CA TEGORICAL JUDGMENTS AND PROPOSITIONS. 2 1 7 



will express identity, the distinction between subject and predi 

 cate will be merely a distinction of place, the proposition may be 

 read equally well backward and forward, will be perfectly sym 

 metrical, and may always be simply converted (i 18). 



In regard to the propriety of expressing ordinary judgments 

 as logical equations as announcing the identity of the two classes 

 we may say at once that it is inappropriate, that it does not ex 

 press what we are really thinking of, except in the rare cases, 

 already referred to, of propositions like &quot; Civilization and Chris 

 tianity are coextensive &quot;. 



In order to have an equational proposition, the subject and predicate 

 must be taken collectively, as names of definite classes, as two single collec 

 tions, which are either coextensive, and accordingly identical, or are not. It 

 has been claimed that the eight Hamiltonian propositions are equational. But 

 how is this so in the propositions containing &quot; some ? &quot; If &quot; some &quot; be taken 

 indefinitely as &quot; some at least &quot; the proposition cannot be equational ; for an 

 indefinite portion of a class is itself a class indefinite in its denotation, and 

 there is no meaning in making an indefinite class &quot; identical &quot; with any other 

 class, definite or indefinite. 



&quot; Some &quot; therefore, must be taken as expressing a definite portion not all 

 of a larger class ; i.e. as &quot; some only &quot;. Nor will it even now yield a strictly 

 equational proposition : except on the further understanding ;that it indicates 

 a specified group ; and this reservation does not appear in the form of the 

 proposition. In order that the A proposition^// S is some P be simply 

 convertible to Some P is all 5 it is necessary that the &quot; Some P &quot; be the 

 same particular group in both propositions, and that it be one special, singu 

 lar group, not any group taken at random. The converse of &quot; All men are 

 some animals &quot; viz. &quot; Some animals are all men &quot; is not true except of 

 the one particular group of animals referred to in the former proposition, and 

 there is nothing in the form of that proposition to show such limitation. 



Attempts have been made to find equational forms corresponding to the 

 four forms of the traditional scheme (9I). 1 Jevons expresses the U and A 

 forms by the sign of equality, writing the former &quot; S = P &quot; and the latter 

 &quot; 5 = SP &quot;. The form S = P he calls a simple identity. It expresses the U 

 proposition All S } s are all P s. But, clearly, we cannot write the proposition 

 All S s are P s in the form S = P, for S may include more than P. But it 

 can be written S = SP, where SP denotes the S s that are P s. For instance, 

 All men are mortal may be written Men = mortal men. 



Now, since particulars contradict universals (112), if we express the latter 

 as equalities we should express the former as inequalities by the signs &amp;gt; and 

 &amp;lt;. Thus S&amp;gt;SP would imply that the class S includes more than SP, i.e. 

 that Some S s are not P s. &quot;If we further introduce the symbol o as ex 

 pressing nonentity, No S is P may be written SP = o, and its contradictory, 

 i.e. Some S is P, may be written SP &amp;gt; o. We shall then have the following 

 scheme (where p = not-P} : 



1 C/. KEYNES, o^ t cit., p. 193 : whose treatment is here followed. 



