Two chief problems of the theory of knowledge are the question of meaning and the question of verification. The first question asks under what conditions a sentence has meaning, in the sense of cognitive, factual meaning. The second one asks how we get to know something, how we can find out whether a given sentence is true or false. The second question presupposes the first one. Obviously we must understand a sentence, i.e. we must know its meaning, before we can try to find out whether it is true or not. But, from the point of view of empiricism, there is a still closer connection between the two problems. In a certain sense, there is only one answer to the two questions. If we knew what it would be for a given sentence to be found true then we would know what its meaning is. And if for two sentences the conditions under which we would have to take them as true are the same, then they have the same meaning. Thus the meaning of a sentence is in a certain sense identical with the way we determine its truth or falsehood; and a sentence has meaning only if such a determination is possible.

One is tempted to look upon the positive and the logical somewhat as one looks upon the quick and the dead. Yet the issue is hardly that sharp. Viability has strange possibilities and varied forms, and must often be appraised with an eye directed as much towards the environment as towards the claimant organism. Stretching the application of the word ‘viable’ to complexes of behavior such as the philosophies and theories of knowledge, we may ask: Is the combination of the positive and the logical in logical positivism viable? And is this logical positivism itself as it stands either positive or logical?

Mathematical continuity, in the technical sense, is a precisely definable mathematical notion which refers to certain properties of numbers and number sequences. The continuity of the physical world, on the other hand, is rather different from mathematical continuity, since it is a directly experienced attribute of nature and does not require, for being understood, any mathematical theory of properties of numbers.

Ockham realized the value of a careful study and application of the mathematical theory of continuity, even though he possessed no conclusive empirical evidence that it adequately explained the continuity experienced in the world of sense. He also clearly perceived a relation of continuity to infinity, since continuity, as a property of an order which is possible only to a series of terms, must have an infinite number of terms. “A continuous series must have an infinite number of terms.”

Attention has been drawn, particularly since Kant, to propositions which can not have negative instances. They used to be called a priori, axioms, first principles. Today, they are usually called postulates—C. I. Lewis uses both the old and new terminology—because there is a growing recognition of the fact that at least some of them are not “necessary” in the traditional sense. Kant placed a limitation on the apriorism of the continental rationalists. Current epistemologists and logicians have outstripped Kant in the direction signalled by him. Both he and they, however, concur in the opinion that apriori propositions are subject to negation of a sort. The Kantian dinge-an-sich, though they can not contradict these propositions by being negative instances of them, do nevertheless stand to them in the relation of insubordination, thereby exerting on them a “negative action” of a sort. And, according to current views, there are instances which, though they fall outside the range of significance of a proposition or system and hence can not contradict it, do nevertheless tend to bring about a revision of it by a negative action of some sort. The purpose of this brief essay is to coin a convenient name for such instances and to describe certain properties of their peculiar negative action.

The analyses thus far undertaken of cost theory and utility theory have viewed these doctrines as they originally were, namely, as the contentions of two conflicting groups of value theorists. And had the two rival groups been allowed to continue to match strength on the plane on which Macvane, for example, contended, the result would probably have been the destruction of them both. Underneath surface conflicts, however, the rival theories had much in common. As the twentieth century got under way, this state of affairs was discerned, with the result that in the end there was hand-shaking and backslapping all around and the inauguration of a live-and-let-live policy. Although at first regarding each other as mortal enemies, these two groups of price-determination theorists later came to see in one another something resembling blood kinship. But are the underlying assumptions which classicism and neoclassicism have in common any more realistic and scientific than the specific assumptions already examined? This is a question to which we now turn.

It is proposed in this paper to develop a method by which the most general problem of the algebra of propositions is solved. This problem is to construct all propositions whose truth is independent of the form of the variables. As might be expected this method will enable us to determine without the use of matrices the consistency and independence of propositions, except in the case of those fundamental properties, which taken together define consistency itself. In the discussion which follows no limitations are placed on the meaning of the terms and the word proposition is used simply in the sense of a variable. If we were to say, for example, that our variables must be restricted to zero or one values, we would be introducing a limitation which would reduce the generality of our science. A logic may be false in the sense that its postulates lead to contradictory theorems, or it may be false in the sense that it lacks generality, and the situation is not saved by calling it a two-valued, or a three-valued logic, or a logic of “propositions,” or by designating it by some other euphuism.