4.2
The sense of a proposition is its agreement and disagreement with the possibilities of the existence and non-existence of the atomic facts.

4.21
The simplest proposition, the elementary proposition, asserts the existence of an atomic fact.

4.211
It is a sign of an elementary proposition, that no elementary proposition can contradict it.

4.22
The elementary proposition consists of names. It is a connection, a concatenation, of names.

4.221
It is obvious that in the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination.

The question arises here, how the propositional connection comes to be.

4.2211
Even if the world is infinitely complex, so that every fact consists of an infinite number of atomic facts and every atomic fact is composed of an infinite number of objects, even then there must be objects and atomic facts.

4.23
The name occurs in the proposition only in the context of the elementary proposition.

4.24
The names are the simple symbols; I indicate them by single letters (x, y, z).

The elementary proposition I write as function of the names, in the form "fx", " (x, y)", etc.

Or I indicate it by the letters p, q, r.

4.241
If I use two signs with one and the same meaning, I express this by putting between them the sign "=".

"a=b" means then, that the sign "a" is replaceable by the sign "b".

(If I introduce by an equation a new sign "b", by determining that it shall replace a previously known sign "a", I write the equation -- definition -- (like Russell) in the form "a=b Def.". A definition is a symbolic rule.)

4.242
Expressions of the form "a=b" are therefore only expedients in presentation: They assert nothing about the meaning of the signs "a" and "b".

4.243
Can we understand two names without knowing whether they signify the same thing o r two different things? Can we understand a proposition in which two names occur, without knowing if they mean the same or different things?

If I know the meaning of an English and a synonymous German word, it is impossible for me not to know that they are synonymous, it is impossible for me not to be able to translate them into one another.

Expressions like "a=a", or expressions deduced from these are neither elementary propositions nor otherwise significant signs. (This will be shown later.)

4.25
If the elementary proposition is true, the atomic fact exists; if it is false the atomic fact does not exist.

4.26
The specification of all true elementary propositions describes the world completely. The world is completely described by the specification of all elementary propositions plus the specification, which of them are true and which false.

4.27
With regard to the existence of n atomic facts there are possibilities.

It is possible for all combinations of atomic facts to exist, and the others not to exist.

4.28

To these combinations correspond the same number of possibilities of the truth -- and falsehood -- of n elementary propositions.

4.3

The truth-possibilities of the elementary propositions mean the possibilities of the existence and non-existence of the atomic facts.

4.31
The truth-possibilities can be presented by schemata of the following kind ("T" means "true", "F" "false". The rows of T's and F's under the row of the elementary propositions mean their truth-possibilities in an easily intelligible symbolism).

p	q	r		p	q		p
T	T	T		T	T		T
F	T	T		F	T		F
T	F	T		T	F
T	T	F		F	F
F	F	T
F	T	F
T	F	F
F	F	F

4.4
A proposition is the expression of agreement and disagreement with the truth-possibilities of the elementary propositions.

4.41
The truth-possibilities of the elementary propositions are the conditions of the truth and falsehood of the propositions.

4.411
It seems probable even at first sight that the introduction of the elementary propositions is fundamental for the comprehension of the other kinds of propositions. Indeed the comprehension of the general propositions depends palpably on that of the general propositions.

4.42
With regard to the agreement and disagreement of a proposition with the truth-possibilities of n elementary propositions there are possibilities.

4.43
Agreement with the truth-possibilities can be expressed by co-ordinating with them in the scheme the mark "T" (true).

Absence of this mark means disagreement.

4.431
The expression of the agreement and disagreement with the truth-possibilities of the elementary propositions expresses the truth-conditions of the proposition.

The proposition is the expression of its truth-conditions.

(Frege has therefore quite rightly put them at the beginning, as explaining the signs of his logical symbolism. Only Frege's explanation of the truth-concept is false: if "the true" and "the false" were real objects and the arguments in ~p, etc., then the sense of ~p would by no means be determined by Frege's determination.)


4.44
The sign which arises from the co-ordination of that mark "T" with the truth-possibilities is a propositional sign.

4.441
It is clear that to the complex of the signs "F" and "T" no object (or complex of objects) corresponds; any more than to horizontal and vertical lines or to brackets. There are no "logical objects".

Something analogous holds of course for all signs, which express the same as the schemata of "T" and "F".

4.442
Thus e.g. ``

p	q
T	T	T
F	T	T
T	F
F	F	F

'' is a propositional sign.

(Frege's assertion sign " " is logically altogether meaningless; in Frege (and Russell) it only shows that these authors hold as true the propositions marked in this way.

" " belongs therefore to the propositions no more than does the number of the proposition. A proposition cannot possible assert of itself that it is true.)

If the sequence of the truth-possibilities in the scheme is once for all determined by a rule of combination, then the last column is by itself an expression of the truth-conditions. If we write this column as a row the propositional sign becomes: "(T T - T)(p, q)", or more plainly: "(T T F T)(p, q)".

The number of places in the left-hand bracket is determined by the number of terms in the right-hand bracket.)

4.45
For n elementary propositions there are L_n possible groups of truth-conditions.

The groups of truth-conditions which belong to the truth-possibilities of a number of elementary propositions can be ordered in a series.

4.46
Among the possible groups of truth-conditions there are two extreme cases.

In the one case the proposition is true for all the truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological.

In the second case the proposition is false for all the truth-possibilities. The truth-conditions are self-contradictory.

In the first case we call the proposition a tautology, in the second case a contradiction.

4.461
The proposition shows what it says, the tautology and the contradiction that they say nothing.

The tautology has no truth-conditions, for it is unconditionally true; and the contradiction is on no condition true.

Tautology and contradiction are without sense.

(Like the point from which two arrows go out in opposite directions.)

(I know, e.g. nothing about the weather, when I know that it rains or does not rain.)

4.4611
Tautology and contradiction are, however, not nonsensical; they are part of the symbolism, in the same way that "0" is part of the symbolism of Arithmetic.

4.462
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.

In the tautology the conditions of agreement with the world -- the presenting relations -- cancel one another, so that it stands in no presenting relation to reality.

4.463
The truth-conditions determine the range, which is left to the facts by the proposition.

(The proposition, the picture, the modem, are in a negative sense like a solid body, which restricts the free movement of another: in a positive sense, like the space limited by solid substance, in which a body may be placed.)

Tautology leaves to reality the whole infinite logical space; contradiction fills the whole logical space and leaves no point to reality. Neither of them, therefore, can in any way determine reality.

4.464
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.)

4.465
The logical product of a tautology and a proposition says the same as the proposition. Therefore that product is identical with the proposition. For the essence of the symbol cannot be altered without altering its sense.

4.666
To a definite logical combination of signs corresponds a definite logical combination of their meanings; every arbitrary combination only corresponds to the unconnected signs.

That is, propositions which are true for every stat of affairs cannot be combinations of signs at all, for otherwise there could only correspond to them definite combinations of objects.

(And to no logical combination corresponds no combination of the objects.)

Tautology and contradiction are the limiting cases of the combination of symbols, namely their dissolution.

4.4661
Of course the signs are also combined with one another in the tautology and contradiction, i.e. they stand in relations to one another, but these relations are meaningless, unessential to the symbol.

4.5
Now it appears to be possible to give the most general form of proposition; i.e. to give a description of the propositions of some one sign language, so that every possible sense can be expressed by a symbol, which falls under the description, and so that every symbol which falls under the description can express a sense, if the meanings of the names are chosen accordingly.

It is clear that in the description of the most general form of proposition only what is essential to it may be described -- otherwise it would not be the most general form.

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.

4.51
Suppose all elementary propositions were given me: then we can simply ask: what propositions I can build out of them. And these are all propositions and so are they limited.

4.52
The propositions are everything which follows from the totality of all elementary propositions (of course also from the fact that it is the totality of them all). (So, in some sense, one could say, that all propositions are generalizations of the elementary propositions.)

4.53
The general proposition form is a variable.

5
Propositions are truth-functions of elementary propositions.

(An elementary proposition is a truth-function of itself.)

5.0

5.01
The elementary propositions are the truth-arguments of propositions.

5.02
It is natural to confuse the arguments of functions with the indices of names. For I recognize the meaning of the sign containing it from the argument just as much as from the index.

In Russell's "+_c", for example, "c" is an index which indicates that the whole sign is the addition sign for cardinal numbers. But this way of symbolizing depends on arbitrary agreement, and could choose a simple sign instead of "+_c": but in "~p" "p" is not an index but an argument; the sense of "~p" cannot be understood, unless the sense of "p" has previously been understood. (In the name Julius Caesar, Julius is an index. The index is always part of a description of the object to whose name we attach it, e.g. The Caesar of the Julian gens.)

The confusion of argument and index is, if I am not mistaken, at the root of Frege's theory of the meaning of propositions and functions. For Frege the propositions of logical were names and their arguments the indices of these names.

5.1
The truth-functions can be ordered in series.

That is the foundation of the theory of probability.

5.10

5.101
The truth-functions of every number of elementary propositions can be written in a scheme of the following kind:

(T T T T)(p, q)	Tautology (if p then p, and if q then q) [p p . q q]
(F T T T)(p, q)	in words: Not both p and q. [~(p . q)]
(T F T T)(p, q)	'' '' If q then p. [q p]
(T T F T)(p, q)	'' '' If p then q. [p q]
(T T T F)(p, q)	'' '' p or q. [p v q]
(F F T T )(p, q)	'' '' Not q. [~q]
(F T F T)(p, q)	'' '' Not p. [~p]
(F T T F)(p, q)	'' '' p or q, but not both. [p . ~q :v: q . ~p]
(T F F T)(p, q)	'' '' If p, then q; and if q, then p. [p q]
(T F T F)(p, q)	'' '' p
(T T F F)(p, q)	'' '' q
(F F F T)(p, q)	'' '' Neither p nor q. [p . ~q or p | q]
(F F T F)(p, q)	'' '' p and not q. [p . ~q]
(F T F F)(p, q)	'' '' q and not p. [q . ~p]
(T F F F)(p, q)	'' '' p and q. [p . q]
(F F F F)(p, q)	Contradiction (p and not p; and q and not q.) [p . ~p . q . ~q]

Those truth-possibilities of its truth-arguments, which verify the proposition, I shall call its truth-grounds.

5.11
If the truth-grounds which are common to a number of propositions are all also truth-grounds of some one proposition, we say that the truth of this proposition follows from the truth of those propositions.

5.12
In particular the truth of a proposition p follows from that of a proposition q, if all the truth-grounds of the second are truth-grounds of the first.

5.121
The truth-grounds of q are contained in those of p; p follows from q.

5.122
If p follows from q, the sense of "p" is contained in that of "q".

5.123
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.

5.124
A proposition asserts every proposition which follows from it.

5.1241
"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.

5.13
That the truth of one proposition follows from the truth of other propositions, we perceive from the structure of the propositions.

5.131
If the truth of one proposition follows from the truth of others, this expresses itself in relations in which the forms of these propositions stand to one another, and we do not need to put them in these relations first by connecting them with one another in a proposition; for these relations are internal, and exist as soon as, and by the very fact that, the propositions exist.

5.1311
When we conclude from p v q and ~p to q the relation between the forms of the propositions "p v q" and "~p" is here concealed by the method of symbolizing. But if we write, e.g. instead of "p v q" "p | q .|. p | q" and instead of "~p" "p | p" (p | q = neither p nor q), then the inner connection becomes obvious.

(The fact that we can infer fa from (x) . fx shows that generality is present also in the symbol "(x) . fx".

5.132
If p follows from q, I can conclude from q to p; infer p from q.

The method of inference is to be understood from the two propositions alone.

Only they themselves can justify the inference.

Laws of inference, which -- as in Frege and Russell -- are to justify the conclusions, are senseless and would be superfluous.

5.133
All inference takes place a priori.


5.134
From an elementary proposition no other can be inferred.

5.135
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.

5.136
There is no causal nexus which justifies such an inference.

5.1361
The events of the future cannot be inferred from those of the present.

Superstition is the belief in the causal nexus.

5.1362
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 knows that p is the case" is senseless if p is a tautology.)

5.1363
If from the fact that a proposition is obvious to us it does not follow that it is true, then obviousness is no justification for our belief in its truth.

5.14
If a proposition follows from another, then the latter says more than the former, the former less than the latter.

5.141
If p follows from q and q from p then they are one and the same proposition.

5.142
A tautology follows from all propositions: it says nothing.

5.143
Contradiction is something shared by propositions, which no proposition has in common with another. 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.

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

5.15
If T_r is the number of the truth-grounds of the proposition "r", T_rs the number of those truth-grounds of the proposition "s" which are at the same time truth-grounds of "r", then we call the ratio T_rs : T_r the measure of the probability which the proposition "r" gives to the proposition "s".

5.151
Suppose in a scheme like that above in No. 5.101 T_r is the number of the "T"'s in the proposition r, T_rs the number of those "T"'s in the proposition s, which stand in the same columns as "T"'s of the proposition r; then the proposition r gives to the proposition s the probability T_rs : T_r.

5.152
Propositions which have no truth-arguments in common with one another we call independent.

Two elementary propositions give to one another the probability 1/2.

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.

(Application to tautology and contradiction.)

5.153
A proposition is in itself neither probable nor improbable. An even occurs or does not occur, there is no middle course.

5.154
In an urn there are equal numbers of white and black balls (and no others). I draw on 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.

If then, I say, It is equally probable that I should d raw a white and a black ball, this means, All the circumstances known to me (including the natural laws hypothetically assumed) give to the occurrence of the one event no more probability than to the occurrence of the other. That is they give -- as can easily be understood from the above explanations -- to each the probability 1/2.

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.

5.155
The unit of the probability proposition is: The circumstances -- with which I am not further acquainted -- give to the occurrence of a definite event such and such a degree of probability.

5.156
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.

5.21

We can bring out these internal relations in our manner of expression, by presenting a proposition as the result of an operation which produces it from other propositions (the bases of the operation).

5.22

The operation is the expression of a relation between the structures of its result and its bases.

5.23

The operation is that which must happen to a proposition in order to make another out of it.

5.231

And that will naturally depend on their formal properties, on the internal similarity of their forms.

5.232

The internal relation which orders a series is equivalent to the operation by which one term arises from another.

5.233

The first place in which an operation can occur is where a proposition arises from another in a logically significant way; i.e. where the logical construction of the proposition begins.

5.234

The truth-functions of elementary proposition. are results of operations which have the elementary propositions as bases. (I call these operations, truth-operations.)

5.2341

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.)

5.24

An operation shows itself in a variable; it shows how we can proceed from one form of proposition to another.

It gives expression to the difference between the forms.

(And that which is common the bases, and the result of an operation, is the bases themselves.)

5.241

The operation does not characterize a form but only the difference between forms.

5.242

The same operation which makes "q" from "p", makes "r" from "q", and so on. This can only be expressed by the fact that "p", "q", "r", etc., are variables which give general expression to certain formal relations.

5.25

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.

(Operation and function must not be confused with one another.)

5.251

A function cannot be its own argument, but the result of an operation can be its own basis.

5.252

Only in this way is the progress from term to term in a formal series possible (from type to type in the hierarchy of Russell and Whitehead). (Russell and Whitehead have not admitted the possibility of this progress but have made use of it all the same.)

5.2521

The repeated application of an operation to its own result I call its successive application ("O' O' O' a" is the result of the threefold successive application of "O'" to "a").

In a similar sense I speak of the successive application of several operations to a number of propositions.

5.2522

The general term of the formal series a, O' a, O' O' a, . . . I write thus: "[a, x, O' x]". This expression in brackets is a variable. The first term of the expression is the beginning of the formal series, the second the form of an arbitrary term x of the series, and the third the form of that term of the series which immediately follows x.

5.2523

The concept of the success application of an operation is equivalent to the concept "and so on".

5.253

One operation can reverse the effect of another. Operations can cancel one another.

5.254

Operations can vanish (e.g. denial in "~~p". ~~p = p).

5.3

All propositions are results of truth-operations on the elementary propositions.

The truth-operation is the way in which a truth-function arises from elementary propositions.

According to the nature of truth-operations, in the same way as out of elementary propositions arise their truth-functions, from truth-functions arises a new one. Every truth-operation creates from truth-functions of elementary propositions, another truth-function of elementary propositions i.e. a proposition. The result of every truth-operation on the results of truth-operations on elementary propositions is also the result of one truth-operation on elementary propositions.

Every proposition is the result of truth-operations on elementary propositions.

5.31

The Schemata No. 4.31 is also significant, if "p", "q", "r", etc. are not elementary propositions.

And it is easy to see that the propositional sign in No. 4.42 expresses one truth-function of elementary propositions even when "p" and "q" are truth-functions of elementary propositions.

5.32

All truth-functions are results of the successive application of a finite number of truth-operations to elementary propositions.

5.4

Here it becomes clear that there are no such things as "logical objects" or "logical constants" (in the sense of Frege and Russell).

5.41

For all those results of truth-operations on truth-functions are identical, which are one and the same truth-function of elementary propositions.

5.42

That v, , etc., are not relations in the sense of right and left, etc., is obvious.

The possibility of crosswise definition of the logical "primitive signs" of Frege and Russell shows by itself that these are not primitive signs and that they signify no relations.

And it is obvious that the "" which we define by means of "~" and "v" is identical with that by which we define "v" with the help of "~", and that this "v" is the same as the first, and so on.

5.43

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 the propositions of logic say the same thing. That is, nothing.

5.4

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.

And if there was an object called "~", then "~~p" would have to say something other than "p". For the one proposition would then treat of ~, the other would not.

5.441

This disappearance of the apparent logical constants also occurs if "~(x) . ~fx" says the same as "(x) . fx", or "(x) . fx . x=a" the same as "fa".

5.442

If a proposition is given to us then the results of all truth-operations which have it as their basis are given with it.

5.45

If there are logical primitive signs a correct logic must make clear their position relative to one another and justify their existence. The construction of logic out of its primitive signs must become clear.

5.451

If logic has primitive ideas these must be independent of one another. If a primitive idea is introduced it must be introduced in all contexts in which it occurs at all. One cannot therefore introduce it for one context and then again for another. For example, if denial is introduced, we must understand it in propositions of the form "~p", just as in propositions like "~(p v q)", "(x) . ~fx" and others. We may not first introduce it for oone class of cases and then for another, for it would then remain doubtful whether its meaning in the two cases was the same, and there would be no reason to use the same way of symbolizing in the two cases.

(In short, what Frege ("Grundgesetze der Arithmetik") has said about the introduction of signs by definitions holds, mutatis mutandis, for the introduction of primitive signs also.)

5.452

The introduction of a new expedient in the symbolism of logic must always be an event full of consequences. No new symbol may be introduced in logic in brackets or in the margin -- with, so to speak, an entirely innocent face.

(Thus in the "Principia Mathematica" of Russell and Whitehead there occur definitions and primitive propositions in words. Why suddenly words here? This would need a justification. There was none, and can be none for the process is actually not allowed.)

But if the introduction of a new expedient has proved necessary in one place, we must immediately ask: Where is this expedient always to be used? Its position in logic must be made clear.

5.453

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.

5.454

In logic there is no side by side, there can be no classification.

In logic there cannot be a more general and a more special.

5.4541

The solution of logical problems must be neat for they set the standard of neatness.

Men have always thought that there must be a sphere of questions whose answers -- a priori -- are symmetrical and united into a closed regular structure.

A sphere in which the proposition, simplex sigillum veri, is valid.

5.46

When we have rightly introduced the logical signs, the sense of all their combinations has been already introduced with them: therefore not only "p v q" but also "~(p v ~q)", etc. etc. We should then already have introduced the effect of all possible combinations of brackets; and it would then have become clear that the proper general primitive signs are not "p v q", "(x) . fx", etc., but the most general form of their combinations.

5.461

The apparently unimportant fact that the apparent relations like v and need brackets -- unlike real relations -- is of great importance.

The use of brackets with these apparent primitive signs shows that these are not the real primitive signs; and nobody of course would believe that the brackets have meaning by themselves.

5.4611

Logical operation signs are punctuations.

5.47

It is clear that everything which can be said beforehand about the form of all propositions at all can be said on one occasion.

For all logical operations are already contained in the elementary proposition. For "fa" says the same as "(x) . fx . x=a".

Where there is composition, there is argument and function, and where these are, all logical constants already are.

One could say: the one logical constant is that which all propositions, according to their nature, have in common with one another.

That however is the general form of proposition.

5.471

The general form of proposition is the essence of proposition.

5.4711

To give the essence of proposition means to give the essence of all description, therefore the essence of the world.

5.472

The description of the most general propositional form is the description of the one and only general primitive sign in logic.

5.473

Logic must take care of itself.

A possible sign must also be able to signify. Everything which is possible in logic is also permitted. ("Socrates is identical" means nothing because there is no property which is called "identical". The proposition is senseless because we have not made some arbitrary determination, not because the symbol is in itself impermissible.)

In a certain sense we cannot make mistakes in logic.

5.4731

Self-evidence, of which Russell has said so much, can only be discard in logic by language itself preventing every logical mistake. That logic is a priori consists in the fact that we cannot think illogically.

5.4732

We cannot give a sign the wrong sense.

5.47321

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.

Signs which serve one purpose are logically equivalent; signs which serve no purpose are logically meaningless.

5.4733

Frege says: Every legitimately constructed proposition msut have a sense; and I say: Every possible proposition is legitimately constructed, and if it has no sense this can only be because we have given no meaning to some of its constituent parts.

(Even if we believe that we have done so.)

Thus "Socrates is identical" says nothing, because we have given no meaning to the word "identical" as adjective. For when it occurs as the sign of equality it symbolizes in an entirely different way -- the symbolizing relation is another -- therefore the symbol is in the two cases entirely different; the two symbols have the sign in common with one another only by accident.

5.474

The number of necessary fundamental operations depends only on our notation.

5.475

It is only a question of constructing a system of signs of a definite number of dimensions -- of a definite mathematical multiplicity.

5.476

It is clear that we are not concerned here with a number of primitive ideas which must be signified but with the expression of a rule.

5.5

Every truth-function is a result of the successive application of the operation

(- - - - -T) (, . . . .) to elementary propositions.

This operation denies all the propositions in the right-hand bracket and I call it the negation of these propositions.

5.50

5.501

An expression in brackets whose terms are propositions I indicate -- if the order of the terms in the bracket is indifferent -- by a sign of the form "()". "" is a variable whose values are the terms of the expression in brackets, and the line over the variable indicates that it stands for all its values in the bracket.

(Thus if has the 3 values P, Q, R, then () = (P, Q, R).)

The values of the variables must be determined.

The determination is the description of the propositions which the variable stands for.

How the description of the terms of the expression in brackets takes place is unessential.

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.

5.502

Therefore I write instead of "(- - - - - T)(, . . . .)", "N()".

N() is the negation of all the values of the propositional variable .

5.503

As it is obviously easy to express how propositions can be constructioned by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.

5.51

If has only one value, then N()=~p (not p), if it has two values then N()=~p . ~q (neither p nor q).