954 



SCIENCE 



[N. S. Vol. XXX. No. 783 



eats from dogs. For one thing material 

 implication subsists only between proposi- 

 tions while formal implication, thotigh it is 

 present in prepositional logic, holds only 

 between prepositional functions. Now a 

 proposition to be such must be true or else 

 false, while a prepositional function, say, 

 a; is a number, though it has the form of 

 a proposition is net one, being neither true 

 nor false, until the unspecified term or 

 terms {x in the example cited) are specified 

 and then we have no longer a function but 

 a preposition. The implication postulated 

 in the primitive propositions is material. 

 The meaning of (1) is that if poq, then 

 poq is a preposition; (2) means that what- 

 ever implies anything is a proposition ; and 

 that of (3) is, whatever is implied is a 

 proposition. Number (4), which does not 

 admit of completely symbolic statement, is 

 the postulate that justifies the advance 

 from the hypothetic to the categoric— the 

 advancement involved in passing from say- 

 ing "such and such a conclusion is true 

 if the premises are true" to saying, once 

 the premises are granted true, "the propo- 

 sition" (not new regarded as a conclusion) 

 "is true." 



One of the most striking facts in the 

 prepositional logic is the theorem that 

 every false proposition implies all propo- 

 sitions and that all true propositions are 

 implied by every proposition. The shock- 

 ing character of the theorem— which refers, 

 of course, to material implication only— 

 disappears on reflecting that the proposi- 

 tion, p implies g, means simply "q or 

 not-p"— means, that is, "g is true or p is 

 false" and nothing else; for surely it is 

 nothing shocking to affirm that a proposi- 

 tion that is not contradicted by any propo- 

 sition in the class of true propositions is a 

 member of the class; and that affirmation 

 seems equivalent to asserting that "p im- 

 plies q" is true unless q is false and p 



true. If you assert of two prepositions p 

 and q that p implies q, thereby meaning 

 simply and solely that q can not be false 

 and p true, then unless it happens that at 

 once q is false and p true, there would 

 seem to be in the arsenal of refutation no 

 weapon with which your assertion may be 

 struck down. The primitive prepositions 

 are some of them far from being "self- 

 evident." It is not essential that they 

 should be. They are chosen with reference 

 to their sufficiency and look for justification 

 to the body of their consequences. In these 

 they shine— not a priori but a postenori. 

 Neither can they be proved true by de- 

 ducing them from a theorem that is itself 

 deduced from them— to say which is, of 

 course, but to utter a commonplace. As 

 an exercise, however, it is legitimate as well 

 as interesting and instructive to assume 

 the foregoing theorem as a postulate and 

 as such to apply it as a test to the primi- 

 tive propositions in question. Thus, to 

 take a single example, the procedure in 

 the case of (8) would be as follows. Let r 

 be true and p and q either or both be false 

 or true; then qor is ti-ue, hence ^jo.^d/' is 

 true, hence (8) is true. Let r be false and 

 p and g be true ; then pop and qor are both 

 true, pg is true, p^'or is false, hence what 

 precedes the colon is false, hence (8) is 

 true. And so en for the remaining possible 

 suppositions respecting p, q and r. 



Two propositions are equivalent if each 

 implies the other, and we write p^q. 

 Two propositions are equivalent when and 

 only when both are true or both are false. 

 The fundamental operations of preposi- 

 tional multiplication and summation are 

 definable as follows: We may first define 

 the logical product of the two special prop- 

 ositions—a is a proposition, & is a propo- 

 sition— to be the proposition, a is a propo- 

 sition and b is a preposition. Then, 

 denoting this special product by aoa.bob. 



