' 



xliv Appendix. 



thing is spoken of, and to that subject-thing there is ascribed an attribute. 

 "What, then, are we to iindei-stand by a proposition being self-evident? It 

 must be this, that the perception of the subject-thing compi'ises the perception 

 of the predicated attribute ; that we cannot perceive the one without per- 

 ceiving the other. In other words, a self-evident proposition is such that no 

 adequate definition or explanation of the subject of the proposition can be 

 given without its comprising the ascription of the predicated attribute. If, on 

 the other hand, we take any inductive conclusion — e.g., All matter gravitates, 

 All animals having horns on the forehead are ruminant. All revolutionary 

 democracies tend to military monarchy — in such a case we find that the 

 predicated attribute is additional to, outside of, the definition of the subject. 

 But if we take the proposition, 4 is 3 + 1, the mere definition of what we 

 mean by /our gives us the predicate. The attribute, in such an instance, is 

 not additional to, but is included in, the definition of the subject. Again, let 

 us take the axiom. If equals be added to equals, the sums are equal : we 

 cannot give any definition or explanation of what is meant by equals, and of 

 what is meant by adding, without implying the equality of the sums. Let us, 

 for further illustration, refer to the science of quantity in space. The funda- 

 mental axioms peculiar to Geometry are two, of which the first is, that Two 

 straight lines cannot enclose space. This proposition may be resolved into two 

 alternative propositions. Any two lines must either meet one another, or not. 

 As to the first alternative, it is obvious that the perception of two lines not 

 meeting one another comprises the perception of their not enclosing space. 

 And so also as to the second alternative : we cannot have the perception 

 of two lines diverging from a point, and of each of those lines being straight, 

 without our perceiving that the divergence is perpetual. The term " straight 

 line " may perhaps be considered to be undefinable as being the expression of 

 a simple conception ; but we may doubtless explain the conception, if requisite, 

 by variation of language, or by whatever means of illustration. We cannot, 

 however, give any explanation of straightness, nor any definition or explana- 

 tion of " diverging," without implying the perpetuity of the divergence. 



In like manner we might explain why the following propositions are self- 

 evident : Two diverging straight lines are not parallel to the same third ; 

 Every event has a cause ; Causes perfectly similar produce perfectly similar 

 effects. 



Such, then, is our definition of self-evident truth, viz., A self-evident 

 proposition is a proposition such that the perception of the subject-thing com- 

 prises the perception of the predicated attribute ; or, it is a proposition such 

 that any adequate definition or explanation of the subject implies the j^redicate. 

 Having this definition, we have no difficulty in completing our definition of 

 Necessary Truth. A necessary truth is that which is either a self-evident 



