86 OF INTERPRETATION. [CHAP. VI. 



2nd. Unclean beasts dividing the hoof, but not chewing the 

 cud. 



3rd. Unclean beasts chewing the cud, but not dividing the 

 hoof. 



4th. Things which are neither clean beasts, nor chewers of 

 the cud, nor dividers of the hoof. 



This form of conclusion may be termed the form of " Single 

 or Disjunctive Affirmation," single when but one constituent 

 appears in the final equation ; disjunctive when, as above, more 

 constituents than one are there found. 



Any equation, V= 0, wherein V satisfies the law of duality, 

 may also be made to yield this form of interpretation by reducing 

 it to the form 1 - F= 1, and developing the first member. The 

 case, however, is really included in the next general form. Both 

 the previous forms are of slight importance compared with the 

 following one. 



FORM III. 



8. In the two preceding cases the functions to be developed 

 were equated to and to 1 respectively. In the present case I 

 shall suppose the corresponding function equated to any logical 

 symbol w. We are then to endeavour to interpret the equation 

 V w, V being a function of the logical symbols x, y, z, &c. In 

 the first place, however, I deem it necessary to show how the 

 equation V=w, or, as it will usually present itself, w = F, arises. 



Let us resume the definition of " clean beasts," employed in 

 the previous examples, viz., "Clean beasts are those which both 

 divide the hoof and chew the cud," and suppose it required to de- 

 termine the relation in which "beasts chewing the cud" stand to 

 " clean beasts" and " beasts dividing the hoof." The equation 

 expressing the given proposition is 



and our object will be accomplished if we can determine z as an 

 interpretable function of x and y. 



Now treating #, y, z as symbols of quantity subject to a pe- 

 culiar law, we may deduce from the above equation, by solution, 



x 



z - -. 



