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From The Journal of Philosophy, Vol. 23, No. 16, August 5, 1926, 435-438.  “A logical form is always relative to a system; a logical term or complex of terms without reference to any particular system is as meaningless as a word or phrase without reference to any particular language.”

June 31, 2009


Form and Content: A Study in Paradox

Susanne K. Langer

The acceptance of paradoxes has played a pecu-liar role in philosophy.  It has led some thinkers to deny the reality of the world, and others to reject the laws of logic; it has delighted the Sophists and made mystics of the erudite saints; it is the great weapon of idealism and the reproach of rationalism.

But, in truth, there is no metaphysical virtue in paradox.  The laws of logic have not produced it; the world does not contain it.  The presence of a true paradox1 in any proposition is essentially an index of non-significance, and therefore it is a symptom of some philosopher’s muddle-mindedness, not an indictment of Reality or of logic.  It is only what we say about Reality or about logic that is absurd, not the subject of the assertion.

The discovery that paradox is always indicative of a false premise is due to Bertrand Russel.2  It was he who noted that the solution of Bolzano’s paradoxes of the infinite, of Bradley’s relation-antinomies, of the medieval doctors’ “insolubilia” and the more modern contradictions in mathematical theory, all require the abandonment of some tacitly accepted premise.  But the test of ingenuity is to find this premise.  It is usually so deep-rooted in common sense that it is hard to explicate and harder yet to doubt.  It requires great intellectual candor to question the validity of old implicit axioms, such as that the whole is always greater than any of its parts, or that any entities whatever may be combined to form a logical class. Fortunately, Mr. Russell is blessed with both candor and acumen, which led him to discover the fallacy of “illegitimate totalities.”3  This discovery has obviated some of the classical paradoxes of logic; the most delightful thing about it, however—the surest mark of its subtility and depth—is its aptness for killing metaphysical as well as logical chimæras.  Its auth-or’s comments on Mr. Wittgenstein’s mysticism are an instance of such philosophical solution by a pertinent inquiry into the possibility of some gener-ally accepted notions.4  The assumption of “lan-guage,” which is the class of all symbolisms, or of the “world,” which is the totality of facts, is not a safe assumption, as we learn by scrutinizing such com-mon-sense constructs as “language” and “world.”

Now there are few philosophers who are mystics concerning language, and few, perhaps, who are disturbed by accepting the unity of the world (in fact, Monism is frequently recommended as the answer to all mundane riddles); but one problem seems yet to reduce even “scientific philosophers,” i.e., logicians, to a sort of mysticism: that is the problem of relating the abstract form of anything to its specific content. That a logical form as such is an abstraction, is generally recognized; less generally, perhaps, the fact that every concrete object of our experience is an interpretation of familiar forms.  Interpretation is the converse of abstraction; given a purely formal system of concepts, we are now in search of the form with a content.  Interpretation seems to involve a going outside of logic for part of our material.  When we interpret an abstract form, we are no longer dealing with this form alone, but find it in conjunction with some non-logical element, the “content.”

This relation of form and content raises an interesting and difficult problem.  At first sight it appears obvious that there can be such a relation; but if there is, then it can be expressed symbolically, as R(f,c) ; and thereby we have transformed our empirical content into a term of the formal structure, i.e., we have formalized it, and are no longer dealing with the non-logical content.  Thus it seems there can be no such thing as the relation between the form of a thing and the content of that form, since this relation would entail a true paradox.

The false premise which engenders this paradox is the supposition that there is such a thing as the form of anything.  A logical form is always relative to a system; a logical term or complex of terms without reference to any particular system is as meaningless as a word or phrase without reference to any particular language.  Thus the form of an object, if taken to denote a single absolute notion, suffers from the same sort of non-significance as the name of an object, which Messrs. Whitehead and Russell deal with in Principia Mathematica.6  But, whereas a name is necessarily just one definite name, quite obviously dependent for its significance upon some principle of naming, the “form” is a more difficult notion and its relations are not so evident.  What we have in mind when we speak of the form of anything, is probably the class of all possible forms under which the object in question can be conceived; these forms being of various degrees of simplicity, i.e., abstraction, from the emptiest to the most differentiated.  Only in some such sense as this could we speak of the form of an object.

But such a class of forms proves upon inspection to be just one more “illegitimate totality” of the sort discussed in Principia Mathematica.  For a hierarchy of forms, each presupposing and supplementing the one beneath it in order of complexity, assumes a single system wherein all these forms are conceived. But an object may be analyzed into forms which do not presuppose each other, and which are neither supplementary nor contradictory one to another, but simply incommensurable.  Professor Whitehead has called attention to this common fallacy of systema-tization and pointed out the fact that different princi-ples of analysis give rise to radically diverse forms, and that these do not form a hierarchy of abstrac-tions.6  Every form is relative to a system, or—which is the same thing—to a logical language.

The problem of language is a troublesome one in logic.  Language (verbal or non-verbal) determines by its structure just the sorts of concept, i.e., the forms, which can be expressed in it.  And whatever object we are talking about, we are limited to some definite language, with its idiosyncrasies of structure, and consequently we are limited as to the things we can say.  There is no “Interlingua” which is an ab-straction from languages; we can use only one language on each occasion, and we must use just one.  It is this provincialism of language that has made Mr. Wittgenstein despair of any philosophical, that is to say, truly general, propositions.  Since the logical forms we can designate are determined by the language, or medium of designation, what holds for language holds also for logical forms; if, as Mr. Russell maintains, there could be a hierarchy of languages but no totality of them, then even if there were a hierarchy of forms exemplified by any object there yet could be no totality of these.  Therefore we can not take “the form” to denote the class of forms, because this is an illegitimate whole.

Thus the relation of form and content is not the unqualified, ultimate, two-termed relation which it is generally thought to be.  If “form” always means a form, then content always means a content, because content is relative to form; content means “that which is not given as part of this logical structure.”

Our problem, then, boils down to that of the rela-tion between a logical structure and something which it does not include; between a form and the locus wherein it meets or conflicts with other possible forms.  If there is more to “content” than just this penumbra of extraneous and irrelevant other structures, that more is not communicable and not cognizable—since we can not know what can not (in same language) be talked about.  Such a truly alogical content could really not be conceived as entering into a logical relation, even if in some supra-mundane fashion it existed.7  We can not even say that the content is the residue of all possible structures after a chosen one has been set aside, because “all possible structures” do not exist.  We have simply a relation between a form which is explicable in the given language, and the possibility of others which are not.  This is analogous to the relation obtaining between logical systems them-selves, or logical languages as such, which Mr. Russell touched upon in his discussion of Mr. Wittgenstein’s views.  But such a relation, abstruse and complex as it may be, exhibits no true paradox, and therefore does not necessarily invite our mystical contemplation.



1 I say a true paradox in distinction from those contradictions which rest upon a (none too subtle) change of viewpoint or meaning.  Thus in Principia Mathematica “the greatest ordinal number” involves a true paradox, whereas “the least integer not nameable in fewer than nineteen syllables” does not.

2 Statements of this discovery are scattered throughout his earlier metaphysical writings, especially those collected in Mysticism and Logic and Our Knowledge of the External World.

3 For the evolution of this idea, compare Russell’s Principles of Mathematics, Ch. X and App. B, with Principia Mathematica, Introd. to 2d ed., Ch. II, and App. C.

4 L. Wittgenstein, Tractatus Logico-Philosophicus, pp. 22-23 (Russell's Introd.).

5 Ch. II.

6 An Enquiry Concerning the Principles of Natural Knowledge, pp. 59-60 (1st ed.): “It is not merely that one mode of diversification of nature is incomplete and leaves out some entities which another mode supplies.  The entities which are yielded by different modes of diversification are radically different . . . . Thus each mode of diversification produces natural elements of a type peculiar to itself.

“One mode of diversification is not necessarily more abstract than another.”

7 Just such a content is Kant’s Thing-in-itself; it can not really be related to the phenomenon because all the categories of form and relation are foreign to it. That is why many philosophers have condemned it as an unprofitable notion.

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