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Tractatus Logico-Philosophicus (English): Difference between revisions
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A sphere in which the proposition, simplex sigillum veri, is valid. | A sphere in which the proposition, simplex sigillum veri, is valid. | ||
{{ParTLP|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''∨''q''" but also "~(''p''∨~''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''∨''q''", "(∃''x'') . '' | {{ParTLP|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 {{nowrap|"''p'' ∨ ''q''"}} but also {{nowrap|"~(''p'' ∨ ~''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 {{nowrap|"''p'' ∨ ''q''", "(∃ ''x'') . ''f x''",}} etc., but the most general form of their combinations. | ||
{{ParTLP|5.461}} The apparently unimportant fact that the apparent relations like ∨ and ⊃ need brackets—unlike real relations—is of great importance. | {{ParTLP|5.461}} The apparently unimportant fact that the apparent relations like ∨ and ⊃ need brackets—unlike real relations—is of great importance. | ||
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{{ParTLP|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''. | {{ParTLP|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 "''f a''" says the same as "(∃''x'') . '' | For all logical operations are already contained in the elementary proposition. For {{nowrap|"''f a''"}} says the same as {{nowrap|"(∃''x'') . ''f x'' . ''x'' = ''a''"}}. | ||
Where there is composition, there is argument and function, and where these are, all logical constants already are. | Where there is composition, there is argument and function, and where these are, all logical constants already are. | ||
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{{ParTLP|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. | {{ParTLP|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. | ||
{{ParTLP|5.5}} Every truth-function is a result of the successive application of the operation ( | {{ParTLP|5.5}} Every truth-function is a result of the successive application of the operation {{nowrap|(– – – – – T)(''ξ'', ....)}} to elementary propositions. | ||
{{ParTLP|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 "<math>( \bar{\xi} )</math>". "''ξ''" 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. | {{ParTLP|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 "<math>( \bar{\xi} )</math>". "''ξ''" 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. | ||
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How the description of the terms of the expression in brackets takes place is unessential. | How the description of the terms of the expression in brackets takes place is unessential. | ||
We may distinguish 3 kinds of description: Direct enumeration. In this case we can place simply its constant values instead of the variable. Giving a function ''f x'' whose values for all values of ''x'' are the propositions to be described. 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. | We may distinguish 3 kinds of description: Direct enumeration. In this case we can place simply its constant values instead of the variable. Giving a function {{nowrap|''f x''}} whose values for all values of ''x'' are the propositions to be described. 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. | ||
{{ParTLP|5.502}} Therefore I write instead of "( | {{ParTLP|5.502}} Therefore I write instead of {{nowrap|"(– – – – – T)(''ξ'', ....)",}} "<math>N ( \bar{\xi} )</math>". | ||
<math>N ( \bar{\xi} )</math> is the negation of all the values of the propositional variable ξ. | <math>N ( \bar{\xi} )</math> is the negation of all the values of the propositional variable ξ. | ||
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The positive ''proposition'' must presuppose the existence of the negative ''proposition'' and conversely. | The positive ''proposition'' must presuppose the existence of the negative ''proposition'' and conversely. | ||
{{ParTLP|5.52}} 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>. | {{ParTLP|5.52}} If the values of ''ξ'' are the total values of a function {{nowrap|''f x''}} for all values of ''x'', then <math>N ( \bar{\xi} ) = \sim (\exist x) . f x</math>. | ||
{{ParTLP|5.521}} I separate the concept ''all'' from the truth-function. | {{ParTLP|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 "(∃''x'') . ''f x''" and "(''x'') . ''f x''" 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 {{nowrap|"(∃ ''x'') . ''f x''" and "(''x'') . ''f x''"}} in which both ideas lie concealed. | ||
{{ParTLP|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. | {{ParTLP|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. | ||
{{ParTLP|5.525}} It is not correct to render the proposition "(∃''x'') . ''f x''"—as Russell does—in words "''f x'' is ''possible''". | {{ParTLP|5.525}} It is not correct to render the proposition {{nowrap|"(∃ ''x'') . ''f x''"}}—as Russell does—in words {{nowrap|"''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'', | ||
{{ParTLP|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.) | {{ParTLP|5.5261}} A completely generalized proposition is like every other proposition composite. (This is shown by the fact that in {{nowrap|"(∃ ''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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{{ParTLP|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. | {{ParTLP|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. | ||
{{ParTLP|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''. | {{ParTLP|5.5301}} That identity is not a relation between objects is obvious. This becomes very clear if, for example, one considers the proposition {{nowrap|"(''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'' ''a'' has this relation to ''a'' but in order to express this we should need the sign of identity itself. | One could of course say that in fact ''only'' ''a'' has this relation to ''a'' but in order to express this we should need the sign of identity itself. | ||
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{{ParTLP|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. | {{ParTLP|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. | ||
{{ParTLP|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'')" | {{ParTLP|5.531}} I write therefore not {{nowrap|"f(''a'', ''b'') . ''a''}} = ''b''" but {{nowrap|"''f'' (''a'', ''a'')"}} (or {{nowrap|"''f'' (''b'', ''b'')").}} And not {{nowrap|"''f'' (''a'', ''b'') . ~''a''}} = ''b''" but {{nowrap|"''f'' (''a'', ''b'')"}}. | ||
{{ParTLP|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'')". | {{ParTLP|5.532}} And analogously: not {{nowrap|"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ''x''}} = ''y''", but {{nowrap|"(∃''x'') . ''f'' (''x'', ''x'')";}} and not {{nowrap|"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ~''x''}} = ''y''" but {{nowrap|"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')".}} | ||
(Therefore instead of Russell's "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')": "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ∨ . (∃''x'') . ''f'' (''x'', ''x'')".) | (Therefore instead of Russell's {{nowrap|"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')"}}: {{nowrap|"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ∨ . (∃''x'') . ''f'' (''x'', ''x'')"}}.) | ||
{{ParTLP|5.5321}} Instead of "(''x'') : ''f'' ''x'' ⊃ ''x'' = ''a''" we therefore write ''e.g.'' "(∃''x'') . ''f'' ''x'' . ⊃ . ''f'' ''a'' : ~(∃ ''x'', ''y'') . ''f'' ''x'' . ''f'' ''y''". | {{ParTLP|5.5321}} Instead of {{nowrap|"(''x'') : ''f'' ''x'' ⊃ ''x''}} = ''a''" we therefore write ''e.g.'' {{nowrap|"(∃''x'') . ''f'' ''x'' . ⊃ . ''f'' ''a'' : ~(∃ ''x'', ''y'') . ''f'' ''x'' . ''f'' ''y''"}}. | ||
And the proposition "''only'' one ''x'' satisfies ''f''()" reads: "(∃''x'') . ''f'' ''x'' : ~(∃ ''x'', ''y'') . ''f'' ''x'' . ''f'' ''y''". | And the proposition "''only'' one ''x'' satisfies {{nowrap|''f''( )}}" reads: {{nowrap|"(∃ ''x'') . ''f'' ''x'' : ~(∃ ''x'', ''y'') . ''f'' ''x'' . ''f'' ''y''"}}. | ||
{{ParTLP|5.533}} The identity sign is therefore not an essential constituent of logical notation. | {{ParTLP|5.533}} The identity sign is therefore not an essential constituent of logical notation. | ||
{{ParTLP|5.534}} And we see that apparent propositions like: "''a'' = ''a''", "''a'' = ''b'' . ''b'' = ''c'' . ⊃ ''a'' = ''c''", "(''x'') . ''x'' = ''x''", "(∃''x'') . ''x'' = ''a''", etc. cannot be written in a correct logical notation at all. | {{ParTLP|5.534}} And we see that apparent propositions like: <span class="nowrap">"''a'' = ''a''"</span>, <span class="nowrap">"''a'' = ''b'' . ''b'' = ''c'' . ⊃ ''a'' = ''c''"</span>, <span class="nowrap">"(''x'') . ''x'' = ''x''"</span>, <span class="nowrap">"(∃ ''x'') . ''x'' = ''a''"</span>, etc. cannot be written in a correct logical notation at all. | ||
{{ParTLP|5.535}} So all problems disappear which are connected with such pseudo-propositions. | {{ParTLP|5.535}} So all problems disappear which are connected with such pseudo-propositions. | ||
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(It is nonsense to place the hypothesis ''p'' ⊃ ''p'' 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.) | (It is nonsense to place the hypothesis ''p'' ⊃ ''p'' 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.) | ||
{{ParTLP|5.5352}} Similarly it was proposed to express "There are no things" by "(∃''x'') . ''x'' = ''x''" But even if this were a proposition—would it not be true if indeed "There were things", but these were not identical with themselves? | {{ParTLP|5.5352}} Similarly it was proposed to express "There are no things" by <span class="nowrap">"(∃ ''x'') . ''x'' = ''x''"</span> But even if this were a proposition—would it not be true if indeed "There were things", but these were not identical with themselves? | ||
{{ParTLP|5.54}} In the general propositional form, propositions occur in a proposition only as bases of the truth-operations. | {{ParTLP|5.54}} In the general propositional form, propositions occur in a proposition only as bases of the truth-operations. | ||
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In order that propositions connected together in a definite way may give a tautology they must have definite properties of structure. That they give a tautology when ''so'' connected shows therefore that they possess these properties of structure. | In order that propositions connected together in a definite way may give a tautology they must have definite properties of structure. That they give a tautology when ''so'' connected shows therefore that they possess these properties of structure. | ||
{{ParTLP|6.1201}} That ''e.g.'' the propositions "''p''" and "~''p''" in the connexion "~(''p'' . ''~p'')" give a tautology shows that they contradict one another. That the propositions "''p'' ⊃ ''q''", "''p''" and "''q''" connected together in the form "(''p'' ⊃ ''q'') . (''p'') : ⊃ : (''q'')" give a tautology shows that "''q''" follows from "''p''" and "''p'' ⊃ ''q''". That "(''x'') . ''f x'' : ⊃ : ''f a''" is a tautology shows that '' | {{ParTLP|6.1201}} That ''e.g.'' the propositions "''p''" and "~''p''" in the connexion {{nowrap|"~(''p'' . ''~p'')"}} give a tautology shows that they contradict one another. That the propositions {{nowrap|"''p'' ⊃ ''q''"}}, "''p''" and "''q''" connected together in the form {{nowrap|"(''p'' ⊃ ''q'') . (''p'') : ⊃ : (''q'')"}} give a tautology shows that "''q''" follows from "''p''" and {{nowrap|"''p'' ⊃ ''q''"}}. That {{nowrap|"(''x'') . ''f x'' : ⊃ : ''f a''"}} is a tautology shows that {{nowrap|''f a''}} follows from {{nowrap|(''x'') . ''f x''}}, etc. etc. | ||
{{ParTLP|6.1202}} It is clear that we could have used for this purpose contradictions instead of tautologies. | {{ParTLP|6.1202}} It is clear that we could have used for this purpose contradictions instead of tautologies. | ||
{{ParTLP|6.1203}} In order to recognize a tautology as such, we can, in cases in which no sign of generality occurs in the tautology, make use of the following intuitive method: I write instead of "''p", "q", "r''", etc., "'' | {{ParTLP|6.1203}} In order to recognize a tautology as such, we can, in cases in which no sign of generality occurs in the tautology, make use of the following intuitive method: I write instead of "''p", "q", "r''", etc., {{nowrap|"''T p F''"}}, {{nowrap|"''T q F''"}}, {{nowrap|"''T r F''"}}, etc. The truth-combinations I express by brackets, ''e.g.'': | ||
[[File:TLP 6.1203a-en.png|300px|center|link=]] | [[File:TLP 6.1203a-en.png|300px|center|link=]] | ||