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# Relations between the class and truth-functional systems

DOI link for Relations between the class and truth-functional systems

Relations between the class and truth-functional systems book

# Relations between the class and truth-functional systems

DOI link for Relations between the class and truth-functional systems

Relations between the class and truth-functional systems book

## ABSTRACT

If we can show tha t the class law is, in some sense, a special case of the truth-functional law, this will help to replace the general picture of two independent interpreted systems, miraculously having the same formal structure, by the picture of one of the systems (or, more exactly, a certain part of it) as simply a less general version of the other, having as a matter of course the same formal structure. To show this in the case of our example will be to show how the class law can be derived from the t ru thfunctional law merely by the use of the principle of substitution and by reminding ourselves of the interpretation given to the symbols of the two systems. As a first step, we may substitute class-membership formulae for the statement-variables in

(1) p.q pvq obtaining, say. the expression

(2) xΕΑ . yεβ xΕΑ V yεβ Since (1) holds good whatever sentences are substituted for ' p ' and ' q ', so long as identical substitutions are made for identical variables throughout, it will hold for the restricted class of cases in which the substituted sentences exemplify simple classmembership formulae. Law (2) is therefore simply a special case of law (1). Since (2) holds good whatever values we give to the variables of the class-membership formulae, provided we give identical values to identical variables throughout, it will

PT. I] E L E M E N T S OF P R E D I C A T I V E SYSTEM 127 hold for the restricted class of cases in which the variables ' x ' and ' y ' have the same values in the two class-membership formulae. Therefore

(3) xΑ . xεβ xΕΑ v xεβ is simply a special case of (2) and hence of (1). Now, by the interpretational rules given for the logical product and logical sum symbols of the class system, (3) is equivalent to

(4) xΕΑΒ xε(α + β) And by the interpretational rules for the ' * ' symbol of the class system, (4) is equivalent to

(5) xε(αβ * α + β) Now to assert (5) as a law or analytic formula, as we are here doing, is to assert tha t an analytic sentence results from it whatever significant word-substitutions we make for ' x ', ' α ', and ' β ', provided we make the same substitutions for the same variables throughout. I t is precisely the point of variables to ensure this generality. Now let us, as regards the variable ' x ', write this permitted generality of substitution into the formula itself, by the use of the word ' everything '. Thus we obtain

(6) Everything is a member of the class of αβ * α + β which is the meaning given by the interpretational rules for ' = 1 ' to

(7) αβ * α + β = 1 which is equivalent to

(7a) αβ α + β These steps show in what sense I wish to say tha t (7), or (7a), is a special case of (1). Although something more than the bare use of the principle of substitution is involved, it does not seem a very unnatural use of the words ' a special case '.