Only two types of compound propositions are here recognised, the conjunctive and the alternative. Pure hypothetical propositions are compound, but (except in so far as we interpret hypotheticals and alternatives differently in respect of modality) they are equivalent to alternative propositions, and may be regarded as constituting one mode of expressing an alternative synthesis. Thus (taking x and y as symbols representing propositions, and x and y as their contradictories) the hypothetical proposition If x then y expresses an alternative between x and y and is, therefore, equivalent to the alternative proposition x or y. Combinations of the true disjunctive type (for example, not both x and y) may also be regarded as a mode of expressing an alternative synthesis; thus, the true disjunctive proposition just given is equivalent to the alternative proposition x or y.[486]
[486] The above may seem to imply that an alternative synthesis may be expressed in a greater number of ways than a conjunctive synthesis. This, however, is not the case. It has been shewn that an alternative synthesis may be expressed by a hypothetical or by the denial of a conjunctive (that is, by a true disjunctive). But corresponding to this, a conjunctive synthesis may be expressed by the denial of a hypothetical or by the denial of an alternative. Thus, representing the denial of a proposition by a bar drawn across it, we have
xy = x̅ or y̅ = If x, y̅ ;
xy = x or y = If x, y.
Mr Johnson shews that any ordinary proposition with a general term as subject may be regarded as a compound proposition resulting from the conjunctive or alternative combination of singular (molecular) propositions, with a common predication, but different subjects. Let S1, S2, … S∞ represent a number of different individual subjects; and let S represent the aggregate collection of individuals S1, S2, … S∞. Then
S1 and S2, and S3 … and S∞ = Every S ;
S1 or S2, or S3 …… or S∞ = Some S.
480 “Thus we arrive at the common logical forms, Every S is P, Some S is P. The former is an abbreviation for a determinative, the latter for an alternative, synthesis of molecular propositions.”[487]
[487] Mind, 1892, p. 25. Mr Johnson of course recognises that a quantified subject-term (all S) is not usually a mere enumeration of individuals first apprehended and named. But he points out that “however the aggregate of things, to which the universal name applies, is mentally reached, the propositional force for purposes of inference or synthesis in general is the same” (p. 28).
In other words,
Every S is P = S1 is P and S2 is P and S3 is P … and S∞ is P ;
Some S is P = S1 is P or S2 is P or S3 is P … or S∞ is P.
443. The Opposition of Compound propositions.—The rule for obtaining the contradictory of a complex term given in section [426] may be applied also to compound propositions. Thus, the contradictory of a compound proposition is obtained by replacing the constituent propositions by their contradictories and everywhere changing the manner of their combination, that is to say, substituting conjunctive combination for alternative and vice versâ.[488] The following are examples: All A is B and some P is Q has for its contradictory Either some A is not B or no P is Q ; Either some A is both B and C, or all B is either C or both D and E has for its contradictory No A is both B and C, and some B is not either C or both D and E.