(b) If conditionals are interpreted modally, then the apodeictic form takes the place of the universal, and the 260 problematic takes the place of the particular. On this basis, the converse of If any P is Q that P must be R would be If any P is R that P may be Q, and the contrapositive would be If any P is not R that P cannot be Q.
Are these inferences legitimate? On the interpretation that a modal proposition implies nothing as to the possibility of its antecedent, then our answer must be in the affirmative, as regards the contraposition of Am. The full import both of the original proposition and of the contrapositive is to deny the possibility of the combination P and Q without R. On the same interpretation, however, the conversion of Am is not valid. For the converse implies that if PR is possible then PQ is possible, while the possibility of PR combined with the impossibility of PQ is compatible with the truth of the original proposition. It can be shewn similarly that, while the conversion of Em is valid, its contraposition is invalid.
If we were to vary the interpretation, the results would be different.
The correspondence between the results shewn above and our results respecting the conversion and contraposition of the assertoric A and E propositions, on the interpretation that no proposition implies the existence of its subject (see page [225]), is obvious. The truth is that the interpretation of modals in respect to the possibility of their antecedents gives rise to problems precisely analogous to those arising out of the interpretation of assertoric propositions in respect to the actuality of their subjects. It is unnecessary that we should work out the different cases in detail.
Amongst immediate inferences from a conditional proposition, its reduction to categorical form, so far as this is valid, is generally included. This is a case of what has been called change of relation, meaning thereby an immediate inference in which we pass from a given proposition to another which belongs to a different category in the division of propositions according to relation (see section [54]). The more convenient term transversion is used by Miss Jones for this process.
How far conditionals can be inferred from categoricals and vice versâ depends on their interpretation. If both types of 261 propositions are interpreted assertorically or both modally, and if they are interpreted similarly as regards the implication of the existence (or possibility) of their subjects (or antecedents), then the validity of passing from either type to the other cannot be called in question. Some doubt may, however, be raised as to whether in this case we have an inference at all or merely a verbal change. This is a distinction to which attention will be called later on.
If conditionals are interpreted modally and categoricals assertorically then (apart from any complications that may arise from existential implications) A can be inferred from Am or E from Em, but not vice versâ. On the other hand, Im can be inferred from I, or Om from O, but not vice versâ.
We have another case of transversion when we pass from conditional to disjunctive, or from disjunctive to conditional. The consideration of this case must be deferred until we have discussed disjunctives.
178. The Import of Hypothetical Propositions.—The pure hypothetical may be written symbolically in the form If A is true then C is true, or more briefly, If A then C, where A and C stand for propositions of independent import. It is clear that this proposition affirms nothing as regards the truth or falsity of either A or C taken separately. We may indeed frame the proposition, knowing that C is false, with the express object of showing that A is false also. What we have is of course a judgment not about either A or C taken separately, but about A and C in relation to one another.