- Implication and the Algebra of Logic.
Mind, New Series 21 (84) (1912), 522-531.The development of the algebra of logic brings to light two somewhat startling theorems: (1) a false proposition implies any proposition, and (2) a true proposition is implied by any proposition. These are not the only theorems of the algebra which seem suspicious to common sense, but their sweeping generality has attracted particular attention. In themselves, they are neither mysterious sayings, nor great discoveries, nor gross absurdities. They exhibit only, in sharp outline, the meaning of "implies" which has been incorporated into the algebra. What this meaning is, what are its characteristics and limitations, and its relation to the "implies" of ordinary valid inference, it is the object of this paper briefly to indicate. Such an attempt might be superfluous were it not that certain confusions of interpretation are involved, and that the expositors of the algebra of logic have not always taken pains to indicate that there is a difference between the algebraic and the ordinary meanings of implication. One may suspect that some of them would deny the divergence, or at least would maintain that the technical use is preferable and ought generally to be adopted. As a result, symbolic logic appears to the uninitiated somewhat as an enfant terrible, which intimidates one with its array of exact demonstrations, and demands the acceptance of incomprehensible results.
- A New Algebra of Implications and Some Consequences.
The Journal of Philosophy, Psychology and Scientific Methods 10 (16) (1913), 428-438.
[Note: This paper was read at a meeting of the San Francisco Section of the American Mathematical Society on 26 October 1912.]The development of the algebra of logic has done more than emphasize the close relation of logic and mathematics. It has helped to show the possibility of an ideal development of pure mathematics in general, free - or nearly free - from tacit assumptions, parsimonious in its postulates, and absolutely rigorous in its methods of proof. In this ideal development, the algebra of implications, or "calculus of propositions," appears as the organon of proof in general, and hence as the necessary first step. The work of Russell and Whitehead and others has called attention to this method of procedure. It is the logical outcome of the denial that mathematics must appeal to "construction" or any other empirical datum, once its postulates are laid down. From this point of view, the drawing of conclusions is not a process in which premises retire into somebody's reasoning faculty and emerge in the form of the result; nor is the conclusion obtained through any subtle appeal to the perceptual character of space or of collections of marbles or arrays of fingers. Proof takes place through the collusion of two factors; first, postulates or propositions of the particular mathematical system in hand; secondly, postulates or theorems which state implication relations between premises of that logical or mathematical type and the desired conclusion. A mathematical operation is ideally no more than this: the substitution of the variables or functions of variables of the particular system - say, of cardinal number - for the logical variables in some proposition about implications. This proposition is more than a rule for inference; when the substitution is made, it states the implication. The result is the statement of what the variables or functions of the cardinal number system imply-a proposition in cardinal arithmetic. This result is not strictly the theorem to be proved, but only the statement that certain expressions or relations of variables imply certain others. Thus pure mathematics does not seek to prove theorems, but only, in the last analysis, that certain postulates imply certain theorems. And the proposition which states the particular implication relation - in more general form, because its variables have a wider range of meaning - is itself a mathematical proposition, in the algebra of logic.
- Interesting Theorems in Symbolic Logic.
The Journal of Philosophy, Psychology and Scientific Methods 10 (9) (1913), 239-242.Two propositions in the algebra of implication or "calculus of propositions" have been much discussed. They are: "A false proposition implies any proposition," and "A true proposition is implied by any proposition." These theorems have been hailed as discoveries and repudiated as absurdities. But on all sides the impression seems to prevail that these two are sui generis in the algebra. For this reason, it may be worth while to present a partial list of propositions which are of the same kind, involve the same principles, and can be proved from the same assumptions. Comparatively few such theorems have been printed, but their number is apparently infinite. ... Any one of these theorems can be proved from the postulates of "Principia Mathematica," from those of Peano, from Schröder's, and from any of the sets given by Huntington. They can also be proved, in somewhat different form, from the assumptions of Mrs Ladd-Franklin's algebra, if the variables of that system are taken to symbolize propositions or propositional functions. What these theorems reveal is the divergence of the meaning of "implies" in the algebra of logic from the "implies" of valid inference. ... The consequences of this difference between the "implies" of the algebra and the "implies" of valid inference are most serious. Not only does the calculus of implication contain false theorems, but all its theorems are not proved. ... The postulates of the "Principia" imply the "consequences" thereafter set down in exactly the same fashion that "Socrates was a solar myth" implies "All triangles have two or more sides."
- Some Logical Considerations Concerning the Mental.
The Journal of Philosophy 38 (9) (1941), 225-233.It is a conception as old as Socrates and as modern as our current I logical analyses that the central task in philosophic discussion of any topic is to arrive at and elucidate a definition. That, I take it, is what is properly meant by a philosophic "theory"; a theory of X is a more or less elaborate definitive statement having "X" as subject, together with such exposition as will remove difficulties of understanding and serve to show that this definition covers the phenomena to be taken into account. I fear that what this occasion calls for is such a theory of mind. But if so, then I am unprepared for it. I am unable to present any statement of the form "Mind is ..." (where "is" would express the relation of equivalence of meaning) which would satisfy me or which I should expect would satisfy you. I can only put forward certain statements intended to formulate attributes which are essential to mind; to point to phenomena of which we can say, "Whatever else is or is not comprehended under 'mind,' at least it is intended to include these." In particular, I shall wish to emphasize that whatever is called "content of consciousness" is so included, and to consider certain consequences of that simple fact. ... We significantly believe in other minds than our own, but we can not know that such exist. This belief is a postulate. At least I should have said this earlier; and did say it. But I now think this statement was a concession to an over-rigorous conception of what deserves the name of "knowledge." For empirical knowledge, in distinction from merely meaningful belief, verification is required. But there is what we call "indirect verification" as well as "direct"; and there is "complete" or "decisive verification" and also "incomplete verification" or "confirmation" as more or less probable. There are reasons to think that these two distinctions - direct or indirect and complete or incomplete - reduce to one: that in any distinctive sense of "directly verifiable," that and that only is directly verifiable which is also completely and decisively verifiable. (The plausibility of this may be suggested by the thought that whatever is incompletely verified does not present itself in its full nature but is observed only in certain manifestations.) Most of what we call "knowledge" is not only incompletely verified at any time but - when the matter is considered carefully - must remain so forever. (It may be completely verifiable, or completely confirmable, in the sense that there is nothing which the truth of it requires which could not, given the conditions of verification, be found true or found false; but it is not completely verifiable in the sense that verification of it can be completed: - somewhat as there is no whole number which can not be counted, but counting of the whole numbers can not be completed.) In view of these facts (if these suggestions indicate fact), it may be that there is no fundamental difference, by reference to its verifiability, between the belief in other minds and the belief, for example, in ultra-violet rays or in electrons. It might even be that the belief in other minds, though always incompletely verified and incapable of becoming otherwise, is supported by inductive evidence so extensive as to be better confirmed than some of the accepted theses of physical science.