### Summary

All the "laws of logic" must be given in advance, and all at once, not, as in Frege and Russell, as a hierarchical axiomatic system. For instance, "*p* and *q*" means the same thing as "not (not *p* or not *q,*)" and "*fa*" means the same thing as "there exists an *x* such that *fx* and *x* is *a*" (5.47). If these propositions are equivalent, the meaning of one must be contained in the meaning of the other. That is, the meaning of "not" and "or" must be contained in "*p* and *q*" and the meaning of "there exists," "such that," and the sign for identity must be contained in "*fa.*" In effect, all the "logical constants" must be given all at once if they are to be given at all.

All propositions share in common a general propositional form, which Wittgenstein calls "the essence of a proposition" (5.471). This general form should function as "the sole logical constant," rendering all other constants superfluous.

Wittgenstein says that "logic must look after itself" (5.473): we don't need external "laws" or "rules" to tell us how logic works. Logic is the realm of everything that is possible and conceivable. Whatever is ruled out by logic is ruled out because it is impossible and inconceivable: we do not need laws to tell us what falls outside the bounds of logic. Any proposition that lacks sense does so because we have not given a meaning to the signs in the proposition. For instance, "Socrates is identical" says nothing because we have not given meaning to the word "identical" when used as an adjective (5.4733).

Wittgenstein observes that all propositions can be derived by means of successive applications of the operation (*——T*)(*ξ,….*), that is by means of negating all the terms in the right hand pair of brackets. For instance, "*p*" would become "*~p,*" "*p*" and "*q*" would be combined to form "*~p.~q,*" and so on. Wittgenstein abbreviates this terminology to *N*(*‾ξ*), where the "*N*" stands for negation and the "*‾ξ*" stands collectively for all the propositions in the right hand pair of brackets (5.502).

It is now clear that, for instance, the various permutations of ~p are are not different propositions (5.512). They are all different ways of expressing the same proposition, which suggests that the signs for "not" and "and" are not the signs for objects.

Wittgenstein attempts to rid logical notation of the signs for generality and identity. Whenever a variable is given, that variable expresses all objects that can take that variable place, so generality is already given when a variable is given (5.524). We don't need an additional sign to denote generality. As for identity, "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 at all" (5.5303). We do not need an "=" sign to say two signs are identical: we need only use the same sign twice. Often, the inclination to use the "=" sign comes from a temptation to say something general about the nature of propositions themselves (5.5351).