December 31, 1909] 



SCIENCE 



955 



the logical product, pq or j^-q^ of any two 

 propositions, p and q, may be formally de- 

 fined by the definition : 



pDp.qoq.ror.o:.pq = •.po{q3r).dr. 



This definition of the notion— \Tilgarly 

 called the joint assertion of p and g— may 

 be rendered thus: p, q, r being proposi- 

 tions, the product of p and q is the proposi- 

 tion—any proposition r such that p im- 

 plies that q implies it, is true. Tlie logical 

 sum, p~q, of two propositions p and q 

 admits of the definition : 



j)dp .qoq.nr.3:ji~q = •.pdr.qsr.or ; 



that is, p, q and r being propositions, q-^ji 

 is the proposition equivalent to the propo- 

 sition that r is implied by the product of 

 por and qsr. Such is the definition of the 

 phrase, p or q. It is noteworthy that, 

 whilst pq is true when and only when p 

 and q are hotli true, the sum p^q is true 

 whenever either p or g is true. Among 

 cardinal theorems I will, further, mention 

 the laws of tautology, commutation, asso- 

 ciation and distribution : 



pp(orp=)=p, p-p=p; 



p-q = 9-p. p-1 = <i-p ; 



{p^q)^r = p^(q^r), (p^)~r = p-{q~r); 

 p-{q-r) = {p~q)-{p~r), 



pAq-r) = (j)^q)^(p-^r) = p-q-p-r. 



The negative, — p, of p is a proposition 

 definable thus : 



pop.qdq.o: — p — -poq, 



which states that — p is the proposition 

 equivalent to the proposition that p im- 

 plies all propositions; and we have the 

 theorem of douile negatives: — ( — p)^P- 

 Also the theorems of contradiction and ex- 

 cluded middle: — p-<Z is false; — p-q is 

 true. 



Definitions and Theorems in Class Logic. 

 —As already pointed out, a prepositional 

 function— say. a: is a pragmatist, or 



tan a; = 2/— though a proposition in form, 

 is not one in fact, being neither true nor 

 false. But such a function yields a propo- 

 sition whenever the indeterminate terms, 

 as X, y, are replaced by determinate terms. 

 Thus any such function is a sort of en- 

 velope of a limitless number of propo.si- 

 tions. A function being given, those terms 

 that on being substituted for its indeter- 

 uiinates yield true propositions are said to 

 constitute a class. Tlie .symboli.sm a-?<^(j-, 

 means "the class of tenns x such that 

 <^(a;) is true," and primitive proposition 

 (1) a.s.serts that, if the individual fe is a 

 member of the class, <t>{k) is true. Two 

 functions <t>{x) and *(«) are said to be 

 equivalent when the propositions of every 

 pair of propositions obtainable by substi- 

 tuting definite terms for x are equivalent; 

 and (2) states that wlien two functions are 

 equivalent the corresponding classes are 

 the same— composed of the same individ- 

 uals. If the propositions derivable from 

 <t>{x) are all of them false, the function is 

 said to determine a null-class; and it read- 

 ily follows that all null-classes are exten- 

 sionally the same, so that we can, in this 

 sense, speak of the null-class. The defi- 

 nition and symbolic expression of "x is 

 identical with y," x and y being individ- 

 uals, is x = y.^■.xe^(.^u.yeu, where o,, 

 means " implies for every (class) u." The 

 relation in question is symmeti-ic, a fact 

 involved in the theorem, x = y.'=.y = x. 

 A singular class u (class of but one term) 

 is defined to be such that 



xeu.yeu.o.x = y; 



and a singular class u is symbolically dis- 

 tinguished from its term a by writing ?a to 

 denote u, and iu to denote a; so we have 

 la = u, ni = a, and nu = u, but not u = a. 

 The notion of inchtsion of the terms of a 

 class !( by a class v is denoted by iidv 

 (where o is the symbol for "implies" in 

 propositional logic) and is defined to be 



