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Tractatus Logico-Philosophicus (English): Difference between revisions
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5.521 I separate the concept ''all'' from the truth-function. | 5.521 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. | 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 "(∃''x'') . ''f x''" and "(''x'') . ''f x''" in which both ideas lie concealed. | ||
5.522 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. | 5.522 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. | ||
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If the elementary propositions are given, then therewith ''all'' elementary propositions are also given. | If the elementary propositions are given, then therewith ''all'' elementary propositions are also given. | ||
5.525 It is not correct to render the proposition "" — as Russell does — in words "'' | 5.525 It is not correct to render the proposition "(∃''x'') . ''f x''" — as Russell does — in words "''f x'' 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. | 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. | ||
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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'', | 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'', | ||
5.5261 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.) | 5.5261 A completely generalized proposition is like every other proposition composite. (This is shown by the fact that in "(∃''x'', ''φ'') . ''φ x''" we must mention "''φ''" and "''x''" 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. | A characteristic of a composite symbol: it has something: in common with ''other'' symbols. | ||
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5.53 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. | 5.53 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. | ||
5.5301 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 . | 5.5301 That identity is not a relation between objects is obvious. This becomes very clear if, for example, one considers the proposition "(''x'') : ''f x'' . ⊃ . ''x'' = ''a''" What this proposition says is simply that ''only'' ''a'' satisfies the function ''f'', and not that only such things satisfy the function ''f'' which have a certain relation to ''a''. | ||
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. | 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. | ||
5.5302 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''.) | 5.5302 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''.) | ||
5.5303 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. | 5.5303 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. | ||
5.531 I write therefore not "" but "" (or ""). And not "" but "" | 5.531 I write therefore not "f(''a'', ''b'') . ''a'' = ''b''" but "''f'' (''a'', ''a'')" (or "''f'' (''b'', ''b'')"). And not "''f'' (''a'', ''b'') . ~''a'' = b" but "''f'' (''a'', ''b'')" | ||
5.532 And analogously : not "", but ""; and not "" but "". | 5.532 And analogously : not "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ''x'' = ''y''", but "(∃''x'') . ''f'' (''x'', ''x'')"; and not "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ~''x'' = ''y''" but "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')". | ||
(Therefore instead of Russell's "": "".) | (Therefore instead of Russell's "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')": "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ∨ . (∃''x'') . ''f'' (''x'', ''x'')".) | ||
5.5321 Instead of "" we therefore write ''e.g.'' "" | 5.5321 Instead of "" we therefore write ''e.g.'' "" | ||