By a two-valued truth-function, we may understand simply a function, of which the independent variables range over a domain of two objects, and of which the value of the dependent variable for each set of arguments is taken from the same domain.  For a given set of n distinct variables as the independent variables, there are exactly 2 2 n such functions, each of them being describable by a truth-table which gives the function value for each of the 2 n sets of arguments.  In the truth-table method, the primitives of the propositional calculus are interpreted by truth-functions; in other words, truth-tables are assigned to them.

An extensive bibliography is given there. 1G4 Post: A General Theory of Elementary Propositions. are about the logic of propositions but are not included therein. More particularly, whereas the propositions of 'Principia' are particular assertions ns introduced for their interest and usefulness in later portions of the work, those of the present paper are about the set of all such possible , assertions. Our most important theorem gives a uniform method foii testing the truth of any proposition of the system; and by means of this) theorem it becomes possible to exhibit certain general relations which exist between these propositions. These relations definitely show that the postulates of ' Principia ' are capable of developing the complete system of the logic of propositions without ever introducing results extraneous to that; systema conclusion that could hardly have been arrived at by the particun lar processes used in that work. Further development suggests itself in two directions. On the one hand this general procedure might be extended to other portions of ' Principia, and we hope at some future time to present the beginning of such an attempt pt. On the other hand we might take cognizance of the fact that the. system of 'Principia' is but one particular development of the theoryparticular in the primitive functions it employs and in the postulates it imposes on those functionsand so might construct a general theory of such developments. This we have tried to do in the other portions of the paper. Our first generalization leads to systems which are essentially equivalent to that of 'Principia' and connects up with the work of Sheffer* and Nicodj in reducing the number of primitive functions and of primitive propositions .respectively. The second generalization, on the other hand, while including the first also seems to introduce essentially new systems. One class of such systems, and we study these in detail, seems to have the same relation to ordinary logic that geometry in a space of an arbitrary number of dimensions has to the geometry of Euclid. Whether these '' non-Aristotelian ' logics and the general development which includes them will have a direct application we do not know; but we believe that inasmuch as the theory of elementary propositions is at the base of the complete system of ' Principia,' this broadened outlook upon the theory will serve to prepare us foi a similar analysis of that complete system, and so ultimately of mathematics. Finally a word must be said about the viewpoint that is adopted in this paper and the method that is used. We have consistently regarded the system of ' Principia ' and the generalizations thereof as purely formal de

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Introduction. Recent developments of symbolic logic have considerable importance for mathematics both with respect to its philosophy and practice. That mathematicians generally are oblivious to the importance of this work of Gödel, Church, Turing, Kleene, Rosser and others as it affects the subject of their own interest is in part due to the forbidding, diverse and alien formalisms in which this work is embodied. Yet, without such formalism, this pioneering work would lose most of its cogency. But apart from the question of importance, these formalisms bring to mathematics a new and precise mathematical concept, that of the general recursive function of Hërbrand-Gödel-Kleene, or its proved equivalents in the developments of Church and Turing. 1 It is the purpose of this lecture to demonstrate by example that this concept admits of development into a mathematical theory much as the group concept has been developed into a theory of groups. Moreover, that stripped of its formalism, such a theory admits of an intuitive development which can be followed, if not indeed pursued, by a mathematician, layman though he be in this formal field. It is this intuitive development of a very limited portion of a sub-theory of the hoped for general theory that we present in this lecture. We must emphasize that, with a few exceptions explicitly so noted, we have obtained formal proofs of all the consequently mathematical theorems here developed informally. Yet the real mathematics involved must lie in the informal development. For in every instance the informal "proof" was first obtained; and once gotten, transforming it into the formal proof turned out to be a routine chore. 2 We shall not here reproduce the formal definition of recursive function of positive integers. A simple example of such a function is an