6.1 The propositions of logic are tautologies.

6.11 Therefore the propositions of logic say nothing. (They are the analytic propositions.)

6.111 All theories that make a proposition of logic appear to have content are false.  One might think, for example, that the words ‘true’ and ‘false’ signified two properties among other properties, and then it would seem to be a remarkable fact that every proposition possessed one of these properties.  On this theory it seems to be anything but obvious, just as, for instance, the proposition, ‘All roses are either yellow or red’, would not sound obvious even if it were true.  Indeed, the logical proposition acquires all the characteristics of a proposition of natural science and this is the sure sign that it has been construed wrongly.

6.112 The correct explanation of the propositions of logic must assign to them a unique status among all propositions.

6.113 It is the peculiar mark of logical propositions that one can recognize that they are true from the symbol alone, and this fact contains in itself the whole philosophy of logic.  And so too it is a very important fact that the truth or falsity of non-logical propositions cannot be recognized from the propositions alone.

6.12 The fact that the propositions of logic are tautologies shows the formal—­logical—­properties of language and the world.  The fact that a tautology is yielded by this particular way of connecting its constituents characterizes the logic of its constituents.  If propositions are to yield a tautology when they are connected in a certain way, they must have certain structural properties.  So their yielding a tautology when combined in this shows that they possess these structural properties.

6.1201 For example, the fact that the propositions ‘p’ and ‘Pp’ in the combination ‘(p .  Pp)’ yield a tautology shows that they contradict one another.  The fact that the propositions ‘p z q’, ‘p’, and ‘q’, combined with one another in the form ‘(p z q) . (p) :z:  (q)’, yield a tautology shows that q follows from p and p z q.  The fact that ‘(x) . fxx :z:  fa’ is a tautology shows that fa follows from (x) . fx.  Etc. etc.

6.1202 It is clear that one could achieve the same purpose by using contradictions instead of tautologies.

6.1203 In order to recognize an expression as a tautology, in cases where no generality-sign occurs in it, one can employ the following intuitive method:  instead of ‘p’, ‘q’, ‘r’, etc.  I write ‘TpF’, ‘TqF’, ‘TrF’, etc.  Truth-combinations I express by means of brackets, e.g. and I use lines to express the correlation of the truth or falsity of the whole proposition with the truth-combinations of its truth-arguments, in the following way So this sign, for instance, would represent the proposition p z q.  Now, by way of example, I wish to examine the proposition P(p .Pp) (the law of contradiction)