To begin with Categorical Syllogisms, of which the following is an example: 

      All authors are vain;
      Cicero is an author: 
    .’.  Cicero is vain.

Here we may suppose that there are no direct means of knowing that Cicero is vain; but we happen to know that all authors are vain and that he is an author; and these two propositions, put together, unmistakably imply that he is vain.  In other words, we do not at first know any relation between ‘Cicero’ and ‘vanity’; but we know that these two terms are severally related to a third term, ‘author,’ hence called a Middle Term; and thus we perceive, by mediate evidence, that they are related to one another.  This sort of proof bears an obvious resemblance (though the relations involved are not the same) to the mathematical proof of equality between two quantities, that cannot be directly compared, by showing the equality of each of them to some third quantity:  A = B = C .’.  A = C. Here B is a middle term.

We have to inquire, then, what conditions must be satisfied in order that a Syllogism may be formally conclusive or valid.  A specious Syllogism that is not really valid is called a Parasyllogism.

Sec. 2.  General Canons of the Syllogism.

(1) A Syllogism contains three, and no more, distinct propositions.

(2) A Syllogism contains three, and no more, distinct univocal terms.

These two Canons imply one another.  Three propositions with less than three terms can only be connected in some of the modes of Immediate Inference.  Three propositions with more than three terms do not show that connection of two terms by means of a third, which is requisite for proving a Mediate Inference.  If we write—­

    All authors are vain;
    Cicero is a statesman—­

there are four terms and no middle term, and therefore there is no proof.  Or if we write—­

      All authors are vain;
      Cicero is an author: 
    .’.  Cicero is a statesman—­

here the term ‘statesman’ occurs without any voucher; it appears in the inference but not in the evidence, and therefore violates the maxim of all formal proof, ‘not to go beyond the evidence.’  It is true that if any one argued—­

      All authors are vain;
      Cicero wrote on philosophy: 
    .’.  Cicero is vain—­

this could not be called a bad argument or a material fallacy; but it would be a needless departure from the form of expression in which the connection between the evidence and the inference is most easily seen.

Still, a mere adherence to the same form of words in the expression of terms is not enough:  we must also attend to their meaning.  For if the same word be used ambiguously (as ‘author’ now for ‘father’ and anon for ’man of letters’), it becomes as to its meaning two terms; so that we have four in all.  Then, if the ambiguous term be the Middle, no connection is shown between the other two; if either of the others be ambiguous, something seems to be inferred which has never been really given in evidence.