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Tractatus Logico-Philosophicus (English): Difference between revisions

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{{ParTLP|3.344}} What signifies in the symbol is what is common to all those symbols by which it can be replaced according to the rules of logical syntax.
{{ParTLP|3.344}} What signifies in the symbol is what is common to all those symbols by which it can be replaced according to the rules of logical syntax.


{{ParTLP|3.3441}} We can, for example, express what is common to all notations for the truth-functions as follows: It is common to them that they all, for example, ''can be replaced'' by the notations of “~''p''” (“not ''p''”) and “''p''∨''q''” (“''p'' or ''q''”).
{{ParTLP|3.3441}} We can, for example, express what is common to all notations for the truth-functions as follows: It is common to them that they all, for example, ''can be replaced'' by the notations of “~''p''” (“not ''p''”) and {{nowrap|“''p'' ∨ ''q''”}} (“''p'' or ''q''”).


(Herewith is indicated the way in which a special possible notation can give us general information.)
(Herewith is indicated the way in which a special possible notation can give us general information.)
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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.
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.


{{ParTLP|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''".
{{ParTLP|5.441}} This disappearance of the apparent logical constants also occurs if {{nowrap|"~(∃ ''x'') . ~''f x''"}} says the same as {{nowrap|"(''x'') . ''f x''"}}, or <span class="nowrap">"(∃ ''x'') . ''fx'' . ''x'' = ''a''"</span> the same as {{nowrap|"''f a''"}}.


{{ParTLP|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.
{{ParTLP|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.
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{{ParTLP|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.
{{ParTLP|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.


{{ParTLP|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.
{{ParTLP|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 {{nowrap|"~(''p'' ∨ ''q'')",}} {{nowrap|"(∃ ''x'') . ~''f x''"}} 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.)
(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.)
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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 {{nowrap|"''f a''"}} says the same as {{nowrap|"(∃''x'') . ''f x'' . ''x'' = ''a''"}}.
For all logical operations are already contained in the elementary proposition. For {{nowrap|"''f a''"}} says the same as <span class="nowrap">"(∃''x'') . ''f x'' . ''x'' = ''a''"</span>.


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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That which denies in "~''p''" is however not "~" but that which all signs of this notation, which deny ''p'', have in common.
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.
Hence the common rule according to which "~''p''", "~''~~p''", {{nowrap|"~''p'' ∨ ''~p''"}}, {{nowrap|"~''p . ~p''",}} etc. etc. (to infinity) are constructed. And this which is common to them all mirrors denial.


{{ParTLP|5.513}} 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''".
{{ParTLP|5.513}} 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 {{nowrap|"''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.
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.
Thus in Russell's notation also it appears evident that {{nowrap|"''q'' : ''p'' ∨ ''~p''"}} says the same as "''q''"; that {{nowrap|"''p'' ∨ ''~p''"}} says nothing.


{{ParTLP|5.514}} 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.
{{ParTLP|5.514}} 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.
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{{ParTLP|5.515}} It must be recognized in our symbols that what is connected by "∨", ".", etc., must be propositions.
{{ParTLP|5.515}} 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.
And this is the case, for the symbols "''p''" and "''q''" presuppose "∨", "''~''", etc. If the sign "''p''" in {{nowrap|"''p'' ∨ ''q''"}} does not stand for a complex sign, then by itself it cannot have sense; but then also the signs {{nowrap|"''p'' ∨ ''p''"}}, {{nowrap|"''p'' . ''p''"}} etc. which have the same sense as "''p''" have no sense. If, however, {{nowrap|"''p'' ∨ ''p''"}} has no sense, then also {{nowrap|"''p'' ∨ ''q''"}} can have no sense.


{{ParTLP|5.5151}} 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.)
{{ParTLP|5.5151}} 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.)
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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 {{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|5.531}} I write therefore not <span class="nowrap">"f(''a'', ''b'') . ''a'' = ''b''"</span> but {{nowrap|"''f'' (''a'', ''a'')"}} (or {{nowrap|"''f'' (''b'', ''b'')").}} And not <span class="nowrap">"''f'' (''a'', ''b'') . ~''a''</span> = ''b''" but {{nowrap|"''f'' (''a'', ''b'')"}}.


{{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'')".}}
{{ParTLP|5.532}} And analogously: not <span class="nowrap">"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ''x'' = ''y''"</span>, but {{nowrap|"(∃''x'') . ''f'' (''x'', ''x'')"}}; and not <span class="nowrap">"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ~''x'' = ''y''"</span> but {{nowrap|"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')".}}


(Therefore instead of Russell's {{nowrap|"(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')"}}: {{nowrap|"(∃ ''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 {{nowrap|"(''x'') : ''f'' ''x'' ⊃ ''x''}} = ''a''" we therefore write ''e.g.'' {{nowrap|"(∃''x'') . ''f'' ''x'' . ⊃ . ''f'' ''a'' : ~(∃ ''x'', ''y'') . ''f'' ''x'' . ''f'' ''y''"}}.
{{ParTLP|5.5321}} Instead of <span class="nowrap">"(''x'') : ''f'' ''x'' ⊃ ''x'' = ''a''"</span> we therefore write ''e.g.'' {{nowrap|"(∃''x'') . ''f'' ''x'' . ⊃ . ''f'' ''a'' : ~(∃ ''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''"}}.
And the proposition "''only'' one ''x'' satisfies {{nowrap|''f''( )}}" reads: {{nowrap|"(∃ ''x'') . ''f'' ''x'' : ~(∃ ''x'', ''y'') . ''f'' ''x'' . ''f'' ''y''"}}.
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