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April 6 - April 20, 2023
The proposition shows what it says, the tautology and the contradiction that they say nothing.
Tautology and contradiction are not pictures of the reality. They present no possible state of affairs. For the one allows every possible state of affairs, the other none.
The truth of tautology is certain, of propositions possible, of contradiction impossible. (Certain, possible, impossible: here we have an indication of that gradation which we need in the theory of probability.)
That there is a general form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed). The general form of proposition is: Such and such is the case.
Propositions are truth-functions of elementary propositions.
The elementary propositions are the truth-arguments of propositions.
The truth-functions can be ordered in series.
The truth-grounds of q are contained in those of p; p follows from q.
If p follows from q, the sense of “p” is contained in that of “q”.
If a god creates a world in which certain propositions are true, he creates thereby also a world in which all propositions consequent on them are true. And similarly he could not create a world in which the proposition “p” is true without creating all its objects.
p . q” is one of the propositions which assert “p” and at the same time one of the propositions which assert “q”.
Two propositions are opposed to one another if there is no significant proposition which asserts them both.
Every proposition which contradicts another, denies it.
From an elementary proposition no other can be inferred.
In no way can an inference be made from the existence of one state of affairs to the existence of another entirely different from it.
The events of the future cannot be inferred from those of the present. Superstition is the belief in the causal nexus.
The freedom of the will consists in the fact that future actions cannot be known now. We could only know them if causality were an inner necessity, like that of logical deduction.—The connexion of knowledge and what is known is that of logical necessity.
A tautology follows from all propositions: it says nothing.
Tautology is that which is shared by all propositions, which have nothing in common with one another.
Contradiction vanishes so to speak outside, tautology inside all propositions.
Independent propositions (e.g. any two elementary propositions) give to one another the probability ½.
If p follows from q, the proposition q gives to the proposition p the probability 1. The certainty of logical conclusion is a limiting case of probability.
A proposition is in itself neither probable nor improbable. An event occurs or does not occur, there is no middle course.
In an urn there are equal numbers of white and black balls (and no others). I draw one ball after another and put them back in the urn. Then I can determine by the experiment that the numbers of the black and white balls which are drawn approximate as the drawing continues. So this is not a mathematical fact.
What I can verify by the experiment is that the occurrence of the two events is independent of the circumstances with which I have no closer acquaintance.
Probability is a generalization. It involves a general description of a propositional form. Only in default of certainty do we need probability. If we are not completely acquainted with a fact, but know something about its form. (A proposition can, indeed, be an incomplete picture of a certain state of affairs, but it is always a complete picture.)
The probability proposition is, as it were, an extract from other propositions.
5.2 The structures of propositions stand to one another in internal relations.
The sense of a truth-function of p is a function of the sense of p.
Denial, logical addition, logical multiplication, etc. etc., are operations. (Denial reverses the sense of a proposition.)
The occurrence of an operation does not characterize the sense of a proposition. For an operation does not assert anything; only its result does, and this depends on the bases of the operation.
A function cannot be its own argument, but the result of an operation can be its own basis.
5.3 All propositions are results of truth-operations on the elementary propositions.
All truth-functions are results of the successive application of a finite number of truth-operations to elementary propositions.
Here it becomes clear that there are no such things as “logical objects” or “logical constants” (in the sense of Frege and Russell).
For all those results of truth-operations on truth-functions are identical, which are one and the same truth-function of elementary propositions.
That from a fact p an infinite number of others should follow, namely ∼∼p, ∼∼∼∼p, etc., is indeed hardly to be believed, and it is no less wonderful that the infinite number of propositions of logic (of mathematics) should follow from half a dozen “primitive propositions”. But all propositions of logic say the same thing. That is, nothing.
Truth-functions are not material functions. If e.g. an affirmation can be produced by repeated denial, is the denial—in any sense—contained in the affirmation? Does “∼∼p” deny ∼p, or does it affirm p; or both? The proposition “∼∼p” does not treat of denial as an object, but the possibility of denial is already prejudged in affirmation.
If logic has primitive ideas these must be independent of one another.
All numbers in logic must be capable of justification. Or rather it must become plain that there are no numbers in logic. There are no pre-eminent numbers.
The general form of proposition is the essence of proposition.
5.473 Logic must take care of itself.
Occam’s razor is, of course, not an arbitrary rule nor one justified by its practical success. It simply says that unnecessary elements in a symbolism mean nothing.
Every truth-function is a result of the successive application of the operation (– – – – –T)(ξ,....) to elementary propositions.
We may distinguish 3 kinds of description: 1. Direct enumeration. In this case we can place simply its constant values instead of the variable. 2. Giving a function fx, whose values for all values of x are the propositions to be described. 3. Giving a formal law, according to which those propositions are constructed. In this case the terms of the expression in brackets are all the terms of a formal series.
How can the all-embracing logic which mirrors the world use such special catches and manipulations? Only because all these are connected into an infinitely fine network, to the great mirror.
Two propositions are opposed to one another when they have nothing in common with one another; and every proposition has only one negative, because there is only one proposition which lies altogether outside it.
I separate the concept all from the truth-function.
If the objects are given, therewith are all objects also given.
The truth or falsehood of every proposition alters something in the general structure of the world. And the range which is allowed to its structure by the totality of elementary propositions is exactly that which the completely general propositions delimit. (If an elementary proposition is true, then, at any rate, there is one more elementary proposition true.)
