esthetic consideration of neatness, a basis consisting of only two primitives is no better than a basis consisting of twenty or two hundred primitives of the same sort. But if this position is taken seriously, the system-builder might well take as primitive all the terms or concepts he wants in his entire system, and save himself the trouble of finding definitions. The motives for seeking economy are inseparable from the motives for seeking a system a t all. But granting that economy is wanted, how shall we measure the extent to which it is achieved? The first off-hand .answer is, "count the primitives," but this is obviously an unreliable test. For we can ordinarily replace all the primitives of a given system by a single one, of sufficient complexity. Suppose for example that our primitives are "L" meaning "is a line," "F" meaning "is faster than," and "H" meaning "is halfway betmeen in weight," and suppose that each predicate actually applies to a t least some elements. The three could then be replaced as primitive by "P" where " P ( x , y, r, s, t, w)" means "x is a line, and y is faster than r, and s is halfway between t and w in weight." The original predicates ' ( L , "F," and "H" are then readily definiable from "P." In measuring economy, then, we have to consider the formal complexity as well as the number of the primitives, but this is not altogether easy. I t might seem that all we have to do is to caunt the number of predicate-places rather than the number of predicates, but this turns out to be unsatisfactory. Working from such elementary principles as that no genuine economy is achieved where replacement is purely automatic, we soon find, for example, that a basis consisting of two two-place predicates is in general simpler than one consisting of one four-place predicate; and that n-place predicates-depending upon their symmetry and other characteristics-far from having a uniform complexity-value of n, have complexity-values running all the way from I to 2n-1. This is not the occasion to go into further details. I wanted only to indicate that here again the use of modern logic leads to new problems and some~vhatunexpected results. The problem of economy of postulates is in some respects parallel to that of the economy of primitives. But whereas me found we could normally reduce all extralogical primitives to one, we find that we can reduce all extralogical postulates to none. Prof. Quine and I showed in a paper

a few years ago that any set of extralogical postulates can be made deducible theorems of the system-and therefore unnecessary as postulates-merely by constructing the definitions of the system with malice aforethought. For example, suppose that we have as primitive a txvo-place predicate for the relation of intersection among lines in our diagram, and a postulate to the effect that this relation is symmetrical. We can take as primitive, instead, another word for the same relation, such as the predicate "Crosses," and define "Intersects" as applying to any two lines x and y if and only if x crosses y, and y crosses x. That "intersects" is symmetrical can now be proven from its definition and need no longer be postulated. This procedure can be generalized to eliminate virtually any set of extralogical postulates however complicated. his result is highly paradoxical because i t seems to show that we can deduce all extralogical facts-for instance, all scientific truths-without making any extralogical assumptions. The fact that the definitions will be more complex is unimportant, since definitions are by their very nature eliminable from the system. A sufficiently painstaking examination of the matter-such as was undertaken by Prof. Langford in his note on our paper, and by Prof. Hempel in his review of that note-will remove some of the sting of this paradox. But we are left with the problem how to measure what might be called the assertive economy of a system. Counting postulates, or even counting them and considering their complexity is obviously futile. What we have to do is to develop some method of measuring the basic assertive content of our r kthis problem has hardly begun. system. So far as I know, ~ ~ o on Finally, before we construct any system, we have to decide what logical apparatus we shall use. I t might seem that here efficiency and convenience need be our only considerations, that logic is neutral machinery that cannot affect the philosophical content of a system. But this is quite untenable. In the first place, the line between logic and extralogic is so vague that there is good reason to doubt that it can be drawn a t all except by arbitrary enumeration of certain terms as logical. Accordingly, unless we rigorously restrict what we shall acknowledge as logic, we may find ourselves casually smuggling into our system as logic the means for solving problems that we find it hard to deal with more constructively. The most common kind of complaint against modern logic is that it fails to provide a ready answer to whatever troublesome problem happens to arise. If we find it difficult to explain the meaning of counterfactual conditionals, someone is sure to suggest that we just slip into our logic a new so-called "logical" sign for the kind of sentential connection we need. Obviously this gets us nowhere. This is one reason \vhy some of us take a puritanical attitude towards our logical apparatus, and refuse to admit such notions as intensional implication and the modalities. But even to restrict ourselves

to the familiar extensional logic may not be severe enough. For this logic, as Prof. Quine has shown, is platonistic; it commits us to acceptance of a vast-indeed infinite-realm of abstract entities, and it imports these entities into the scope of any system in which this logic is used. For example, let us suppose that in a system dealing with our diagram, only lines are recognized as individuals. If we use ordinary classial logic in this system, we are thereby also recognizing as entities all classes of these lines, all classes of such classes, and so on; and therefore all sequences and relations of these lines. Possibly, we may be willing to open our arms to this multitude: but if so, we must a t least be aware that our logic is responsible for more of our ontology than is the extralogical part of our system. And some of us are not willing to countenance such abstract entities a t all (if we can help it) either because we are nominalists or because, for the sake of economy, we want to commit ourselves to as little as possible. If either nominalism or plain parsimony leads us to insist upon a logic that is not committed to abstract entities,'then we shall have to forego a large part of the usual modern logic-namely, most of the theory of classes and relations. This will indeed make the going hard, for it then becomes very difficult to express even so simple and fundamental a fact as that there are more cats than dogs. The difficulty of doing without a philosophically objectionable technique is not, ho~vever,any sufficient reason for retaining it. Perhaps it will appear from what I have said that as soon as we apply logic to philosophy, or philosophy to logic, we become embroiled in new difficulties and paradoxes, some of which we are still unable to solve. Indeed I cannot hold the logical philosopher up to you as a man who has found a magic key to all the riddles of the universe; rather, he seems to have found a way to cause himself a good deal of trouble. I t is true, as the unlogical philosopher and the unphilosophical logician often point out, that the way of the logical philosopher is much like that of any transgressor.


Aunque la amarga experiencia del pasado ha creado una animadversi6n por el sistema en filosofia, hoy nos estamos percatando cada vez mfis de que ciertas cuestiones, tales como "2QuB cualidades son simples?", o bien "CQuB proposiones son analiticas?" s610 pueden resolverse dentro de un sistema dado. Gn problema cardinal en la teoria general de 10s sistemas es: Cmediante quB criterio tenemos que juzgar si una definici6n es aceptable dentro del

sistema? Con frecuencia se argumenta diciendo que la identidad de la extensi6n es un criterio demasiado lato, y que se requiere la identidad de comprensi6n. Por el contrario, la identidad de extensi6n es un criterio demasiado estricto, que excluiria muchas definiciones claramente admisibles. Supongamos que tres lineas a, c y f se cruzan en el punto k; k puede entonces definirse, por ejemplo, como la clase de dos lineas a y c, o como la clase de las tres lineas. Pero, como quiera que estas dos clases no son idhticas, es evidente que k no puede ser id6ntico a ambas. De parecido modo, resultan insuficientes otras versiones del criterio extensivo, como el de la mdtua convertibilidad del dejiniendum y el dejiniens sin alteraci6n del valor de verdad. Se requiere manifiestamente otro criterio m&sd6bi1, probablemente en tbrminos de isomorfismo sistembtico. Por otra parte 2qu6 consideraciones deben regular la elecci6n de 10s predicados extral6gicos primarios que hay que usar en un sistema? Resulta que la economia es un factor nada trivial, sino central. Con todo, las medidas de economia que se adoptan usualmente,, sin mfis, resultan inservibles. La complejidad de las bases no puede determinarse meramente contando el nGmero de predicados primarios, ni siquiera el de posiciones predicativas primarias. Para esto, sin embargo, se ha elaborado un nuevo m6todo de computaci6n de la complejidad. En 61 se ve, por ejemplo, que n predicados de posici6n tienen valores complejos que van de 1 a 2 n - 1. E n cuanto a la economia de 10s postulados, Quine y yo hemos seiialado la manera de eliminar enteramente 10s postulados extral6gicos, sin aumentar la complejidad de las bases, mediante la simple selecci6n estratkgica de 10s primarios y la formulaci6n de definiciones. Las definiciones son m&slargas; per0 es que las definiciones, por su misma naturaleza, tambi6n son eliminables. El problema de medir la economia b&sica asertiva de un sistema sigue sin resolver. Finalmente, el constructor de un sistema debe considerar qu6 aparato 16gico va a emplear. Puesto que la distinci6n entre lo 16gico y lo extral6gico es arbitraria, debe tener cuidado en no introducir subrepticiamente como algo "16gico" f6rmulas ad hoc para resolver problemas que resulta m&sdificil plantear de una manera constructiva. Y aunque se limite a la habitual Mgica extensiva, se encontrarh comprometido por ello mismo con una ontologia que abarcarh todas las clases, las clases de clases, etc., de todo aquello que tome como individuos. Si, en virtud de la economia o del nominalismo, se resiste a admitir tales entidades abstractas, entonces tiene que limitarse a una parte angosta de la 16gica ordinaria.