189. The terms Disjunctive and Alternative as applied to Propositions.—Propositions of the form Either X or Y is true are ordinarily called disjunctive. It has been pointed out, however, that two propositions are really disjoined when it is denied that they are both true rather than when it is asserted that one or other of them is true; and the term alternative, as suggested by Miss Jones (Elements of Logic, p. 115), is obviously appropriate to express the latter assertion. We should then use the terms conjunctive, disjunctive, alternative, remotive, for the four following combinations respectively: X and Y are both true, X and Y are not both true, Either X or Y is true,[295] Neither X nor Y is true.
[295] Some writers indeed regard the proposition Either X or Y is true as expressing a relation between X and Y which is disjunctive in the above sense as well as alternative; but the disjunctive character of this proposition as regards X and Y is at any rate open to dispute, whilst its alternative character is unquestionable (see section [191]).
Whilst, however, the name alternative is preferable to disjunctive for the proposition Either X or Y is true, the latter name has such an established position in logical nomenclature that it seems inadvisable altogether to discontinue its use in the old sense. It may be pointed out further that an alternative contains a veiled disjunction (namely, between not-X and not-Y) even in the stricter sense; for the statement that Either X or Y is true is equivalent to the statement that Not-X and not-Y are not both true. Hence, although generally using the term alternative, I shall not entirely discard the term disjunctive as synonymous with it.
276 190. Two types of Alternative Propositions.—In the case of propositions which are ordinarily described as simply disjunctive a distinction must be drawn similar to that drawn in the preceding chapter between conditionals and true hypotheticals. For the alternatives may be events or combinations of properties one or other of which it is affirmed will (always or sometimes) occur, e.g., Every blood vessel is either a vein or an artery, Every prosperous nation has either abundant natural resources or a good government ; or they may be propositions of independent import whose truth or falsity cannot be affected by varying conditions of time, space, or circumstance, and which must therefore be simply true or false, e.g., Either there is a future life or many cruelties go unpunished, Either it is no sin to covet honour or I am the most offending soul alive.
Any proposition belonging to the first of the above types may be brought under the symbolic form All (or some) S is either P or Q, and may, therefore, be regarded as an ordinary categorical proposition with an alternative term as predicate. It is usual and for some reasons convenient to defer the discussion of the import of alternative terms until propositions of this type are being dealt with. Such propositions might otherwise be dismissed after a very brief consideration.[296]
[296] It should be particularly observed that although the proposition Every S is P or Q may be said to state an alternative, it cannot be resolved into a true alternative combination of propositions. Such a resolution is, however, possible if the proposition (while remaining affirmative and still having an alternative predicate) is singular or particular: for example, This S is P or Q = This S is P or this S is Q ; Some S is P or Q = Some S is P or some S is Q.
Corresponding to this, we may note that an affirmative categorical proposition with a conjunctive predicate is equivalent to a conjunction of propositions if it is singular or universal, but not if it is particular. Thus, This S is P and Q = This S is P and this S is Q ; All S is P and Q = All S is P and all S is Q. From the proposition Some S is P and Q we may indeed infer Some S is P and some S is Q ; but we cannot pass back from this conclusion to the premiss, and hence the two are not equivalent to one another.
It may be added that a negative categorical proposition with an alternative predicate cannot be said to state an alternative at all, since to deny an alternation is the same thing as to affirm a conjunction. Thus the proposition No S is either P or Q can only be resolved into a conjunctive synthesis of propositions, namely, No S is P and no S is Q.
277 Alternative propositions of the second type are compound (as defined in section [55]). They contain an alternative combination of propositions of independent import: and they have for their typical symbolic form Either X is true or Y is true, or more briefly, Either X or Y, where X and Y are symbols representing propositions (not terms). So far as it is necessary to give them a distinctive name, they have a claim to be called true alternative propositions, since they involve a true alternative synthesis of propositions, and not merely an alternative synthesis of terms.
It will be convenient to speak of P and Q as the alternants of the alternative term P or Q, and of X and Y as the alternants of the alternative proposition Either X or Y.