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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.'': | 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.'': | ||
[[File:TLP 6.1203a-en.png|300px|center|link=]] | |||
and the co-ordination of the truth or falsity of the whole proposition with the truth-combinations of the truth-arguments by lines in the following way: | |||
[[File:TLP 6.1203b-en.png|300px|center|link=]] | |||
This sign, for example, would therefore present the proposition "p⊃q". Now I will proceed to inquire whether such a proposition as ~(''p. ~p'') (The Law of Contradiction) is a tautology. The form "~ ''ξ''" is written in our notation | |||
[[File:TLP 6.1203c-en.png|250px|center|link=]] | |||
the form "''ξ . η''" thus : — | |||
[[File:TLP 6.1203d-en.png|300px|center|link=]] | |||
Hence the proposition ''~(p . ~ q)'' runs thus:— | |||
[[File:TLP 6.1203e-en.png|250px|center|link=]] | |||
If here we put "''p''" instead of "''q''" and examine the combination of the outermost T and F with the innermost, it is seen that the truth of the whole proposition is co-ordinated with ''all'' the truth-combinations of its argument, its falsity with none of the truth-combinations. | |||
6.121 The propositions of logic demonstrate the logical properties of propositions, by combining them into propositions which say nothing. | |||
This method could be called a zero-method. In a logical proposition propositions are brought into equilibrium with one another, and the state of equilibrium then shows how these propositions must be logically constructed. | |||
6.122 Whence it follows that we can get on without logical propositions, for we can recognize in an adequate notation the formal properties of the propositions by mere inspection. | |||
6.1221 If for example two propositions "''p''" and "''q''" give a tautology in the connexion "''p⊃a''", then it is clear that "''q''" follows from "''p''". | |||
E.g. that "''q''" follows from "''p⊃q.p''" we see from these two propositions themselves, but we can also show it by combining them to "''p⊃q.p:⊃:q''" and then showing that this is a tautology. | |||