This theory of logical necessity is so simple and elegant that it attracts all the attention, and Wittgenstein’s next step sometimes goes unnoticed. His next step is to argue that, though the propositions of logic are tautologies and not theories about the nature of reality, the fact that logic exists does indicate something about the nature of reality—it indicates that reality consists ultimately of simple things. His argument for this connection between the existence of logic and his ontology can be broken down into stages. The existence of logic depends on the possibility of combining factual propositions to form tautologies. But that requires the possibility of first constructing factual propositions, without which there would be nothing to combine; and this, in its turn, involves the possibility of elementary propositions and the ultimate granulation of reality. Read in this direction, the argument is a transcendental one, which, in the spirit of Kant, seeks to show how the a priori propositions of logic are possible. They are possible only because reality consists ultimately of simple things. From this point the argument can be traced back in the reverse direction—from simple things to elementary propositions, and thence by the application of the truth-functional formula to the limit of language, which is fixed by the possible permutations and combinations of elementary propositions, however much this may be disguised by the convenient grossness of ordinary factual discourse.
So the task which Wittgenstein undertook in the Tractatus was really an investigation of the foundations of logic. His Notebooks show that this was how it began, and the other aspect of it, which he emphasizes in the Preface to the Tractatus—the fixing of the limit of language—emerged later. The two aspects are connected because logic reveals the structure of factual discourse, and so reveals the structure of reality, which factual discourse reflects. It is for logic to reveal these two structures, which are really one, because they are given in advance of experience—a priori. Experience can only give us a world of facts—everything that is contingently the case—but this world floats in a space of possibilities, which is given a priori. The limit of the space of possibilities, which is reflected in the limit of factual discourse, is given by logic. For the point of origin from which the limit is calculated is determined by logic, and the truth-functional formula, according to which it is calculated, is a logical formula.
So though the propositions of logic are tautologies and lack factual sense, it is logic that reveals the essential structure of reality. The point is sometimes overlooked, because it might be thought that, if the propositions of logic are tautologies, that is only because we happen to have chosen a language in which tautologies are produced when certain propositions are combined. So it might appear that a different choice would have produced a different result, and that all logical truths are true merely by convention. But this was not Wittgenstein’s view. He held that, although we have certain options—that this word should have this meaning, and that word that meaning—the general framework of any factual language is fixed objectively in advance. The general framework is a truth-functional structure based on elementary propositions. When human beings devise a particular factual language, they must connect it up to this pre-existing structure. They have certain options about the ways in which they make the connection, but the structure itself is rigid.
The Tractatus is a philosophical study of this structure, and the medium through which it works is logic. This explains why the book contains so little detailed analysis of particular types of proposition. Wittgenstein is concerned with the general theory of factual language, and with the general theory of reality which he believed that he could deduce from it. He was not much concerned with what seemed to him to be the comparatively trivial details of particular analyses. Now the general theory of language and the ontology do not themselves belong to factual discourse. That is one connection between the darkness at the center of Wittgenstein’s theory of factual discourse and the mysteriousness of certain kinds of non-factual discourse—such things cannot be expressed in factual propositions. But there is also another connection. Wittgenstein’s ontology and part of his theory of language—the part which deals with elementary propositions—are intrinsically mysterious. It is not that they can be fully and clearly expounded in language whose only disadvantage—if it is one—is that it is not factual language. The exposition of them is necessarily sketchy and impossible to illustrate with examples, because they are speculative theories. Even if his arguments proved that these things must be so, they could not be seen to be so.
Ten years later Wittgenstein had begun to dismantle parts of this system, but not in full view of the usual public. In fact, out of the mass of written work which he produced between the late 1920s and his death in 1951 only one brief article was published in his lifetime. In this period his ideas became known through discussions with small groups and lectures to restricted audiences. The result was that, at a time when some of the ideas of the Tractatus were being modified, they continued to exert an influence which was, perhaps, greater than it would otherwise have been. The theory of analysis, or at least the usable part of it, was taken as a model by analytical philosophers in this period.
The model suggested that all factual propositions are truth-functional compounds of simpler propositions. So it appeared that the way to deal with a type of factual proposition which raises a philosophical problem would be to analyze it into its simpler and more explicit components. Thus in the realm of contingency philosophy became the search for translations which would reveal all the implications of the various types of factual proposition and the real nature of their subject matter. Factual propositions whose meaning appears on first inspection to be rather nebulous, such as propositions about social or political groups, or propositions about electrons, would be analyzed in a way that made their implications absolutely clear. Similarly, in the realm of necessity, if all a priori propositions are reducible to tautologies, the way to deal with a puzzling proposition, which is evidently a priori, and yet which does not seem to be tautological, would be to translate it into a form in which it was demonstrably tautological. For example, it is evidently an a priori truth that we cannot travel backwards in time, and, though it does not seem to be a tautology in this form, it could be hoped that it would be possible to translate it into a tautology if an adequate analysis of temporal terms could be found.
Wittgenstein’s modification of the second of these two ideas began in 1929, and can be seen in Philosophische Bemerkungen. All a priori propositions would be reducible to tautologies only if the kind of final analysis which is described in the Tractatus could be achieved. But in the Bemerkungen he admits that it cannot be achieved. He had always conceded that he had not been able to produce final analyses of that kind. But he had thought that they existed and awaited discovery. This is what he retracted in 1929.
In order to understand the retractation, it is necessary to remember the specification of elementary propositions: they are logically independent of one another. Now ordinary factual propositions are not logically independent of one another. For example, a proposition describing a certain area as blue is incompatible with another proposition describing it as yellow. In the Tractatus Wittgenstein had explained this incompatibility by saying that colors have a certain internal complexity. His idea was that propositions in which colors are mentioned could be analyzed down into elementary propositions which would mention things with no internal complexity. If this kind of analysis could be achieved, an a priori proposition such as “Nothing can be blue and yellow simultaneously” would be demonstrated to be a tautology, and its negation, “Something can be blue and yellow simultaneously,” would be demonstrated to be a contradiction. For in the final analysis of the proposition “This thing is blue” there would be found one or more elementary propositions the negation of which would be found in the final analysis of the proposition “This thing is yellow.” Now if a proposition is combined with its negation, the result is a contradiction. So in the final analysis the proposition which is produced when the two color propositions are combined—“This thing is both blue and yellow simultaneously”—would turn out to be a contradiction, and its negation would therefore be a tautology.
If this program could be carried out, the necessary truth of all a priori propositions, except those which give the ontology of the Tractatus, would be shown to be merely the result of the fact that certain combinations of elementary propositions are tautologies. The necessary truth of an a priori proposition would never depend on the specific natures of the things mentioned in elementary propositions because those things would have no internal complexity. This is why the elementary propositions which mention those things would be logically independent of one another. However, the fact that certain combinations of them are tautologies would reveal something about the nature of reality—it would reveal the general structure of reality, which is objectively necessary. To put the same point the other way round, the general structure of reality would necessarily be reflected in the logical grammar of any factual language which human beings may devise. But, according to Wittgenstein’s earlier theory, there would be no further sources of necessity in the nature of things. If we put on one side the necessary truths about the general structure of reality which give the ontology of the Tractatus, there would be no residual sources of necessity left in the nature of things.
In the Bemerkungen he admits that this theory of necessity will not work in all cases. He had come to think that the necessary truth of certain specific a priori propositions cannot be explained by reducing them to tautologies. His reason for this retractation was that he no longer found it possible to believe that there would be no incompatibilities between the words contained in the elementary propositions into which he thought that all factual propositions are analyzable. Suppose, for example, that incompatibilities of colors were explained by analyzing propositions about them into propositions about the velocities of particles. Then the same situation would recur at a lower level of analysis, because there would be a wide range of velocities which would be incompatible with one another. So Wittgenstein conceded that elementary propositions would not be logically independent of one another. They would contain words which produced certain incompatibilities between them, just as color words do, and it would not be possible to reduce these incompatibilities to contradictions by further analysis. All that the philosopher can do is to note the incompatibilities and treat them as irreducible features of the logical grammar of the words.