Tractatus Logico-Philosophicus (English): Difference between revisions

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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.

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'').''fx'':⊃:''fa''" is a tautology shows that ''fa'' follows from (''x'') ,''fx'', etc. etc.

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 ''fa'' follows from (''x'') . ''f x'', etc. etc.

6.1202 It is clear that we could have used for this purpose contradictions instead of tautologies.

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., "''TpF", "TqF", "TrF''", etc. The truth-combinations I express by brackets, ''e.g.'':

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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.
-.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'').''fx'':⊃:''fa''" is a tautology shows that ''fa'' follows from (''x'') ,''fx'', etc. etc.
+.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 ''fa'' follows from (''x'') . ''f x'', etc. etc.
 .1202 It is clear that we could have used for this purpose contradictions instead of tautologies.
 .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., "''TpF", "TqF", "TrF''", etc. The truth-combinations I express by brackets, ''e.g.'':