181. Hypothetical Propositions and Categorical Propositions.—A true hypothetical proposition has been defined as a proposition expressing a relation between two other propositions of independent import, not between two terms; and it follows that a true hypothetical is not, like a conditional, easily reducible to categorical form. So far as we can obtain an equivalent categorical, its subject and predicate will not correspond with the antecedent and consequent of the hypothetical. Thus, the proposition If A then C may, according to our interpretation of it, be expressed in one or other of the following forms; A is a proposition the truth of which is incompatible with the falsity of C ; A is a proposition from the truth of which the truth of C necessarily follows. It will be observed that, apart from the fact that these propositions are not of the ordinary categorical type,[293] the predicate is not in either of them equivalent to the consequent of the hypothetical.[294] No doubt a hypothetical proposition may be based on a categorical proposition of the ordinary type. But that is quite a different thing from saying that the two propositions are equivalent to one another.
[293] Since they are compound, not simple, propositions. The expression of compound propositions in categorical form is not convenient, and it is better to reserve the hypothetical and disjunctive forms for such propositions, the categorical and conditional forms being used for simple propositions.
[294] Amongst other differences the contrapositives of both these propositions differ from the contrapositive of the hypothetical. For, on either interpretation of the hypothetical, its contrapositive is If C is not true then A is not true, whilst the contrapositives of the above propositions are respectively,—A proposition whose truth is compatible with the falsity of the proposition C is not the proposition A, A proposition from which the proposition C is not a necessary consequence is not the proposition A.
The relation between hypothetical and disjunctive propositions will be discussed in the following [chapter].
182. Alleged Reciprocal Character of Conditional and Hypothetical Judgments.—Mr Bosanquet argues that the hypothetical judgment (and under this designation he would include the conditional as well as what we have called the true 271 hypothetical) “when ideally complete must be a reciprocal judgment. If A is B, it is C must justify the inference If A is C, it is B. We are of course in the habit of dealing with hypothetical judgments which will not admit of any such conversion, and the rules of logic accept this limitation … If in actual fact … AB is found to involve AC while AC does not involve AB, it is plain that what was relevant to AC was not really AB but some element αβ within it … Apart from time on the one hand and irrelevant elements on the other, I cannot see how the relation of conditioning differs from that of being conditioned … In other words, if there is nothing in A beyond what is necessary to B, then B involves A just as much as A involves B. But if A contains irrelevant elements, then of course the relation becomes one-sided … The relation of Ground is thus essentially reciprocal, and it is only because the ‘grounds’ alleged in every-day life are burdened with irrelevant matter or confused with causation in time, that we consider the Hypothetical Judgment to be in its nature not reversible” (Logic, I. pp. 261–3).
The question here raised is analogous to that of the possibility of plurality of causes which is discussed in inductive logic. It may perhaps be described as a wider aspect of the same question. So long as a given consequence has a plurality of grounds, it is clear that the hypothetical proposition affirming it to be a consequence of a particular one of these grounds cannot admit of simple conversion, for the converted proposition would hold good only if the ground in question were the sole ground.
Mr Bosanquet urges that the relation between ground and consequence will become reciprocal by the elimination from the antecedent of all irrelevant elements. It should be added that we can also secure reciprocity by the expansion of the consequent so that what follows from the antecedent is fully expressed. Thus, if we have the hypothetical If A then γ, which is not reciprocal, it is possible that A may be capable of analysis into αβ, and γ of expansion into γδ, so that either of the hypotheticals If α then γ, If αβ then γδ, is reciprocal. In the former case we have a more exact statement of the ground, all extraneous 272 elements being eliminated; in the latter case we have a more complete statement of the consequence. Sometimes, moreover, the latter of these alternatives may be practicable while the former is not.
This may be tested by reference to a formal hypothetical. The proposition If all S is M and all M is P, then all S is P is not reciprocal. We may make it so by expanding the consequent so that the proposition becomes If all S is M and all M is P, then whatever is either S or M is P and is also M or not S. But how in this case it would be possible to eliminate the irrelevant from the antecedent it is difficult to see. Our object is to eliminate M from the consequent, and if in advance we were to eliminate it from the antecedent the whole force of the proposition would be lost. And the same is true of non-formal hypotheticals, at any rate in many cases. Instances of reciprocal conditionals may be given without difficulty, for example, If any triangle is equilateral, it is equiangular. Such propositions are practically U propositions. We may also find instances of pure hypotheticals that are reciprocal; but, on the whole, while agreeing with a good deal that Mr Bosanquet says on the subject, I am disposed to demur to his view that the reciprocal hypothetical represents an ideal at which we should always aim. We have seen that there are two possible ways of securing reciprocity, whether or not they are always practicable; but the expansion of the consequent would generally speaking be extremely cumbrous and worse than useless, while the elimination from the antecedent of everything not absolutely essential for the realisation of the consequent would sometimes empty the judgment of all practical content for a given purpose. With reference to the case where AB involves AC, while AC does not involve AB, Mr Bosanquet himself notes the objection,—“But may not the irrelevant element be just the element which made AB into AB as distinct from AC, so that by abstracting from it AB is reduced to AC, and the judgment is made a tautology, that is, destroyed?” (p. 261). This argument, although somewhat overstated, deserves consideration. The point upon which I should be inclined to lay stress is that in criticising a judgment we ought to have regard 273 to the special object with which it has been framed. Our object may be to connect AC with AB, including whatever may be irrelevant in AB. Consider the argument,—If anything is P it is Q, If anything is Q it is R, therefore, If anything is P it is R. It is clear that if we compare the conclusion with the second premiss, the antecedent of the conclusion contains irrelevancies from which the antecedent of the premiss is free. Yet the conclusion may be of the greatest value to us while the premiss is by itself of no value. If our aim were always to get down to first principles, there would be a good deal to be said for Mr Bosanquet’s view, though it might still present some difficulties; but there is no reason why we should identify the conditional or the hypothetical proposition with the expression of first principles.
It is to be added that, if Mr Bosanquet’s view is sound, we ought to say equally that the A categorical proposition is imperfect, and that in categoricals the U proposition is the ideal at which we should aim. In categoricals, however, we clearly distinguish between A and U; and so far as we give prominence to the reciprocal modal, whether conditional or hypothetical, we ought to recognise its distinctive character. We may at the same time assign to it the distinctive symbol Um.