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Notes on Logic: Difference between revisions
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Indefinables are of two sorts: names and forms. Propositions cannot consist of names alone, they cannot be classes of names. [''Cf '' 3.142.] A name cannot only occur in two different propositions, but can occur in the same way in both. Propositions, which are symbols having reference to facts, are themselves facts (that this inkpot is on this table may express that I sit in this chair). We must be able to understand propositions we have never heard before. But every proposition is a new symbol. Hence we must have ''general'' indefinable symbols; these are unavoidable if propositions are not all indefinable. Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads us to an impenetrable thicket. | Indefinables are of two sorts: names and forms. Propositions cannot consist of names alone, they cannot be classes of names. [''Cf '' 3.142.] A name cannot only occur in two different propositions, but can occur in the same way in both. Propositions, which are symbols having reference to facts, are themselves facts (that this inkpot is on this table may express that I sit in this chair). We must be able to understand propositions we have never heard before. But every proposition is a new symbol. Hence we must have ''general'' indefinable symbols; these are unavoidable if propositions are not all indefinable. Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads us to an impenetrable thicket. | ||
A proposition must be understood when '' | A proposition must be understood when ''all'' its indefinables are understood. The indefinables in "aRb" are introduced as follows: (1) "a" is indefinable, (2) "b" is indefinable, (3) whatever "x" and "y" may mean, "xRy" says something indefinable about their meaning. We are not concerned in logic with the relation of any specific name to its meaning and just as little with the relation of a given proposition to reality. We do want to know that our names have meanings and propositions sense, and we thus introduce an indefinable concept "A" by saying "'A' denotes something indefinable", or the form of pro positions aRb by saying: "For all meanings of 'x' and 'y', 'xRy' expresses something indefinable about x and y." | ||
The form of a proposition may be symbolized in the following way: Let us consider symbols of the form "xRy", to which correspond primarily pairs of objects of which one has the name "x", the other the name "y". The x's and y's stand in various relations to each other, and among other relations the relation R holds between some but not between others. I now determine the sense of "xRy" by laying down the rule: when the facts behave in regard to "xRy" so that the meaning of "x" stands in the relation R to the meaning of " y", 2 then I say that these facts are "of like sense" (''gleichsinnig'') with the proposition "xRy"; otherwise, "of opposite sense" (''entgegengesetzt'') | The form of a proposition may be symbolized in the following way: Let us consider symbols of the form "xRy", to which correspond primarily pairs of objects of which one has the name "x", the other the name "y". The x's and y's stand in various relations to each other, and among other relations the relation R holds between some but not between others. I now determine the sense of "xRy" by laying down the rule: when the facts behave in regard to "xRy" so that the meaning of "x" stands in the relation R to the meaning of " y", 2 then I say that these facts are "of like sense" (''gleichsinnig'') with the proposition "xRy"; otherwise, "of opposite sense" (''entgegengesetzt''). I correlate the facts to the symbol "xRy" by thus dividing them into those of like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form "xRy" when I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I extract from all possible relations the relation R, as by a name, I extract its meaning from among all possible things. | ||
There is no ''thing'' which is the ''form'' of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell's theory of judgment. | There is no ''thing'' which is the ''form'' of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell's theory of judgment. | ||
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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 | 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 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 " | 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 "|" do ''not'' contain the symbol "p|q". 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 | ||