# Bertrand Russell

 The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify. From the Introduction to the second edition of The Principles of Mathematics ([Russell37])

### On Defining Logic

It seems clear that there must be some way of defining logic otherwise than in relation to a particular logical language. The fundamental characteristic of logic, obviously, is that which is indicated when we say that logical propositions are true in virtue of their form. The question of demonstrability cannot enter in, since every proposition which, in one system, is deduced from the premises, might, in another system, be itself taken as a premise. If the proposition is complicated, this is inconvenient, but it cannot be impossible. All the propositions that are demonstrable in any admissible logical system must share with the premises the property of being true in virtue of their form; and all propositions which are true in virtue of their form ought to be included in any adequate logic. Some writers, for example Carnap in his "Logical Syntax of Language," treat the whole matter as being more a matter of linguistic choice than I can believe it to be. In the above mentioned work, Carnap has two logical languages, one of which admits the multiplicative axiom and the axiom of infinity, while the other does not. I cannot myself regard such a matter as one to be decided by our arbitrary choice. It seems to me that these axioms either do, or do not, have the characteristic of formal truth which characterises logic, and that in the former event every logic must include them, while in the latter every logic must exclude them. I confess, however, that I am unable to give any clear account of what is meant by saying that a proposition is "true in virtue of its form." But this phrase, inadequate as it is, points, I think, to the problem which must be solved if an adequate definition of logic is to be found.
From the Introduction to the second edition of The Principles of Mathematics ([Russell37])

### Defining Mathematics and Logic

Generality
Certain characteristics of the subject are clear. To begin with, we do not in this subject deal with particular things or particular properties: we deal formally with what can be said about any thing or any property. We are prepared to say that one and one are two, but not that Socrates and Plato are two...
 Formality Thus the absence of all mention of particular things or properties in logic or pure mathematics is a necessary result of the fact that this study is, as we say, "purely formal". Form The "form" of a proposition is that, in it, which remains unchanged when every constituent of the proposition is replaced by another.
 a priori Logical propositions are such as can be known a priori without study of the actual world. Analytic It is clear that the definition of "logic" or "mathematics" must be sought by trying to give a new definition of the old notion of "analytic" propositions.
from Chapter XVIII of Introduction to Mathematical Philosophy ([Russell19])

### The cult of "common usage"

The doctrine, as I understand it, consists in maintaining that the language of daily life, with words used in their ordinary meanings, suffices for philosophy, which has no need of technical terms or of changes in the significance of common terms. I find myself totally unable to accept this view. I object to it:
1. Because it is insincere;
2. Because it is capable of excusing ignorance of mathematics, physics and neurology in those who have had only a classical education;
3. Because it is advanced by some in a tone of unctuous rectitude, as if opposition to it were a sin against democracy;
4. Because it makes philosophy trivial;
5. Because it makes almost inevitable the perpetuation amongst philosophers of the muddle-headedness they have taken over from common sense.
Portraits from Memory, also in [Russell61]

### Philosophy as Logic (1)*1 *2

In the first place a philosophical proposition must be general. It must not deal specially with things on the surface of the earth, or within the solar system, or with any other portion of space and time. ... This brings us to a second characteristic of philosophical propositions, namely that they must be a priori. A philosophical proposition must be such as can neither be proved nor disproved by empirical evidence. ...
Philosophy, if what has been said is correct, becomes indistinguishable from logic as that word has now come to be used.
On Scientific Method in Philosophy, in [Russell17]

### Philosophy as Logic (2)

The study of logic becomes the central study in philosophy: it gives the method of research in philosophy, just as mathematics gives the method in physics.... All this supposed knowledge in the traditional systems must be swept away, and a new beginning must be made. . . .
To the large and still growing body of men engaged in the pursuit of science, . . . the new method, successful already in such time-honored problems as number, infinity, continuity, space and time, should make an appeal which the older methods have wholly failed to make. The one and only condition, I believe, which is necessary in order to secure for philosophy in the near future an achievement surpassing all that has hitherto been accomplished by philosophers, is the creation of a school of men with scientific training and philosophical interests, unhampered by the traditions of the past, and not misled by the literary methods of those who copy the ancients in all except their merits.
Our Knowledge of the External World, as a Field For Scientific Method in Philosophy

### The Mechanisation of Deduction

I am delighted to know that Principia Mathematica can now be done by machinery. . . I am quite willing to believe that anything in deductive logic can be done by machinery.
In a Letter to Herbert Simon, Nov. 2nd 1956; quoted in [MacKenzie95]

*1 By contrast, see Robert Nozick on Explanation versus Proof.

*2 In sympathy, see Alfred Ayer affirming that philosophy is a department of logic.

© created 1997-5-10 modified 2002-9-24