Tractatus Logico-Philosophicus: the original authoritative edition

by Ludwig Wittgenstein

Russell’s definition of “=” won’t do; because according to it one cannot say that two objects have all their properties in common. (Even if this proposition is never true, it is nevertheless significant.)

I write therefore not “f(a, b) . a = b”, but “f(a, a)” (or “f(b, b)”). And not “f(a, b) . ∼a = b”, but “f(a, b)”.

And analogously: not “(∃x, y).f(x, y).x = y”, but “(∃x).f(x, x)”; and not “(∃x, y) . f(x, y) . ∼x = y”, but “(∃x, y) . f(x, y)”. (Therefore instead of Russell’s “(∃x, y) . f(x, y)”: “(∃x, y) . f(x, y) . ∨ . (∃x) . f(x, x)”.) 5.5321 Instead of “(x) : fx ⊃ x = a” we therefore write e.g. “(∃x).fx. ⊃ .fa : ∼(∃x, y) . fx . fy”. And the proposition “only one x satisfies f()” reads: “(∃x) . fx : ∼(∃x, y) . fx . fy”.

So all problems disappear which are connected with such pseudo-propositions. This is the place to solve all the problems which arise through Russell’s “Axiom of Infinity”. What the axiom of infinity is meant to say would be expressed in language by the fact that there is an infinite number of names with different meanings.

So Russell in the Principles of Mathematics has rendered the nonsense “p is a proposition” in symbols by “p ⊃ p” and has put it as hypothesis before certain propositions to show that their places for arguments could only be occupied by propositions. (It is nonsense to place the hypothesis p ⊃ p before a proposition in order to ensure that its arguments have the right form,

The application of logic decides what elementary propositions there are.

5.63 I am my world. (The microcosm.) 5.631 The thinking, presenting subject; there is no such thing. If I wrote a book “The world as I found it”, I should also have therein to report on my body and say which members obey my will and which do not, etc. This then would be a method of isolating the subject or rather of showing that in an important sense there is no subject: that is to say, of it alone in this book mention could not be made.

6 The general form of truth-function is: [p, ξ, N(ξ)]. This is the general form of proposition.

6.021 A number is the exponent of an operation.

6.031 The theory of classes is altogether superfluous in mathematics. This is connected with the fact that the generality which we need in mathematics is not the accidental one.

6.1 The propositions of logic are tautologies.

6.12 The fact that the propositions of logic are tautologies shows the formal—logical—properties of language, of the world. That its constituent parts connected together in this way give a tautology characterizes the logic of its constituent parts.

It now becomes clear why we often feel as though “logical truths” must be “postulated” by us. We can in fact postulate them in so far as we can postulate an adequate notation.

also becomes clear why logic has been called the theory of forms and of inference.

The mark of logical propositions is not their general validity. To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologous just as well as a generalized one.

Propositions like Russell’s “axiom of reducibility” are not logical propositions, and this explains our feeling that, if true, they can only be true by a happy chance.

6.13 Logic is not a theory but a reflexion of the world. Logic is transcendental.

6.2 Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions.

6.21 Mathematical propositions express no thoughts.

6.234 Mathematics is a method of logic.

6.32 The law of causality is not a law but the form of a law.∗

6.33 We do not believe a priori in a law of conservation, but we know a priori the possibility of a logical form.

All propositions, such as the law of causation, the law of continuity in nature, the law of least expenditure in nature, etc. etc., all these are a priori intuitions of possible forms of the propositions of science.

Mechanics is an attempt to construct according to a single plan all true propositions which we need for the description of the world.

Laws, like the law of causation, etc., treat of the network and not of what the network described.

Hence the description of the temporal sequence of events is only possible if we support ourselves on another process.

The Kantian problem of the right and left hand which cannot be made to cover one another already exists in the plane, and even in one-dimensional space; where the two congruent figures a and b cannot be made to cover one another without moving them out of this space. The right and left hand are in fact completely congruent. And the fact that they cannot be made to cover one another has nothing to do with it.

What can be described can happen too, and what is excluded by the law of causality cannot be described.

The process of induction is the process of assuming the simplest law that can be made to harmonize with our experience. 6.3631 This process, however, has no logical foundation but only a psychological one.

That the sun will rise to-morrow, is an hypothesis; and that means that we do not know whether it will rise.

6.4 All propositions are of equal value.

It is clear that ethics cannot be expressed. Ethics are transcendental. (Ethics and æsthetics are one.)

The temporal immortality of the soul of man, that is to say, its eternal survival also after death, is not only in no way guaranteed, but this assumption in the first place will not do for us what we always tried to make it do. Is a riddle solved by the fact that I survive for ever? Is this eternal life not as enigmatic as our present one? The solution of the riddle of life in space and time lies outside space and time.

6.432 How the world is, is completely indifferent for what is higher. God does not reveal himself in the world.

For an answer which cannot be expressed the question too cannot be expressed. The riddle does not exist.

Scepticism is not irrefutable, but palpably senseless, if it would doubt where a question cannot be asked.

6.52 We feel that even if all possible scientific questions be answered, the problems of life have still not been touched at all. Of course there is then no question left, and just this is the answer. 6.521

Whereof one cannot speak, thereof one must be silent.