Sec. 1.  Having now discussed Terms, Propositions, Immediate and Mediate Inferences, and investigated the conditions of formal truth or consistency, we have next to consider the conditions of material truth:  whether (or how far) it is possible to arrive at propositions that accurately represent the course of nature or of human life.  Hitherto we have dealt with no sort of proof that gives any such assurance.  A valid syllogism guarantees the truth of its conclusion, provided the premises be true:  but what of the premises?  The relation between the premises of a valid syllogism and its conclusion is the same as the relation between the antecedent and consequent of a hypothetical proposition.  If A is B, C is D:  grant that A is B, and it follows that C is D; and, similarly, grant the premises of a syllogism, and the conclusion follows.  Again, grant that C is not D, and it follows that A is not B; and, similarly, if the conclusion of a valid syllogism be false, it follows that one, or other, or both of the premises must be false.  But, once more, grant that C is D, and it does not follow that A is B; so neither, if the conclusion of a syllogism be true, does it follow that the premises are.  For example:—­

      Sociology is an exact science;
      Mathematics is a branch of Sociology: 
    .’.  Mathematics is an exact science.

Here the conclusion is true although the premises are absurd.  Or again:—­

      Mathematics is an exact science;
      Sociology is a branch of Mathematics: 
    .’.  Sociology is an exact science.

Here the major premise is true, but the minor is false, and the conclusion is false.  In both cases, however, whether the conclusion be true or false, it equally follows from the premises, if there is any cogency in Barbara.  The explanation of this is, that Barbara has only formal cogency; and that whether the conclusion of that, or any other valid mood, shall be true according to fact and experience, depends upon how the form is filled up.  How to establish the premises, then, is a most important problem; and it still remains to be solved.

Sec. 2.  We may begin by recalling the distinction between the denotation and connotation of a general term:  the denotation comprising the things or events which the term is a name for; the connotation comprising the common qualities on account of which these things are called by the same name.  Obviously, there are very few general terms whose denotation is exhaustively known; since the denotation of a general term comprises all the things that have its connotation, or that ever have had, or that ever will have it, whether they exist here, or in Australia, or in the Moon, or in the utmost stars.  No one has examined all men, all mammoths, all crystals, all falling bodies, all cases of fever, all revolutions, all stars—­nor even all planets, since from time to time new ones are discerned.  We have names for animals that existed long before there were men to observe them, and of which we know only a few bones, the remains of multitudinous species; and for others that may continue to exist when men have disappeared from the earth.