6.12 (00)



The fact that the propositions of logic are tautologies shows the formal - logical - properties of language, of the world.

That its constituent parts connected together in this way give a tautology characterizes the logic of its constituent parts. 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 HOOK q", "p" and "q" connected together in the form forma "(p HOOK q). (p): HOOK :(q)" give a tautology shows that q follows from p and p HOOK q.  That "(x).fx: HOOK :fa" is a tautology shows that fa follows from (x) . fx, 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.:

diagram of p/q=F/F F/T T/F T/T
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:
diagram of p/q=(F/F F/T T/T)->T (T/F)->F
This sign, for example, would therefore present the proposition p HOOK q. Now I will proceed to inquire whether such a proposition as ~(p . ~p) (The Law of Contradiction) is a tautology. The form "~" is written in our notation
diagram of xi=(F)->T, (T)->F
the form " .  eta " thus:
diagram of xi/eta=(F/F F/T T/F)->F (T/T)->T
Hence the proposition ~(p . ~q) runs thus:
diagram of p/q =(F/F F/T T/T)->T, T/F->F
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.