258 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



many descriptions of the concept is that given by E. V. 

 Huntington: x 



If a class, K, and a relation, < (called the relation of order), 

 satisfy the conditions expressed in the postulates 0, 1-3, below, 

 then the system (K, <) is called a simply ordered class, or a series. 

 The notation a < b (or b > a, which means the same thing) , may 

 be read: "a precedes 6" (or "6 follows a"). The class K is said to 

 be arranged, or set in order, by the relation < , and the relation < 

 is called a serial relation within the class K. 



Postulate 0. The class K is not an empty class, nor a class con- 

 taining merely a single element. 



This postulate is intended to exclude obviously trivial cases. . . . 



Postulate 1. If a and b are distinct elements of K, then either 

 a < b or b < a. 



Postulate 2. If a < b, then a and b are distinct. 



Postulate 3. If a < b and b < c, then a < c. 



These three postulates may be called, respectively, the 

 postulate of connexity, the postulate of irreflexiveness, and 

 the postulate of transitivity. 2 The logical independence of 

 these postulates, i.e., the fact that no one of them can be 

 derived logically from any of the others, is shown in Hunt- 

 ington by illustrations of systems which satisfy each two 

 of the postulates but not the third. As an example of a 

 series satisfying postulates 2 and 3 but not postulate 1, he 

 offers the class of all human beings throughout history, with 

 < defined as "ancestor of"; as an example satisfying 1 and 

 3 but not 2 he gives the class of all the natural numbers with 

 a < b signifying "a is less than or equal to 6"; finally, as an 

 example satisfying 1 and 2 but not 3, he suggests the class 

 of natural numbers, with < meaning "different from." 



The generality of this postulate set is clearly seen in the 

 fact that nothing is said with reference to the number of 



1 The Continuum (Cambridge: Harvard University, 1917), Chap. II. 



2 The equivalence of this postulate set with the set given in Chapter VI, p. 122, 

 may easily be shown. The principle of connexity was not there stated as a specific 

 postulate, though it is equivalent to the assertion that an ordering relation must 

 exist between every two members; the principles of transitivity in the two cases 

 are identical; the principle of asymmetry which was given earlier as a specific 

 postulate may be shown to follow from the postulates 2 and 3 as given by Hunting- 

 ton. See ibid., pp. 10, 11. 



