Tautology and contradiction are, however, not senseless; they are part of the symbolism, in the same way that “0” is part of the symbolism of Arithmetic.

Contradiction is the external limit of the propositions, tautology their substanceless centre.

Yes or no?' said Murphy. The eternal tautology.

6.1 The propositions of logic are tautologies.
6.2 Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions.
6.3 The exploration of logic means the exploration of everything that is subject to law. And outside logic everything is accidental.
6.4 All propositions are of equal value.
6.5 When the answer cannot be put into words, neither can the question be put into words. The riddle does not exist. If a question can be framed at all, it is also possible to answer it.

The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.

When you have excluded the impossible, whatever remains, however improbable, must be the truth.

Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.

The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle. Are we then to admit that the enunciations of all the theorems with which so many volumes are filled, are only indirect ways of saying that A is A?

It is certain because it is impossible

The very possibility of mathematical science seems an insoluble contradiction. If this science is deductive in appearance only, from where does it get its perfect rigor that no one dares to doubt? If, on the contrary, all the propositions it sets forth can be derived from one another by the rules of formal logic, why is mathematics not reducible to an immense tautology? Syllogism can teach us nothing that is essentially new and, if everything originated in the principle of identity, it should also be possible to reduce everything to it. Are we then to concede that the statements of all those theorems filling so many volumes are merely roundabout ways of saying that A is A?

I know three things will never be believed - the true, the probable, and the logical

Truth can be stated in a thousand different ways, yet each one can be true.

1. The world is everything that is the case.
2. What is the case (a fact) is the existence of states of affairs.
3. A logical picture of facts is a thought.
4. A thought is a proposition with a sense.
5. A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)
6. The general form of a proposition is the general form of a truth function, which is: [p, E, N(E)]. This is the general form of a proposition.
7. Whereof one cannot speak, thereof one must be silent.

1. When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.
2. The only way of discovering the limits of the possible is to venture a little way past them into the impossible.
3. Any sufficiently advanced technology is indistinguishable from magic.