READ STUDY GUIDE: 5.47–5.54 |
|
Propositions 5.47-5.54
5.47 It is clear that whatever we can say in advance about the form of all propositions, we must be able to say all at once . An elementary proposition really contains all logical operations in itself. For 'fa' says the same thing as '(dx) . fx . x = a' Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants. One could say that the sole logical constant was what all propositions, by their very nature, had in common with one another. But that is the general propositional form.
5.471 The general propositional form is the essence of a proposition.
5.4711 To give the essence of a proposition means to give the essence of all description, and thus the essence of the world.
5.472 The description of the most general propositional form is the description of the one and only general primitive sign in logic.
5.473 Logic must look after itself. If a sign is possible , then it is also capable of signifying. Whatever is possible in logic is also permitted. (The reason why 'Socrates is identical' means nothing is that there is no property called 'identical'. The proposition is nonsensical because we have failed to make an arbitrary determination, and not because the symbol, in itself, would be illegitimate.) In a certain sense, we cannot make mistakes in logic.
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.
5.502 So instead of '(——-T)(E, ....)', I write 'N(E)'. N(E) is the negation of all the values of the propositional variable E.
5.503 It is obvious that we can easily express how propositions may be constructed with this operation, and how they may not be constructed with it; so it must be possible to find an exact expression for this.
5.51 If E has only one value, then N(E) = Pp (not p); if it has two values, then N(E) = Pp . Pq. (neither p nor g).
5.511 How can logic—all-embracing logic, which mirrors the world—use such peculiar crotchets and contrivances? Only because they are all connected with one another in an infinitely fine network, the great mirror.
5.512 'Pp' is true if 'p' is false. Therefore, in the proposition 'Pp', when it is true, 'p' is a false proposition. How then can the stroke 'P' make it agree with reality? But in 'Pp' it is not 'P' that negates, it is rather what is common to all the signs of this notation that negate p. That is to say the common rule that governs the construction of 'Pp', 'PPPp', 'Pp C Pp', 'Pp . Pp', etc. etc. (ad inf.). And this common factor mirrors negation.
5.513 We might say that what is common to all symbols that affirm both p and q is the proposition 'p . q'; and that what is common to all symbols that affirm either p or q is the proposition 'p C q'. And similarly we can say that two propositions are opposed to one another if they have nothing in common with one another, and that every proposition has only one negative, since there is only one proposition that lies completely outside it. Thus in Russell's notation too it is manifest that 'q : p C Pp' says the same thing as 'q', that 'p C Pq' says nothing.
5.514 Once a notation has been established, there will be in it a rule governing the construction of all propositions that negate p, a rule governing the construction of all propositions that affirm p, and a rule governing the construction of all propositions that affirm p or q; and so on. These rules are equivalent to the symbols; and in them their sense is mirrored.
5.515 It must be manifest in our symbols that it can only be propositions that are combined with one another by 'C', '.', etc. And this is indeed the case, since the symbol in 'p' and 'q' itself presupposes 'C', 'P', etc. If the sign 'p' in 'p C q' does not stand for a complex sign, then it cannot have sense by itself: but in that case the signs 'p C p', 'p . p', etc., which have the same sense as p, must also lack sense. But if 'p C p' has no sense, then 'p C q' cannot have a sense either.
5.5151 Must the sign of a negative proposition be constructed with that of the positive proposition? Why should it not be possible to express a negative proposition by means of a negative fact? (E.g. suppose that "a' does not stand in a certain relation to 'b'; then this might be used to say that aRb was not the case.) But really even in this case the negative proposition is constructed by an indirect use of the positive. The positive proposition necessarily presupposes the existence of the negative proposition and vice versa.
5.52 If E has as its values all the values of a function fx for all values of x, then N(E) = P(dx) . fx.
5.521 I dissociate the concept all from truth-functions. Frege and Russell introduced generality in association with logical productor logical sum. This made it difficult to understand the propositions '(dx) . fx' and '(x) . fx', in which both ideas are embedded.
5.522 What is peculiar to the generality-sign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants.
5.523 The generality-sign occurs as an argument.
5.524 If objects are given, then at the same time we are given all objects. If elementary propositions are given, then at the same time all elementary propositions are given.
5.525 It is incorrect to render the proposition '(dx) . fx' in the words, 'fx is possible ' as Russell does. The certainty, possibility, or impossibility of a situation is not expressed by a proposition, but by an expression's being a tautology, a proposition with a sense, or a contradiction. The precedent to which we are constantly inclined to appeal must reside in the symbol itself.
5.526 We can describe the world completely by means of fully generalized propositions, i.e. without first correlating any name with a particular object.
5.5261 A fully generalized proposition, like every other proposition, is composite. (This is shown by the fact that in '(dx, O) . Ox' we have to mention 'O' and 's' separately. They both, independently, stand in signifying relations to the world, just as is the case in ungeneralized propositions.) It is a mark of a composite symbol that it has something in common with other symbols.
5.5262 The truth or falsity of every proposition does make some alteration in the general construction of the world. And the range that the totality of elementary propositions leaves open for its construction is exactly the same as that which is delimited by entirely general propositions. (If an elementary proposition is true, that means, at any rate, one more true elementary proposition.)
5.53 Identity of object I express by identity of sign, and not by using a sign for identity. Difference of objects I express by difference of signs.
5.5301 It is self-evident that identity is not a relation between objects. This becomes very clear if one considers, for example, the proposition '(x) : fx . z . x = a'. What this proposition says is simply that only a satisfies the function f, and not that only things that have a certain relation to a satisfy the function, Of course, it might then be said that only a did have this relation to a; but in order to express that, we should need the identity-sign itself.
5.5302 Russell's definition of '=' is inadequate, because according to it we cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense .)
5.5303 Roughly speaking, 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.
5.531 Thus I do not write 'f(a, b) . a = b', but 'f(a, a)' (or 'f(b, b)); and not 'f(a,b) . Pa = b', but 'f(a, b)'.
5.532 And analogously I do not write '(dx, y) . f(x, y) . x = y', but '(dx) . f(x, x)'; and not '(dx, y) . f(x, y) . Px = y', but '(dx, y) . f(x, y)'. 5.5321 Thus, for example, instead of '(x) : fx z x = a' we write '(dx) . fx . z : (dx, y) . fx. fy'. And the proposition, 'Only one x satisfies f( )', will read '(dx) . fx : P(dx, y) . fx . fy'.
5.533 The identity-sign, therefore, is not an essential constituent of conceptual notation.
5.534 And now we see that in a correct conceptual notation like 'a = a', 'a = b . b = c . z a = c', '(x) . x = x', '(dx) . x = a', etc. cannot even be written down.
5.535 This also disposes of all the problems that were connected with such pseudo-propositions. All the problems that Russell's 'axiom of infinity' brings with it can be solved at this point. What the axiom of infinity is intended to say would express itself in language through the existence of infinitely many names with different meanings.
5.5351 There are certain cases in which one is tempted to use expressions of the form 'a = a' or 'p z p' and the like. In fact, this happens when one wants to talk about prototypes, e.g. about proposition, thing, etc. Thus in Russell's Principles of Mathematics 'p is a proposition'—which is nonsense- -was given the symbolic rendering 'p z p' and placed as an hypothesis in front of certain propositions in order to exclude from their everything but propositions. (It is nonsense to place the hypothesis 'p z p' in front of a proposition, in order to ensure that its arguments shall have the right form, if only because with a non-proposition as argument the hypothesis becomes not false but nonsensical, and because arguments of the wrong kind make the proposition itself nonsensical, so that it preserves itself from wrong arguments just as well, or as badly, as the hypothesis without sense that was appended for that purpose.)
5.5352 In the same way people have wanted to express, 'There are no things ', by writing 'P(dx) . x = x'. But even if this were a proposition, would it not be equally true if in fact 'there were things' but they were not identical with themselves?
5.54 In the general propositional form propositions occur in other propositions only as bases of truth-operations.
