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Unit 7: Wittgenstein and Analytic Philosophy

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:

1.   The world is everything that is the case.

2.   What is the case, the fact, is the existence of atomic facts.

3.   The logical picture of the facts is the thought.

4.   The thought is the significant proposition.

5.   Propositions are truth-functions of elementary propositions. (An elementary proposition is a truth function of itself.)

6.   The general form of a truth-function is…. This is the general form of a proposition.

7.   Whereof one cannot speak, thereof one must be silent.