[289] This holds good whether we adopt the assertoric or the modal interpretation. On the former interpretation, the import of both the propositions If A then C and If Cʹ then Aʹ is to negative ACʹ ; on the latter interpretation, the import of both is to deny the possibility of the conjunction ACʹ.
From the negative hypothetical If A is true then C is not true we can infer If C is true then A is not true. This is analogous to conversion in the case of categoricals.
269 From the affirmative If A then C, we may obtain by conversion If C then possibly A ; but this is only on the interpretation that both propositions imply the possibility of the truth of their antecedents.[290] The reader will notice that to pass from If A then C to If C then A would be to commit a fallacy analogous to simply converting a categorical A proposition; and this is perhaps the most dangerous fallacy to be guarded against in the use of hypotheticals.[291]
[290] Compare section [158]. The various results obtained in section 158 may be applied mutatis mutandis to modal hypotheticals. The reader may consider for himself the contraposition of Em.
[291] On the assertoric interpretation If A then C merely negatives ACʹ, while If C then A merely negatives AʹC, and hence it is clear that neither of these propositions involves the other; on the modal interpretation the result is the same, for the truth of C may be a necessary consequence of the truth of A, while the converse does not hold good.
A consideration of immediate inferences enables us to shew from another point of view that If A then C and If A then Cʹ are not true contradictories. For the contrapositives If A then Cʹ, If C then Aʹ, are equivalent to one another; and whenever two propositions are equivalent, their contradictories must also be equivalent. But If A then C is not equivalent to If C then A.
If distinctions of quality are admitted, then the process of obversion is applicable to hypotheticals. For example, If A is true then C is not true = If A is true then Cʹ is true. It is nearly always more natural and more convenient to take hypotheticals in their affirmative rather than in their negative form; and hence in the case of hypotheticals more importance attaches to the process of contraposition than to that of conversion.
If the falsity of C is assumed to be possible, then we may pass by inversion from If A then C to It is possible for both A and C not to be true ; or, putting the same thing in a different way, we may by inversion pass from If A then C to If the falsity of C is possible then the falsity of both A and C is possible.[292] It is of course a fallacy to argue from If A then C to If Aʹ then Cʹ.
[292] The inversion of Em may be worked out similarly. Here, as elsewhere, the process of inversion, although of little or no practical importance, raises problems that are of considerable theoretical interest.
Turning to problematic hypotheticals, we find that from the proposition If A is true C may be true, we obtain by conversion If C is true A may be true ; and from the proposition If A is 270 true C need not be true we obtain by contraposition If C is true A need not be true. Here the analogy with categoricals is again very close.