The world consists of facts: facts cannot strictly speaking be defined, but we can explain what we mean by saying that facts are what make propositions true, or false. Facts may contain parts which are facts or may contain no such parts; for example: “Socrates was a wise Athenian,” consists of the two facts, “Socrates was wise,” and “Socrates was an Athenian.” A fact which has no parts that are facts is called by Mr Wittgenstein a Sachverhalt. This is the same thing that he calls an atomic fact. An atomic fact, although it contains no parts that are facts, nevertheless does contain parts. If we may regard “Socrates is wise” as an atomic fact we perceive that it contains the constituents “Socrates” and “wise.” If an atomic fact is analysed as fully as possibly (theoretical, not practical possibility is meant) the constituents finally reached may be called “simples” or “objects.” It is not contended by Wittgenstein that we can actually isolate the simple or have empirical knowledge of it. It is a logical necessity demanded by theory, like an electron. His ground for maintaining that there must be simples is that every complex presupposes a fact. It is not necessarily assumed that the complexity of facts is finite; even if every fact consisted of an infinite number of atomic facts and if every atomic fact consisted of an infinite number of objects there would still be objects and atomic facts (4.2211). The assertion that there is a certain complex reduces to the assertion that its constituents are related in a certain way, which is the assertion of a fact: thus if we give a name to the complex the name only has meaning in virtue of the truth of a certain proposition, namely the proposition asserting the relatedness of the constituents of the complex. Thus the naming of complexes presupposes propositions, while propositions presupposes the naming of simples. In this way the naming of simples is shown to be what is logically first in logic.
The world is fully described if all atomic facts are known, together with the fact that these are all of them. The world is not described by merely naming all the objects in it; it is necessary also to know the atomic facts of which these objects are constituents. Given this total of atomic facts every true proposition, however complex, can theoretically be inferred. A proposition (true or false) asserting an atomic fact is called an atomic proposition. All atomic propositions are logically independent of each other. No atomic proposition implies any other or is inconsistent with any other. Thus the whole business of logical inference is concerned with propositions which are not atomic. Such propositions may be called molecular.
Wittgenstein’s theory of molecular propositions turns upon his theory of the construction of truth-functions.
A truth-function of a proposition p is a proposition containing p and such that its truth or falsehood depends only upon the truth or falsehood of p, and similarly a truth-function of several propositions p, q, r . . . is one containing p, q, r . . . and such that its truth or falsehood depends only upon the truth or falsehood of p, q, r . . . It might seem at first sight as though there were other functions of propositions besides truth-functions; such, for example, would be “A believes p,” for in general A will believe some true propositions and some false ones: unless he is an exceptionally gifted individual, we cannot infer that p is true from the fact that he believes it or that p is false from the fact that he does not believe it. Other apparent exceptions would be such as “p is a very complex proposition” or “p is a proposition about Socrates.” Mr Wittgenstein maintains, however, for reasons which will appear presently, that such exceptions are only apparent, and that every function of a proposition is really a truth-function. It follows that if we can define truth-functions generally, we can obtain a general definition of all propositions in terms of the original set of atomic propositions. This Wittgenstein proceeds to do.
It has been shown by Dr Sheffer (Trans. Am. Math. Soc, Vol. XIV. pp. 481-488) that all truth-functions of a given set of propositions can be constructed out of either of the two functions “not-p or not-q or “not-p and not-q.” Wittgenstein makes use of the latter, assuming a knowledge of Dr Sheffer’s work. The manner in which other truth-functions are constructed out of “not-p and not-q” is easy to see. “Not-p and not-q” is equivalent to “not-p,” hence we obtain a definition of negation in terms of our primitive function: hence we can define “p or q,” since this is the negation of “not-p and not-q,” i.e. of our primitive function. The development of other truth-functions out of “not-p” and “p or q” is given in detail at the beginning of Principia Mathematica. This gives all that is wanted when the propositions which are arguments to our truth-function are given by enumeration. Wittgenstein, however, by a very interesting analysis succeeds in extending the process to general propositions, i.e. to cases where the propositions which are arguments to our truth-function are not given by enumeration but are given as all those satisfying some condition. For example, let fx be a propositional function (i.e. a function whose values are propositions), such as “x is human”—then the various values of fx form a set of propositions. We may extend the idea “not-p and not-q” so as to apply to simultaneous denial of all the propositions which are values of fx. In this way we arrive at the proposition which is ordinarily represented in mathematical logic by the words “fx is false for all values of x.” The negation of this would be the proposition “there is at least one x for which fx is true” which is represented by “(
Mr Wittgenstein’s explanation of his symbolism at this point is not quite fully given in the text. The symbol he uses is (
