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once, from this statement, that if X is not a knave, he must be a fool, and that if he is not a fool, he must be a knave, and thereby emphasize in a fresh and specific way our original contention that X is an objectionable character. The inference that 'If X is a knave, he is not a fool' would be beside the point. It would not emphasize his objectionableness, but qualify and limit it.

Suppose, again, that X is in a perilous situation at the edge of a chasm, and that the categorical requirement of immediate action is presented in the disjunctive form: 'Either jump or starve.' Here the essential inference is that 'If X does not jump, he will starve,' for it is this consideration that supplies the incentive to action. Little can be gained by inferring that 'If X does jump, he will not starve,' and that 'If X starves, he will not have jumped.'

I quote from the article by Mr. Ross the following illustrations in further support of the contention that, under ordinary circumstances, an exhaustivist need not pledge himself either to the exclusivist or to the non-exclusivist theory.

1. 'Planetary orbits* fall either wholly inside or wholly outside the Earth's orbit.'

Here the disjunction admits of being read exclusively. We can infer, if we like, that 'Jupiter's orbit, lying without that of the Earth, cannot lie wholly nearer to the sun than it.' But this is futile. The real force of the disjunction lies, as Mr. Ross points out, in its exhaustiveness, in the implied denial that there are any planets with orbits intersecting that of the Earth. It lies in the possibility of the inference that if a planetary orbit does not fall wholly inside the Earth's orbit, then it must lie wholly outside it.

2. 'Planets whose orbits lie between the Earth and the Sun are, when visible, to be seen either in the morning or in the evening.'

Here we have a statement which we may very profitably utilize without possessing any proof that Venus, for instance, when visible in the evenings, must in the morning rise after the Sun, and so be lost in his light. The force of the disjunction lies, again, in its exhaustiveness, in the certainty which it professes to give us that planets of the kind specified are never to be seen during the middle of the night. It assures us that if we see a starry object of peculiar brilliance at midnight, though it may be Jupiter, it cannot be Venus.

We have so far been considering the Disjunctive Proposition from a point of view that compels no reference to the Exclusivist controversy. But a point is necessarily reached where a decision on this question becomes imperative. The point is reached so soon as the distinction of the alternatives from each other becomes a matter of logical interest. It may become important to decide whether an

* In this statement the Earth is not included among the planets.

objectionable person is a pure knave or a pure fool, the knavery being unadulterated with foolishness, and vice versa, or whether he is at once a knave and a fool. In that case, when we make the statement that 'Either X is a knave or he is a fool,' we are directly interested in knowing whether the alternatives are intended or are not intended to be mutually exclusive.

We must distinguish here between two quite different intereststhe interest in non-ambiguity and the interest in scientific precision.

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1. The statement in question may not be our own, but may be made by some other person, and we may be interested in understanding what that same person precisely means by his statement. In this case, however, it does not lie with us to decide on the sense to be put upon the 'either or.' We must leave that to the 'other person.' The most we can do in defence of logical interests is to insist that he shall make his meaning explicit, and that a rational convention shall be reached which, in respect of the assertion in question, shall secure complete freedom from ambiguity. Thus, we might suggest that, if the exclusive reading be intended, the statement should take the form 'Either he is a non-foolish knave or a non-knavish fool,' or perhaps, more simply, 'Either he is a fool or else he is a knave'; the ordinary form, 'Either he is a knave or he is a fool' being reserved for the non-exclusive reading.

If our interlocutor is stating his disjunctive proposition in symbolic form, it seems simpler to reserve the ordinary form 'Either Por Q' for the non-exclusive reading. The attempt to expand this into 'Either Por Qor PQ' (the three alternatives being taken as mutually exclusive) results in a logical disaster or else in a confusion of symbols. For 'P' here means 'Pand not Q,' and 'Q' means 'Q and not P.' Hence PQ should mean 'P and not Q combined with Q and not P,' which leaves little distance between PQ and logical nonentity. In symbolic language, then, 'Either Por Q' represents most suitably the non-exclusive reading, and 'Either PQ or QP'— or, more simply, 'Either Por else Q'-the exclusive reading. The form Either P only or Q only' is obviously unsatisfactory as an interpretation of the exclusive meaning, for 'Ponly' implies a much greater restriction than 'P but not Q.' There is, indeed, another alternative. The exclusive form may be uniformly adopted. In this case the proposition 'Either P or Q' would be transformed into 'Either PQ or QP or PQ' when 'Either or' implies 'it may be both,' and into 'Either PQ or QP,' when 'Either or' implies 'not both.' But it is very doubtful whether this gain in uniformity would sufficiently compensate for the loss of the simple form 'Either P or Q.'

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2. We come now to the second of the two interests in connexion with which a decision on the Exclusivist question becomes essential. The distinction of alternative possibilities from each other may have a real value for Science, and the interest in exclusiveness may ally itself with the interest in a logical ideal of disjunction. Assuming that this distinction of alternative possibilities is desirable as it is, for instance, in the classification of species under a genus-we have to ask what the logical ideal of disjunction demands of the disjunctive proposition in view of this requirement.

The natural answer would seem to speak wholly in favour of the Exclusivists. For surely it is only when the disjunctive proposition is read exclusively that we obtain this desired distinctness between the various alternatives.

But in this view there is involved an assumption which we must try to make clear. It is this: We are assuming that the way in which the disjunctive proposition can always best further the logical idea of precision is by being itself ideally precise. But this is a fundamental misconception. In the service of logical ideals the various forms of judgment must co-operate. In some cases the ideal that of precise characterization, for instance-will be reached by means of a categorical proposition. As an example of this, we may cite the record of some delicate scientific observation. In other cases, as in the development of a supposition into its consequences, it is the hypothetical proposition that must embody the logical ideal-in this case that of necessary connexion between the parts of the proposition. In other circumstances, again, as in scientific Classification, the ideal, that of the mutual exclusiveness of a number of co-ordinate possibilities, requires for its embodiment a disjunctive proposition.

Now, where the function of embodying the logical ideal falls to the lot of the disjunctive proposition, there can be no doubt that the alternative possibilities which it enumerates must be mutually exclusive as well as exhaustive. Here the function of the disjunctive logically requires that its form shall be perfect. Hence, when scientific results are tabulated in disjunctive form, the disjunctive should be of the exclusive type, or should approximate as closely as possible to that type. In Mathematics these perfect disjunctions are always obtainable, but in the more concrete sciences this is not the case. In that classification of species in which the species are defined 'by type' (vide p. 62) the boundaries between two co-ordinate species will often be somewhat uncertain, though the various types, as centrally defined, are distinct and mutually exclusive.

But where the function of embodying the logical ideal falls, shall we say, to the lot of the categorical proposition, and the disjunctive proposition fulfils its service as a mere means towards ensuring to the categorical the maximum precision of statement, that service may frequently be best fulfilled when the disjunctive is nonexclusive. The ideal functioning of the disjunctive requires here the imperfection of its form. Suppose that, starting from a given categorical basis, we are able to state our alternatives exhaustively in the form 'Either Por Qor R.' A methodical scientific inquiry may cancel Pas a possibility which in the given circumstances cannot be actualized, and we are left with the conclusion 'Either Qor R.' The possibility Q, we suppose, is similarly cancelled, and we are left with the categorical assertion 'R.' In a case of this kind the imperfect, non-exclusive type of disjunction is the most serviceable, and for that reason the most ideal. Where our aim is to reach the truth of one alternative by the elimination of the others, it is otiose to insist on the various alternatives being ab initio mutually exclusive. The successive cancelling of P and Q is a process that is quite independent of the question of exclusiveness or non-exclusiveness. Suppose that the disjunction is given in exclusive form. We are then told that either PQR is true, or QPR is true, or RPQ is true. P and Q are cancelled, and we are left with RPQ. But this is really the same result as that which we reached above, when we started with the imperfect, non-exclusive disjunctive 'Either Por Q or R.' For the result R there obtained might equally well have been expressed 'RPQ,' since it was certainly exclusive of P and Q.

The inconvenience of refusing to utilize a disjunctive proposition at all except when expressed in its most perfect form may best be brought out by a comparison which is much more than a mere analogy. It would be like refusing to use a hypothetical proposition until it had been rendered so precise in both its parts that not only should the affirmation of the antecedent involve that of the consequent, but also the affirmation of the consequent should involve that of the antecedent.

We conclude, then, that the disjunctive, in the service of the categorical, may profitably be left in non-exclusive form, and that this imperfect, non-exclusive form is the form required in the interests of Scientific Explanation. Formal Classification, on the other hand, and the laying out of alternative possibilities in the mathematical sciences, require the service of the disjunctive judgment in its perfect and exclusive form.

A disjunctive proposition, then, may be defined as a statement to the effect that, of a closed number of alternative possibilities, one is taken to be actualized. The one obligatory rule of disjunctionRule I. is that the alternatives shall exhaust the possibilities. Where it falls to the lot of the disjunctive to uphold the ideal of scientific precision, we have further to observe Rule II., that the alternatives shall be reciprocally exclusive. In mathematical inquiry, where scientific precision is as imperative at the beginning of the inquiry as it is at its close, both rules must necessarily hold good for all disjunctive propositions.

Example. Criticize the following disjunction :

'Either triangles are equilateral, or they are isosceles, or they are right-angled.'

This breaks Rule II., for an isosceles triangle may be right-angled. Whether it can be equilateral depends on the precise definition of the isosceles triangle.

It also breaks Rule I. There are triangles which are neither equilateral, isosceles, nor right-angled-namely, the scalene triangles which are not right-angled.

CHAPTER XV.

IV. (v.) THE HYPOTHETICAL PROPOSITION.

THE Categorical Proposition is a proposition which purports to be a statement of fact. The fact to which the statement refers need not be a 'real' fact, or fact of Nature, a fact under Causal Law. It may be a 'formal' fact, a fact that can have actuality only in reference to a certain limited universe of discourse.

The Hypothetical Proposition, on the other hand, may be defined as the statement of a connexion between two possibilities. It contains two clauses, of which the first is called the Antecedent, and the second the Consequent. The Disjunctive Proposition also, as we have seen, states a connexion between possibilities, but in that case the possibilities are regarded as alternative possibilities.

The connexion between the Hypothetical Proposition and the Disjunctive is, from the point of view of the logical development of thought, a very close one. The Hypothetical Proposition, as we have already seen (vide p. 113), takes one of the possibilities which the Disjunctive Proposition specifies, and develops by connecting it with another possibility. Thus the hypothetical selects for its antecedent one of a number of possibilities disjunctively presented. The consequent also may be regarded as having been drawn from the alternatives of another disjunctive series. Thus both the possibilities with which the Hypothetical Proposition is concerned are of the disjunctive type, and therefore real, and not modal, possibilities.

Suppose that we have before us the disjunctive proposition 'Either a triangle is obtuse-angled, or it is right-angled, or it has three acute angles.' Selecting one of these alternative possibilities, we say 'If a triangle is right-angled, a semicircle may be circumscribed to it having its hypotenuse as diameter.'

As the Hypothetical Proposition is concerned with possibilities, and the Categorical with actualities, or with what purport to be actualities, whether formal or real, it is logically impossible to express a hypothetical proposition as a genuine categorical, though it may be equivalently expressed in categorical form. Consider

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