The mediate element of the inference in a Hypothetical Syllogism consists in asserting, or denying, the fulfilment of a given condition; just as in a Categorical syllogism to identify the minor term with the Middle is a condition of the major term’s being predicated of it.  In the hypothetical proposition—­

    If A is B, C is D—­

the Antecedent, A is B, is the conditio sufficiens, or mark, of the Consequent, C is D; and therefore the Consequent, C is D, is a conditio sine qua non of the antecedent, A is B; and it is by means of affirming the former condition, or else denying the latter, that a conclusion is rendered possible.

Indeed, we need not say that the element of mediation consists in affirming, or denying, the fulfilment of a given condition:  it is enough to say ‘in affirming.’  For thus to explain the Modus tollens, reduce it to the Modus ponens (contrapositing the major premise and obverting the minor): 

Celarent.

If A is B, C is D:  The case of C being not-D is
.’.  If C is not-D, A is not B; not a case of A being B;
C is not-D:  This is a case of C being
.’.  A is not B. not-D: 
.’.  This is not a case of A
being B.

The above four forms commonly treated of as Hypothetical Syllogisms, are called by Ueberweg and Dr. Keynes ‘Hypothetico-Categorical.’  Ueberweg restricts the name ‘Hypothetical’ simply (and Dr. Keynes the name ‘Conditional’) to such Syllogisms as the following, having two Hypothetical Premises: 

If C is D, E is F;
If A is B, C is D: 
.’.  If A is B, E is F.

If we recognise particular hypothetical propositions (see chap. v.  Sec. 4), it is obvious that such Syllogisms may be constructed in all the Moods and Figures of the Categorical Syllogism; and of course they may be translated into Categoricals.  We often reason in this hypothetical way.  For example: 

      If the margin of cultivation be extended, rents will rise;
      If prices of produce rise, the margin of cultivation will be extended: 
    .’.  If prices of produce rise, rents will rise.

But the function of the Hypothetical Syllogism (commonly so called), as also of the Disjunctive Syllogism (to be discussed in the next section) is to get rid of the conditional element of the premises, to pass from suspense to certainty, and obtain a decisive categorical conclusion; whereas these Syllogisms with two hypothetical premises leave us still with a hypothetical conclusion.  This circumstance seems to ally them more closely with Categorical Syllogisms than with those that are discussed in the present chapter.  That they are Categoricals in disguise may be seen by considering that the above syllogism is not materially significant, unless in each proposition the word ‘If’ is equivalent to ‘Whenever.’  Accordingly, the name ‘Hypothetical Syllogism,’ is here employed in the older usage.