Tractatus Logico-Philosophicus (English): Difference between revisions

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 <math>N ( \bar{\xi} )</math> is the negation of all the values of the propositional variable ξ.
+.503 As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.
+.51 If ''ξ'' has only one value, then <math>N ( \bar{\xi} )</math> = ~''p'' (not ''p''), if it has two values then <math>N ( \bar{\xi} )</math> = ~''p . ~q'' (neither ''p'' nor ''q'').
+.511 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.
+.512 "~''p''" is true if "''p''" is false. Therefore in the true proposition "~''p''" "''p''" is a false proposition. How then can the stroke "~" bring it into agreement with reality?
+That which denies in "~''p''" is however not "~" but that which all signs of this notation, which deny ''p'', have in common.
+Hence the common rule according to which "~''p''", "~''~~p''", "~''p'' ∨ ''~p''", "~''p . ~p''", etc. etc. (to infinity) are constructed. And this which is common to them all mirrors denial.
+.513 We could say: What is common to all symbols, which assert both ''p'' and ''q'', is the proposition "''p'' . ''q''". What is common to all symbols, which assert either ''p'' or ''q'', is the proposition "''p'' ∨ ''q''".
+And similarly we can say: 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.
+Thus in Russell's notation also it appears evident that "''q'' : ''p'' ∨ ''~p''" says the same as "''q''"; that "''p'' ∨ ''~p''" says nothing.
+.514 If a notation is fixed, there is in it a rule according to which all the propositions denying ''p'' are constructed, a rule according to which all the propositions asserting ''p'' are constructed, a rule according to which all the propositions asserting ''p'' or ''q'' are constructed, and so on. These rules are equivalent to the symbols and in them their sense is mirrored.
+.515 It must be recognized in our symbols that what is connected by "∨", ".", etc., must be propositions.
+And this is the case, for the symbols "''p''" and "''q''" presuppose "∨", "''~''", etc. If the sign "''p''" in "''p'' ∨ ''q''" does not stand for a complex sign, then by itself it cannot have sense; but then also the signs "''p'' ∨ ''p''", "''p'' . ''p''" etc. which have the same sense as "''p''" have no sense. If, however, "''p'' ∨ ''p''" has no sense, then also "''p'' ∨ ''q''" can have no sense.
+.5151 Must the sign of the negative proposition be constructed by means of the sign of the positive? Why should one not be able to express the negative proposition by means of a negative fact? (Like: if "''a''" does not stand in a certain relation to "''b''", it could express that "''aRb''" is not the case.)
+But here also the negative proposition is indirectly constructed with the positive.
+The positive ''proposition'' must presuppose the existence of the negative ''proposition'' and conversely.
+.52 If the values of ''ξ'' are the total values of a function ''f x'' for all values of ''x'', then <math>N ( \bar{\xi} ) = \sim (\exist x) . f x</math>.
+.521 I separate the concept ''all'' from the truth-function.
+Frege and Russell have introduced generality in connexion with the logical product or the logical sum. Then it would be difficult to understand the propositions "" and "" in which both ideas lie concealed.
+.522 That which is peculiar to the "symbolism of generality" is firstly, that it refers to a logical prototype, and secondly, that it makes constants prominent.
+.523 The generality symbol occurs as an argument.
+.524 If the objects are given, therewith are ''all'' objects also given.
+If the elementary propositions are given, then therewith ''all'' elementary propositions are also given.
+.525 It is not correct to render the proposition "" — as Russell does — in words "''fx'' is ''possible''".
+Certainty, possibility or impossibility of a state of affairs are not expressed by a proposition but by the fact that an expression is a tautology, a significant proposition or a contradiction.
+That precedent to which one would always appeal, must be present in the symbol itself.
+.526 One can describe the world completely by completely generalized propositions, ''i.e.'', without from the outset co-ordinating any name with a definite object.
+In order then to arrive at the customary way of expression we need simply say after an expression *'there is one and only one ''x'', which ....": and this ''x'' is ''a'',
+.5261 A completely generalized proposition is like every other proposition composite. (This is shown by the fact that in "" we must mention "" and "" separately. Both stand independently in signifying relations to the world as in the ungeneralized proposition.)
+A characteristic of a composite symbol: it has something: in common with ''other'' symbols.
+.5262 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.)
+.53 Identity of the object I express by identity of the sign and not by means of a sign of identity. Difference of the objects by difference of the signs.
+.5301 That identity is not a relation between objects is obvious. This becomes very clear if, for example, one considers the proposition "" What this proposition says is simply that ''only''  satisfies the function , and not that only such things satisfy the function  which have a certain relation to .
+One could of course say that in fact ''only''  has this relation to  but in order to express this we should need the sign of identity itself.
+.5302 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''.)
+.5303 Roughly speaking: to say of ''two'' things that they are identical is nonsense, and to say of ''one'' thing that it is identical with itself is to say nothing.
+.531 I write therefore not "" but "" (or ""). And not "" but ""
+.532 And analogously : not "", but ""; and not "" but "".
+(Therefore instead of Russell's "": "".)
+.5321 Instead of "" we therefore write ''e.g.'' ""
+And the proposition "''only'' one satisfies " reads : "".
+.533 The identity sign is therefore not an essential constituent of logical notation.
+.534 And we see that apparent propositions like : "", "", "", "", etc. cannot be written in a correct logical notation at all.
+.535 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.
+.5351 There are certain cases in which one is tempted to use expressions of the form *"" or "". As, for instance, when one would speak of the archetype Proposition, Thing, etc. So Russell in the ''Principles of Mathematics'' has rendered the nonsense "''p'' is a proposition" in symbols by "" 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  before a proposition in order to ensure that its arguments have the right form, because the hypothesis for a non-proposition as argument becomes not false but meaningless, and because the proposition itself becomes senseless for arguments of the wrong kind, and therefore it survives the wrong arguments no better and no worse than the senseless hypothesis attached for this purpose.)
+.5352 Similarly it was proposed to express "There are no things" by "" But even if this were a proposition — would it not be true if indeed "There were things", but these were not identical with themselves?
+.54 In the general propositional form, propositions occur in a proposition only as bases of the truth-operations.
+.541 At first sight it appears as if there were also a different way in which one proposition could occur in another.
+Especially in certain propositional forms of psychology, like "A thinks, that  is the case", or "A thinks ", etc.
+Here it appears superficially as if the proposition  stood to the object A in a kind of relation.
+(And in modern epistemology (Russell, Moore, etc.) those propositions have been conceived in this way.)
+.542 But it is clear that "A believes that ", "A thinks ", "A says ", are of the form "<nowiki>''</nowiki> says ": and here we have no co-ordination of a fact and an object, but a co-ordination of facts by means of a co-ordination of their objects.
+.5421 This shows that there is no such thing as the soul — the subject, etc. — as it is conceived in contemporary superficial psychology.
+A composite soul would not be a soul any longer.
+.5422 The correct explanation of the form of the proposition "A judges " must show that it is impossible to judge a nonsense. (Russell's theory does not satisfy this condition.)