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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 connexion 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 ''qp'' 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 external limit of the propositions, tautology their substanceless centre.
5.15 If ''T<sub>r</sub>'' is the number of the truth-grounds of the proposition "''r''", ''T<sub>r</sub>'' 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<sub>rs</sub>:T<sub>r</sub>'' the measure of the ''probability'' which the proposition "''r''" gives to the proposition "''s''".
5.151 Suppose in a schema like that above in No. 5.101 ''T<sub>r</sub>'' is the number of the "T"'s in the proposition ''r'', ''T<sub>rs</sub>'' 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<sub>rs</sub>: T<sub>r</sub>.''
5.1511 There is no special object peculiar to probability propositions.
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 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.
(Application to tautology and contradiction.)
5.153 A proposition is in itself neither probable nor improbable. An event 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 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.
If then, I say, It is equally probable that I should draw 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 under- stood from the above explanations — to each the probability ½.
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 pro- position 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 to 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' ξ''" 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 successive 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 are 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 ∨, ⊃ 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 "∨" is identical with that by which we define "∨" with the help of "~", and that this "∨" 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 all propositions of logic say the same thing. That is, nothing.
5.44 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''∨''q'')", "(∃''x'') . ~''fx''" and others. We may not first introduce it for one 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.
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