Assuming, again, a certain suppositio or universe, to which in a given discussion every argument shall refer, then, any propositions whose terms lie outside that suppositio are irrelevant, and for the purposes of that discussion are sometimes called “false”; though it seems better to call them irrelevant or meaningless, seeing that to call them false implies that they might in the same case be true.  Thus propositions which, according to the doctrine of Opposition, appear to be Contradictories, may then cease to be so; for of Contradictories one is true and the other false; but, in the case supposed, both are meaningless.  If the subject of discussion be Zoology, all propositions about centaurs or unicorns are absurd; and such specious Contradictories as No centaurs play the lyre—­Some centaurs do play the lyre; or All unicorns fight with lions—­Some unicorns do not fight with lions, are both meaningless, because in Zoology there are no centaurs nor unicorns; and, therefore, in this reference, the propositions are not really contradictory.  But if the subject of discussion or suppositio be Mythology or Heraldry, such propositions as the above are to the purpose, and form legitimate pairs of Contradictories.

In Formal Logic, in short, we may make at discretion any assumption whatever as to the existence, or as to any condition of the existence of any particular term or terms; and then certain implications and conclusions follow in consistency with that hypothesis or datum.  Still, our conclusions will themselves be only hypothetical, depending on the truth of the datum; and, of course, until this is empirically ascertained, we are as far as ever from empirical reality. (Venn:  Symbolic Logic, c. 6; Keynes:  Formal Logic, Part II. c. 7:  cf. Wolf:  Studies in Logic.)

CHAPTER IX

FORMAL CONDITIONS OF MEDIATE INFERENCE

Sec. 1.  A Mediate Inference is a proposition that depends for proof upon two or more other propositions, so connected together by one or more terms (which the evidentiary propositions, or each pair of them, have in common) as to justify a certain conclusion, namely, the proposition in question.  The type or (more properly) the unit of all such modes of proof, when of a strictly logical kind, is the Syllogism, to which we shall see that all other modes are reducible.  It may be exhibited symbolically thus: 

      M is P;
      S is M: 
    .’.  S is P.

Syllogisms may be classified, as to quantity, into Universal or Particular, according to the quantity of the conclusion; as to quality, into Affirmative or Negative, according to the quality of the conclusion; and, as to relation, into Categorical, Hypothetical and Disjunctive, according as all their propositions are categorical, or one (at least) of their evidentiary propositions is a hypothetical or a disjunctive.