INTRODUCTION

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 know-

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