84 OF INTERPRETATION. [CHAP. VI. 



6. Let us take as an example the definition of " clean beasts," 

 laid down in the Jewish law, viz., " Clean beasts are those 

 which both divide the hoof and chew the cud," and let us assume 



x = clean beasts ; 

 y = beasts dividing the hoof; 

 z = beasts chewing the cud. 

 Then the given proposition will be represented by the equation 



x=yz t 

 which we shall reduce to the form 



x - yz = 0, 



and seek that form of interpretation to which the present method 

 leads. Fully developing the first member, we have 



xyz + xy (1 - z) + x (1 - y)z + x(\ y) (1 - z) 



Whence the terms, whose coefficients do not vanish, give 



These equations express a denial of the existence of certain classes 

 of objects, viz. : 



1st. Of beasts which are clean, and divide the hoof, but do 

 not chew the cud. 



2nd. Of beasts which are clean, and chew the cud, but do not 

 divide the hoof. 



3rd. Of beasts which are clean, and neither divide the hoof 

 nor chew the cud. 



4th. Of beasts which divide the hoof, and chew the cud, and 

 are not clean. 



Now all these several denials are really involved in the origi- 

 nal proposition. And conversely, if these denials be granted, 

 the original proposition will follow as a necessary consequence. 

 They are, in fact, the separate elements of that proposition. 

 Every primary proposition can thus be resolved into a series of 

 denials of the existence of certain defined classes of things, and 

 may, from that system of denials, be itself reconstructed. It 

 might here be asked, how it is possible to make an assertive pro- 



