5.4731 Self-evidence, which Russell talked about so much, can become dispensable in logic, only because language itself prevents every logical mistake.—­What makes logic a priori is the impossibility of illogical thought.

5.4732 We cannot give a sign the wrong sense.

5,47321 Occam’s maxim is, of course, not an arbitrary rule, nor one that is justified by its success in practice:  its point is that unnecessary units in a sign-language mean nothing.  Signs that serve one purpose are logically equivalent, and signs that serve none are logically meaningless.

5.4733 Frege says that any legitimately constructed proposition must have a sense.  And I say that any possible proposition is legitimately constructed, and, if it has no sense, that can only be because we have failed to give a meaning to some of its constituents. (Even if we think that we have done so.) Thus the reason why ‘Socrates is identical’ says nothing is that we have not given any adjectival meaning to the word ‘identical’.  For when it appears as a sign for identity, it symbolizes in an entirely different way—­ the signifying relation is a different one—­therefore the symbols also are entirely different in the two cases:  the two symbols have only the sign in common, and that is an accident.

5.474 The number of fundamental operations that are necessary depends solely on our notation.

5.475 All that is required is that we should construct a system of signs with a particular number of dimensions—­with a particular mathematical multiplicity

5.476 It is clear that this is not a question of a number of primitive ideas that have to be signified, but rather of the expression of a rule.

5.5 Every truth-function is a result of successive applications to
elementary propositions of the operation ‘(-----T)(E, ....)’.  This
operation negates all the propositions in the right-hand pair of brackets,
and I call it the negation of those propositions.

5.501 When a bracketed expression has propositions as its terms—­and the order of the terms inside the brackets is indifferent—­then I indicate it by a sign of the form ‘(E)’. ‘(E)’ is a variable whose values are terms of the bracketed expression and the bar over the variable indicates that it is the representative of ali its values in the brackets. (E.g. if E has the three values P,Q, R, then (E) = (P, Q, R). ) What the values of the variable are is something that is stipulated.  The stipulation is a description of the propositions that have the variable as their representative.  How the description of the terms of the bracketed expression is produced is not essential.  We can distinguish three kinds of description:  1.Direct enumeration, in which case we can simply substitute for the variable the constants that are its values; 2. giving a function fx whose values for all values of x are the propositions to be described; 3. giving a formal law that governs the construction of the propositions, in which case the bracketed expression has as its members all the terms of a series of forms.