Ludwig Wittgenstein came to Cambridge to study mathematical logic under Russell, but he quickly established himself as his teacher’s intellectual peer. Together, they devised a metaphysical system called “logical atomism.” As discussed at the beginning of Section 2, qua total system, logical atomism seems to have been Wittgenstein’s brainchild. Still, this should not be seen as in any way marginalizing Russell’s significance for the system, which can be described as a metaphysics based on the assumption that an ideal language the likes of which was provided in Principia Mathematica is the key to reality.
According to logical atomism, propositions are built out of elements corresponding to the basic constituents of the world, just as sentences are built out of words. The combination of words in a meaningful sentence mirrors the combination of constituents in the corresponding proposition and also in the corresponding possible or actual state of affairs. That is, the structure of every possible or actual state of affairs is isomorphic with both the structure of the proposition that refers to it and the structure of the sentence that expresses that proposition--so long as the sentence is properly formulated in the notation of symbolic logic. The simplest sort of combination is called an atomic fact because this fact has no sub-facts as part of its structure. An atomic fact for some logical atomists might be something like an individual having a property—a certain leaf’s being green, for instance. Linguistically, this fact is represented by an atomic proposition: for example, “this leaf is green,” or, in logical symbolism “F(a).” Both the fact F(a) and the proposition “F(a)” are called “atomic” not because they themselves are atomic [that is, without structure], but because all their constituents are. Atomic facts are the basic constituents of the world, and atomic propositions are the basic constituents of language.
More complex propositions representing more complex facts are called molecular propositions and molecular facts. The propositions are made by linking atomic propositions together with truth-functional connectives, such as “and,” “or” and “not.” A truth-functional connective is one that combines constituent propositions in such a way that their truth-values (that is, their respective statuses as true or false) completely determine the truth value of the resulting molecular proposition. For instance, the truth value of a proposition of the form “not-p” can be characterized in terms of, and hence treated as determined by, the truth value of “p” because if “p” is true, then “not-p” is false, and if it is false, “not-p” is true. Similarly, a proposition of the form “p and q” will be true if and only if its constituent propositions “p” and “q” are true on their own.
The logic of Principia Mathematica is entirely truth-functional; that is, it only allows for molecular propositions whose truth-values are determined by their atomic constituents. Thus, as Russell observed in the introduction to the second edition of the Principia, “given all true atomic propositions, together with the fact that they are all, every other true proposition can theoretically be deduced by logical methods” (Russell 1925, xv). The same assumption—called the thesis of truth-functionality or the thesis of extensionality—lies behind Wittgenstien’s Tractatus Logico-Philosophicus.
As mentioned previously, Wittgenstein’s Tractatus proved to be the most influential expression of logical atomism. The Tractatus is organized around seven propositions, here taken from the 1922 translation by C. K. Ogden:
The world is everything that is the case.
What is the case, the fact, is the existence of atomic facts.
The logical picture of the facts is the thought.
The thought is the significant proposition.
Propositions are truth-functions of elementary propositions. (An elementary proposition is a truth function of itself.)
The general form of a truth-function is.... This is the general form of a proposition.
Whereof one cannot speak, thereof one must be silent. The body of the Tractatus consists in cascading levels of numbered elaborations of these propositions (1 is elaborated by 1.1 which is elaborated by 1.11, 1.12 and 1.13, and so forth)—except for 7, which stands on its own. Propositions 1 and 2 establish the metaphysical side of logical atomism: the world is nothing but a complex of atomic facts. Propositions 3 and 4 establish the isomorphism between language and reality: a significant (meaningful) proposition is a "logical picture" of the facts that constitute some possible or actual state of affairs. It is a picture in the sense that the structure of the proposition is identical to the structure of the corresponding atomic facts. It is here, incidentally, that we get the first explicit statement of the metaphilosophical view characteristic of early analytic philosophy: “All philosophy is a ‘critique of language’ ...” (4.0031).
Proposition 5 asserts the thesis of truth-functionality, the view that all complex propositions are built out of atomic propositions joined by truth-functional connectives, and that atomic propositions are truth-functional in themselves. Even existentially quantified propositions are considered to be long disjunctions of atomic propositions. It has since been recognized that a truth-functional logic is not adequate to capture all the phenomena of the world; or at least that, if there is an adequate truth-functional system, we haven't found it yet. Certain phenomena seem to defy truth-functional characterization; for instance, moral facts are problematic. Knowing whether the constituent proposition “p” is true, doesn’t seem to tell us whether “It ought to be the case that p” is true. Similarly problematical are facts about thoughts, beliefs, and other mental states (captured in statements such as “John believes that…”), and modal facts (captured in statements about the necessity or possibility of certain states of affairs). And treating existential quantifiers as long disjunctions doesn’t seem to be adequate for the infinite number of facts about numbers since there surely are more real numbers than there are available names to name them even if we were willing to accept infinitely long disjunctions. The hope that truth-functional logic will prove adequate for resolving all these problems has inspired a good bit of thinking in the analytic tradition, especially during the first half of the twentieth century. This hope lies at the heart of logical atomism.
In its full form, Proposition 6 includes some unusual symbolism that is not reproduced here. All it does, however, is to give a general “recipe” for the creation of molecular propositions by giving the general form of a truth-function. Basically, Wittgenstein is saying that all propositions are truth-functional, and that, ultimately, there is only one kind of truth-function. Principia Mathematica had employed a number of truth-functional connectives: “and,” “or,” “not,” and so forth. However, in 1913 a logician named Henry Sheffer showed that propositions involving these connectives could be rephrased (analyzed) as propositions involving a single connective consisting in the negation of a conjunction. This was called the “not and” or “nand” connective, and was supposed to be equivalent to the ordinary language formulation “not both x and y.” It is usually symbolized by a short vertical line ( ) called the Sheffer stroke. Though Wittgenstein uses his own idiosyncratic symbolism, this is the operation identified in proposition 6 and some of its elaborations as showing the general form of a truth-function. Replacing the Principia’s plurality of connectives with the “nand” connective made for an extremely minimalistic system—all one needed to construct a complete picture/description of the world was a single truth-functional connective applied repeatedly to the set of all atomic propositions.
Proposition 7, which stands on its own, is the culmination of a series of observations made throughout the Tractatus, and especially in the elaborations of proposition 6. Throughout the Tractatus there runs a distinction between showing and saying. Saying is a matter of expressing a meaningful proposition. Showing is a matter of presenting something’s form or structure. Thus, as Wittgenstein observes at 4.022, “A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand.”
In the introduction to the Tractatus, Wittgenstein indicates that his overarching purpose is to set the criteria and limits of meaningful saying. The structural aspects of language and the world—those aspects that are shown—fall beyond the limits of meaningful saying. According to Wittgenstein, the propositions of logic and mathematics are purely structural and therefore meaningless—they show the form of all possible propositions/states of affairs, but they do not themselves picture any particular state of affairs, thus they do not say anything. This has the odd consequence that the propositions of the Tractatus themselves, which are supposed to be about logic, are meaningless. Hence the famous dictum at 6.54:
My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must transcend these propositions, and then he will see the world aright. Though meaningless, the propositions of logic and mathematics are not nonsense. They at least have the virtue of showing the essential structure of all possible facts. On the other hand, there are concatenations of words, purported propositions, that neither show nor say anything and thus are not connected to reality in any way. Such propositions are not merely senseless, they are nonsense. Among nonsense propositions are included the bulk of traditional philosophical statements articulating traditional philosophical problems and solutions, especially in metaphysics and ethics. This is the consequence of Wittgenstein’s presumption that meaningfulness is somehow linked to the realm of phenomena studied by the natural sciences (cf. 4.11 ff). Thus, as he claims in 6.53:
The correct method in philosophy would really be the following: to say nothing except what can be said, that is propositions of natural science—that is something that has nothing to do with philosophy—and then, whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions. In the eyes of its author (as he avers in its Introduction), the real accomplishment of the Tractatus was to have solved, or rather dissolved, all the traditional problems of philosophy by showing that they were meaningless conundrums generated by a failure to understand the limits of meaningful discourse.