Either x is false or y is true,
and
Either y is false or z is true.
Suppose, now, that we seek to find the expression of the precise denial of x [which in logical terminology is called the contradictory of x]. Call this X. Then it is necessary and sufficient that X should be true when x is false and false when x is true. We may therefore put
X = u + v − x
or
X = uv/x.
These two expressions are equal, by the equation of excluded middle.
The simplest expression of that proposition which is true if x, y, z are all true and is false if any of them are false is
The simplest expression for the proposition which is true if any of the propositions x, y, z, is true, but is false if all are false is
It is now easy to see that some values of u and v are much more convenient than others. For example, the proposition which asserts that some two at least of the three propositions, x, y, z, are true, is, if u = 2, v = 5,
but if u = − 1, v = + 1, the same statement is simply
x + y + z − xyz.
Perhaps the system which would most readily occur to a mathematician would be to take the true, v, as an odd number, and the false, u, as an even one,3 and not to discriminate between numbers except as odd or even. Thus, we should have
v = 1 = 3 = 5 = 7 = etc.
u = 0 = 2 = 4 = 6 = etc.
In other words, we should measure round a circle, having its circumference equal to 2; so that 2 would fall on 0, 3 on 1, etc. On this system, every possible algebraical expression formed by means of the addition and multiplication of propositions would have a meaning. Thus, x + y + z 1 etc. would mean that some odd number of the propositions, x, y, z, were true. While xyz etc. would mean that the propositions x, y, z, were all true. For these would be the conditions of the expressions representing odd numbers. Subtraction would have the same meaning as addition for we should have − x = x. A quotient, as x/y, would not properly signify a proposition, since it would not necessarily represent any possible whole number. Namely, if x were odd and y even, x/y would be a fraction.
1. In using the conjunctions “either … or,” I always intend to leave open the possibility that both alternatives may hold good. By “either x or y,” I mean “Either x or y or both.”
2. A “rule” in algebra differs from most other rules, in that it requires nothing to be done, but only permits us to make certain transformations.
3. The Pythagorean notion was that odd was good, even bad.
19
Notes on the Question of the Existence of an External World
| c. 1890 | Houghton Library |
1. The idealistic argument turns upon the assumption that certain things are absolutely “present,” namely what we have in mind at the moment, and that nothing else can be immediately, that is, otherwise than inferentially known. When this is once granted, the idealist has no difficulty in showing that that external existence which we cannot know immediately we cannot know, at all. Some of the arguments used for this purpose are of little value, because they only go to show that our knowledge of an external world is fallible; now there is a world of difference between fallible knowledge and no knowledge. However, I think it would have to be admitted as a matter of logic that if we have no immediate perception of a non-ego, we can have no reason to admit the supposition of an existence so contrary to all experience as that would in that case be.
But what evidence is there that we can immediately know only what is “present” to the mind? The idealists generally treat this as self-evident; but, as Clifford jestingly says, “it is evident” is a phrase which only means “we do not know how to prove.” The proposition that we can immediately perceive only what is present seems to me parallel to that other vulgar prejudice that “a thing cannot act where it is not.” An opinion which can only defend itself by such a sounding phrase is pretty sure to be wrong. That a thing cannot act where it is not, is plainly an induction from ordinary experience which shows no forces except such as act through the resistance of materials, with the exception of gravity which, owing to its being the same for all bodies, does not appear in ordinary experience like a force. But further experience shows that attractions and repulsions are the universal types of forces. A thing may be said to be wherever it acts; but the notion that a particle is absolutely present in one part of space and absolutely absent from all the rest of space is devoid of all foundation. In like manner, the idea that we can immediately perceive only what is present, seems to be founded on our ordinary experience that we cannot recall and reexamine the events of yesterday nor know otherwise than by inference what is to happen tomorrow.
Peirce’s “Notes on the Question of the Existence of an External World” was written on the same type of laid paper as “The Architecture of Theories” and several other documents all dated 1890. The paper Peirce used for these notes is an important clue in dating this document circa 1890. (By permission of the Houghton Library, Harvard University.)
20
[Note on Kant’s Refutation of Idealism]
| c. 1890 | Houghton Library |
Kant’s refutation of idealism in the second edition of the Critic of the Pure Reason has been often held to be inconsistent with his main position or even to be knowingly sophistical. It appears to me to be one of the numerous passages in that work which betray an elaborated and vigorous analysis, marred in the exposition by the attempt to state the argument more abstractly and demonstratively than the thought would warrant.
In “Note 1,” Kant says that his argument beats idealism at its own game. How is that? The idealist says that all that we know immediately, that is, otherwise than inferentially, is what is present in the mind; and things out of the mind are not so present. The whole idealist position turns upon this conception of the present. Obviously, then, the first move toward beating idealism at its own game is to remark that we apprehend our own ideas only as flowing in time, and since neither the future nor the past, however near they may be, is present, there is as much difficulty in conceiving our perception of what passes within us as in conceiving external perception. If so, replies the idealist, instead of giving up idealism we must go still further to nihilism. Kant does not notice this retort; but it is clear from his footnote that he would have said: Not so; for it is impossible we should so much as think we think in time unless we do think in time; or rather, dismissing blind impossibility, the mere imagination of time is a clear perception of the past. Hamilton stupidly objects to Reid’s phrase “immediate memory”; but an immediate, intuitive, consciousness of time clearly exists wherever time exists. But once grant immediate knowledge in time, and