This distinction between the usable and the unusable parts of Wittgenstein’s early theory of analysis can be related to his attempt to plot the limit of language. That undertaking requires not only a point of origin but also a formula to fix the limit in relation to it. The exact location of the point of origin—elementary propositions—is something of a mystery, but there is nothing mysterious about the formula. Wittgenstein’s idea was that the way to fix the limit is to find a formula for constructing all the propositions which would fall within the limit—i.e., a formula for constructing any factual proposition. The formula which he adopts is to take a set of elementary propositions, and to form all possible combinations of their truth and falsity, and then to specify which of these combinations shall not be realized if the proposition which is being constructed is true. For he believed that the truth or falsity of any factual proposition depends solely on the truth or falsity of the elementary propositions in its final analysis. To put the point in the usual logical terminology, any factual proposition is a truth-function of elementary propositions, or—to put it in a way that relates it to the previous discussion—the sense of a factual proposition depends on what has to be the case if it is true, and ultimately, when its sense has been ground down to the finest possible grains, on the requirement that certain combinations of the truth and falsity of elementary propositions shall not be realized, if it is true.
Although elementary propositions, to which this formula is applied, are somewhat mysterious, the formula itself is clear enough. It is also clear that it runs into difficulties. For it does not seem to allow for all the ways in which propositions are constructed. It explains the construction of such a proposition as “Either the watch is on the table, or it is on the desk.” But the proposition “I believe that the watch is on the table” seems to be constructed in a different way. Certainly, its truth or falsity does not depend on the truth or falsity of the proposition “The watch is on the table,” which is one of its components.
It is perhaps surprising that Wittgenstein’s elementary propositions should be so inaccessible and mysterious, because they are deduced from a theory of analysis constructed out of material which he got from Russell, and which in Russell’s hands took a less recondite form. Russell’s system is really a version of Hume’s empiricism, presented not as a theory about the mind—which is the way in which Hume presented it—but as a theory about language, and, therefore, based on Russell’s new ideas in logic rather than on the primitive psychology of the eighteenth century. It is clear and cool, and its doctrines and the spirit in which they are developed are Humean, even if the arguments on which they are based are not Humean. The analysis of all factual propositions is supposed to terminate in sense-datum propositions. Nothing could be less recondite.
But when Wittgenstein took over the logical ideas on which Russell’s theory of analysis was based, he developed them in the darker manner of German Idealism. The Tractatus belongs to the same tradition as the work of Freud. It is, of course, neither a psychological treatise nor a philosophical treatise presented in psychological terms. But it offers a speculative theory about something which is usually supposed to take place in the clear light of consciousness—the correlation of words with things through which factual propositions acquire their senses. In the case of elementary propositions this correlation takes place in total obscurity. If it is something that we do, we do not do it consciously or intentionally.
In his Memoir Engelmann connects the darkness at the center of Wittgenstein’s construction of factual discourse with the mysteriousness of certain kinds of non-factual discourse. One connection—an obvious one—is that Wittgenstein could not plot the limit of factual discourse without saying something about at least one kind of non-factual discourse, namely philosophy. What he says about it is that, like religion, morality, and aesthetics, it cannot be expressed in factual propositions, and this is one of the most important findings of the Tractatus. But he also says something more positive about philosophy. It is an organized investigation of a well-defined field. It is not a science, distinguished from other sciences only by its extreme generality. For science studies things that might have been otherwise, and, however general its results may be, they belong to the realm of contingency, whereas philosophy studies the realm of necessity. So philosophical results will be expressed in propositions which are in some way or other necessarily true, a priori propositions. When Wittgenstein claims that all factual propositions are analyzable into elementary propositions, and that the words in elementary propositions are correlated with simple existents, this is a philosophical claim that these things are necessarily so. Although it is a claim about the structure of factual discourse, and ultimately about the structure of reality, it does not itself belong to factual discourse, because a priori propositions do not have factual sense. So at the center of the realm of contingency there are truths of the kind which traditional metaphysicians tried to express in factual propositions, but which cannot be expressed in factual propositions. Inside the world of factual meaning the pressure of other modes of thought can be felt.
This connection between the philosophical study of factual discourse and other non-factual disciplines is a negative one. Neither religion nor morality nor aesthetics nor philosophy can be expressed in factual propositions. Nevertheless it is an important connection, because it shows that, when Wittgenstein places a certain kind of discourse outside factual language, he is not thereby condemning it. Such exclusions are certainly divisive, but they are not intolerant. In fact—to answer one of the questions that was posed earlier—he does distinguish between good and bad nonsense. There is the complete nonsense which is produced when someone tries to take factual propositions beyond their limit. Some of the doctrines of traditional metaphysicians belong to this category: they do not have factual sense, although they purport to have it. But there are also two other categories of “non-sense” which are not condemned. There are a priori propositions which neither have nor purport to have factual sense, and there are propositions such as those of religion and morality which express things that cannot be expressed in a factual way, although they are often cast in a form which suggests, most misleadingly, that their mode of expression is factual.
The peculiarity of philosophy, which differentiates it from other non-factual disciplines, is that its results are expressed in a priori propositions. This peculiarity is, of course, a direct consequence of the thesis that philosophy is linguistic analysis. For the statement that a particular proposition can be analyzed in a certain way is a statement which, given the meaning of the proposition to be analyzed, and the meaning of the analysis, is necessarily true, or a priori. But this is not a complete explanation of the fact that philosophical results are expressed in a priori propositions. For the theory of elementary propositions and the ontology which Wittgenstein deduced from it are also philosophical results expressed in a priori propositions. They are, of course, results of a very different kind. It is one thing to give the analysis of a certain kind of proposition, and quite another thing to offer a general thesis about the structure of factual discourse and the structure of reality. But both alike would be expressed in a priori philosophical propositions.
The distinction between these two kinds of results is, of course, the distinction between the usable and the unusable parts of the theory of analysis which is expounded in the Tractatus. The unusable part was Wittgenstein’s general and highly speculative theory of elementary propositions, and the ontology which he deduced from it. The usable part of the theory of analysis was detached by other philosophers in the period between the two wars and worked out separately. It contained three leading ideas. The first idea is that a factual proposition is a truth-function of the propositions which occur at any stage in its analysis, whether it be the final stage or not; the second idea is that the connection between a proposition and its analysis—sameness of meaning—will be expressed in an a priori philosophical proposition; and the third idea is that all a priori propositions lack factual sense.
It was inevitable that Wittgenstein’s attempt to place philosophy in relation to factual disciplines should raise the question how it is related to a priori disciplines. It was also inevitable that this question—to which the answer given was that it is an a priori discipline of a peculiar kind—should lead on to the question, what the difference between factual and a priori propositions really is. This trail of questions leads to the center of the system of the Tractatus. The easiest way to follow it is to take the three ideas which comprise the usable part of Wittgenstein’s theory of analysis, and to examine them more closely, in order to see why they seemed to him to point unmistakably toward the more speculative part of the theory.
A factual proposition is a truth-function of the propositions which occur in its analysis. This means that a factual proposition, provided that it is not elementary, is a compound proposition, which says what its components say, no more and no less. For example, if a factual proposition, p, can be analyzed as a conjunction of two factual propositions, q and r, then p is true if both q and r are true, but in all other cases it is false. Now there are two other types of truth-function besides factual propositions. There are truth-functions which are true in all cases, and these are tautologies—e.g., the tautology, p or not-p, is always true, whether p itself, which is its only component, is true or false: and there are truth-functions which are false in all cases, and these are contradictions—e.g., the contradiction, p and not-p, is always false.
These two further types of truth-function lack factual sense, because considered as messages they are self-cancelling. Someone who begins by saying, “p…,” and continues by saying, “…and not-p” cancels the sense of the first part of his utterance, and produces a contradiction. The continuation, “…or not-p,” which produces a tautology, cancels the sense of p in a different way. For, whereas a contradiction cannot be true, whatever happens, a tautology must be true, whatever happens. So tautologies and contradictions are not messages. They do not express what is the case, but what must be the case, and what cannot be the case. They themselves lack factual sense, but they express the necessary connections between other propositions which do have factual sense. An argument is valid if the combination of its premises and its conclusion is a tautology: given the premises the conclusion must be true. For example, it is tautological to say that, given p, and given if p, then q, q must be true. To put the same point negatively, an argument is valid if the combination of its premises and the negation of its conclusion is a contradiction. In short, the propositions of logic are not theories about the nature of reality, but tautologies, which lack factual sense.