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Notes on Logic: Difference between revisions
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Names are points, propositions arrows-they have ''sense. [Cf.'' 3.144.] The sense of a proposition is determined by the two poles ''true'' and ''false.'' The form of a proposition is like a straight line, which vides all points of a plane into right and left. The line does this automatically, the form of the proposition only by convention. It is wrong to conceive every proposition as expressing a relation. A natural attempt at such a solution consists in regarding "not-p" as the opposite of "p", where, then, "opposite" would be the indefinable relation. But it is easy to see that every such attempt to replace functions with sense (ab-functions) by descriptions, must fail. | Names are points, propositions arrows-they have ''sense. [Cf.'' 3.144.] The sense of a proposition is determined by the two poles ''true'' and ''false.'' The form of a proposition is like a straight line, which vides all points of a plane into right and left. The line does this automatically, the form of the proposition only by convention. It is wrong to conceive every proposition as expressing a relation. A natural attempt at such a solution consists in regarding "not-p" as the opposite of "p", where, then, "opposite" would be the indefinable relation. But it is easy to see that every such attempt to replace functions with sense (ab-functions) by descriptions, must fail. | ||
When we say "A believes p", this sounds, it is true, as if we could here substitute a proper name for "p". But we can see that here a ''sense,'' not a meaning, is concerned, if we say "A believes that p is true", and in order to make the direction of p even more explicit, we might say "A believes that 'p' is true and 'not-p' is false". Here the bi-polarity of p is expressed, and it seems that we shall only be able to express the proposition "A believes p" correctly by the | When we say "A believes p", this sounds, it is true, as if we could here substitute a proper name for "p". But we can see that here a ''sense,'' not a meaning, is concerned, if we say "A believes that p is true", and in order to make the direction of p even more explicit, we might say "A believes that 'p' is true and 'not-p' is false". Here the bi-polarity of p is expressed, and it seems that we shall only be able to express the proposition "A believes p" correctly by the ab-notation (later explained) by, say, making "A" have a relation to the poles "a" and "b" of a-p-b. The epistemological questions concerning the nature of judgment and belief cannot be solved without a correct apprehension of the form of the proposition. | ||
A proposition is a standard with reference to which facts behave, but with names it is otherwise. Just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition; it is thus bi-polarity and sense come in. In this theory p has the same meaning as not-p but opposite sense. The meaning is the fact. A proper theory of judgment must make it impossible to judge nonsense. [''Cf.'' 5.5422.] The "sense of" an abfunction of a proposition is a function of its sense. [''Cf.'' 5.2341.] In not-p, p is exactly the same as if it stands alone (this point is absolutely fundamental). Among the facts which make "p or q" true there are also facts which make "p and q" true; hence, if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical. But in fact their sense is different, and we have introduced sense by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where their b function stands under a generality sign or enters into another function such as "I believe that", etc., because then the sense enters. | A proposition is a standard with reference to which facts behave, but with names it is otherwise. Just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition; it is thus bi-polarity and sense come in. In this theory p has the same meaning as not-p but opposite sense. The meaning is the fact. A proper theory of judgment must make it impossible to judge nonsense. [''Cf.'' 5.5422.] The "sense of" an abfunction of a proposition is a function of its sense. [''Cf.'' 5.2341.] In not-p, p is exactly the same as if it stands alone (this point is absolutely fundamental). Among the facts which make "p or q" true there are also facts which make "p and q" true; hence, if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical. But in fact their sense is different, and we have introduced sense by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where their b function stands under a generality sign or enters into another function such as "I believe that", etc., because then the sense enters. | ||
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In place of every proposition "p" let us write " ". Let every correlation of propositions to each other or of names to propositions be effected by a correlation of their poles "a" and "b". Let this correlation be transitive. Then accordingly "<nowiki><math></math></nowiki>p" is the same symbol as "<nowiki><math></math></nowiki>p". Let n propositions be given. I then call a "class of poles" of these propositions every class of n members, of which each is a pole of one of the n propositions, so that one member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The sense of the symbolizing fact thus constructed I cannot define, but I know it. | In place of every proposition "p" let us write " ". Let every correlation of propositions to each other or of names to propositions be effected by a correlation of their poles "a" and "b". Let this correlation be transitive. Then accordingly "<nowiki><math></math></nowiki>p" is the same symbol as "<nowiki><math></math></nowiki>p". Let n propositions be given. I then call a "class of poles" of these propositions every class of n members, of which each is a pole of one of the n propositions, so that one member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The sense of the symbolizing fact thus constructed I cannot define, but I know it. | ||
The sense of an | The sense of an ab-function of p is a function of the sense of p. [''Cf.'' 5.2341.] The ab-functions use the discrimination of facts which their arguments bring forth in order to generate new discriminations. The ab-notation shows the dependence of ''or'' and ''not,'' and thereby that they are not to be employed as simultaneous indefinables. | ||
To every molecular function a TF (or ab) scheme corresponds. Therefore we may use the TF scheme itself instead of the function. Now what the TF scheme does is that it correlates the letters T and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another T and F to these poles. In this notation all that matters is the correlation of the outside poles to the poles of the atomic propositions. Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function. As the | To every molecular function a TF (or ab) scheme corresponds. Therefore we may use the TF scheme itself instead of the function. Now what the TF scheme does is that it correlates the letters T and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another T and F to these poles. In this notation all that matters is the correlation of the outside poles to the poles of the atomic propositions. Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function. As the ab (TF)-functions of atomic propositions are bi-polar propositions again, we can perform ab operations on them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic propositions. The symbolizing fact in a-p-b is that ''say'' a is on the left of p and b on the right of p. [This is quite arbitrary, but if we once have fixed on which order the poles have to stand in, we must of course stick to our convention. If, for instance, "apb" says p, then bpa says ''nothing'' (it does ''not'' say ~p). But a-apb-b is the same symbol as apb (here the ab-function vanishes automatically) for here the new poles are related to the same side of p as the old ones. The question is always: how are the new poles correlated to p compared with the way the old poles are correlated to p?] Then, given apb, the correlation of new poles is to be transitive, so that, for instance, if a new pole a in what ever way, i.e. via whatever poles, is correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab-functions by performing one ab-operation repeatedly, and we can therefore talk of all ab-functions as of all those functions which can be obtained by performing this ab-operation repeatedly (''cf.'' Sheffer's work). | ||
Among the facts which make "p or q" true, there are some which make "p and q" true; but the class which makes "p or q" true is different from the class which makes "p and q" true; and only this is what matters. For we introduce this class, as it were, when we intro duce | Among the facts which make "p or q" true, there are some which make "p and q" true; but the class which makes "p or q" true is different from the class which makes "p and q" true; and only this is what matters. For we introduce this class, as it were, when we intro duce ab-functions. | ||
Since the | Since the ab-functions of p are again bi-polar propositions, we can form ab-functions of them, and so on. In this way a series of propositions will arise, in which, in general, the ''symbolizing'' facts will be the same in several members. If now we find an ab-function of such a kind that by repeated applications of it every ab-function can be generated, then we can introduce the totality of ab-functions as the totality of those that are generated by the application of this function. Such a function is ~p ∨ ~q. It is easy to suppose a contradiction in the fact that, on the one hand, every possible complex proposition is a simple ab-function of simple propositions, and that, on the other hand, the repeated application of one ab-function suffices to generate all these propositions. If, e.g., an affirmation can be generated by double negation, is negation in any sense contained in affirmation? Does "p" deny "not-p" or assert "p", or both? [''See'' 5.44.] And how do matters stand with the definition of "⊃" by "∨" and "~", or of "∨" by "~" and "⊃"? And how, e.g., shall we introduce p|q (i.e. ~p ∨ ~q), if not by saying that this expression says something | ||
indefinable about all arguments p and q? But the | indefinable about all arguments p and q? But the ab-functions must be introduced as follows: The function p|q is merely a mechanical instrument for constructing all possible ''symbols'' of ab-functions. The symbols arising by repeated application of the symbol "I" do ''not'' contain the symbol "pIq". We need a rule according to which we can form all symbols of ab-functions, in order to be able to speak of the class of them; and we now speak of them, e.g., as those symbols of functions which can be generated by repeated application of the operation "|". And we say now: For all p's and q's, "p|q" says something indefinable about the sense of those simple propositions which are contained in p and q. | ||
IV. Analysis of General Propositions | IV. Analysis of General Propositions | ||