Studies and Exercises in Formal Logic

by John Neville Keynes

[⁴⁶⁶] The conjunctive combination of terms is in symbolic logic usually represented by the sign of multiplication.

A complex term resulting from the alternative combination of other terms may be called an alternative term; and we may speak of the elements combined in such a term as the alternants of that term. Thus, A and B are the alternants of the alternative term A or B.[467]

[⁴⁶⁷] The alternative combination of terms is in symbolic logic usually represented by the sign of addition.

469 In the following pages, in accordance with the view indicated in section [191], the alternants in an alternative term are not regarded as necessarily exclusive of one another (except of course where they are formal contradictories). Thus, if we speak of anything as being A or B we do not intend to exclude the possibility of its being both A and B. In other words, A or B does not exclude AB.

It is necessary at this point to consider briefly the logical signification of the words and, or. In the predicate of a proposition their signification is clear; they indicate conjunctive and alternative combination respectively; for example, P is Q and R, P is Q or R. But when they occur in the subject of a proposition there is in each case an ambiguity to which attention must be called.

Thus, there would be a gain in brevity if we could write a proposition with an alternative term as subject in the form P or Q is R. This last expression would, however, more naturally be interpreted to mean P is R or Q is R, the force of the or being understood, not as yielding a single categorical proposition with an alternative subject-term, but as a brief mode of connecting alternatively two propositions with a common predicate. Hence, when we intend the former, the more definite mode of statement, Whatever is either P or Q is R, or Anything that is either P or Q is R, should be adopted.

There is also ambiguity in the form P and Q is R. This would naturally be interpreted, not as a single categorical proposition with a conjunctive subject-term (PQ is R), but as a brief mode of connecting conjunctively two propositions with a common predicate, namely, P is R and Q is R. In order, therefore, to express unambiguously a proposition with a conjunctive subject-term, it will be well either to adopt the method of simple juxtaposition without any connecting word as, for example, PQ is R, or else to employ one of the more cumbrous forms, Whatever is both P and Q is R, or Anything that is both P and Q is R.[468]

[⁴⁶⁸] It will be observed that both in this case and in the case of or, we get rid of the ambiguity by making the words occur in the predicate of a subordinate sentence. Mr Johnson expresses the substance of the last three paragraphs in the text by pointing out that “common speech adopts the convention: Subjects are externally synthesised and predicates are internally synthesised” (Mind, 1892, p. 239). In other words, and and or occurring in a predicate are understood as expressing a conjunctive or an alternative term ; but occurring in a subject they are understood as expressing a conjunctive or an alternative proposition.

425. Order of Combination in Complex Terms.—The order of 470 combination in a complex term is indifferent whether the combination be conjunctive or alternative.[469]

[⁴⁶⁹] This is sometimes spoken of as the law of commutativeness. Compare Boole, Laws of Thought, p. 31, and Jevons, Principles of Science, 2, § 8.