[488] It has been shewn in the preceding [section] that the words all and some are abbreviations of conjunctive and alternative synthesis respectively. Hence the rule that, in the ordinarily recognised propositional forms, contradictories differ in quantity as well as in quality is itself only a particular application of the general law here laid down.

It follows, as in section [427], that there is a duality of formal equivalences in the case of compound propositions, each equivalence yielding a reciprocal equivalence in which conjunctive combination is throughout substituted for alternative combination and vice versâ.

444. Formal Equivalences of Compound Propositions.—The laws relating to the conjunctive or alternative synthesis of propositions are practically identical with those relating to the conjunctive or alternative combination of terms; and we have accordingly the following propositional equivalences corresponding to the equivalences of terms given in section [433]. The symbols here stand for propositions, not terms; and negation is represented by a bar over the proposition denied. 481

(1) x (y or z) = xy or xz,⎱Laws of Distribution ;
(2) x or yz = (x or y) (x orz),⎰
(3) xx = x,⎱ Laws of Tautology (Law of Simplicity and Law ofUnity) ;
(4) x or x = x,⎰
(5) x = x or yy = (x or y) (x ory),⎱Laws of Development and Reduction ;
(6) x = x (yor y) = xy or xy,⎰
(7) x or xy = x,⎱Laws of Absorption ;
(8) x (x or y) = x⎰
(9) x or y = x or xy,⎱Law of Exclusion and Law of Inclusion.
(10) xy = x (x or y),[489]⎰

[489] It is not maintained that all the above laws are ultimate or even independent of one another. The synthesis of propositions is admirably worked out by Mr Johnson in his articles on the Logical Calculus (Mind, 1892). He gives five independent laws which are necessary and sufficient for propositional synthesis. These laws are briefly enumerated below; for a more complete exposition the reader must be referred to Mr Johnson’s own treatment of them.
(i) The Commutative Law: The order of pure synthesis is indifferent (xy = yx).
(ii) The Associative Law: The mode of grouping in pure synthesis is indifferent (xy . z = x . yz).
(iii) The Law of Tautology: The mere repetition of a proposition does not in any way add to or alter its force (xx = x).
(iv) The Law of Reciprocity: The denial of the denial of a proposition is equivalent to its affirmation (x̅ = x). “In this principle are included the so-called Laws of Contradiction and Excluded Middle, viz., ‘If x, then not not-x’, and ‘If not not-x, then x’.”
(v) The Law of Dichotomy: The denial of any proposition is equivalent to the denial of its conjunction with any other proposition together with the denial of its conjunction with the contradictory of that other proposition (x = xy xy̅). “This is a further extension of the Law of Excluded Middle, when applied to the combination of propositions with one another. The denial that x is conjoined with y combined with the denial that x is conjoined with not-y is equivalent to the denial of x absolutely. For, if x were true, it must be conjoined either with y or with not-y. This law, which (it must be admitted) looks at first a little complicated, is the special instrument of the logical calculus. By its means we may always resolve a proposition into two determinants, or conversely we may compound certain pairs of determinants into a single proposition.”

445. The Simplification of Complex Propositions.—The terms of a complex proposition may often be simplified by means of the rules given in the preceding chapter, and the force of the assertion will remain unaffected. For the further simplification of complex propositions the following rules may be added:
(1) In a universal negative or a particular affirmative proposition any determinant of the subject may be indifferently introduced or omitted as a determinant of the predicate and vice versâ.

482 To say that No AB is AC is the same as to say that No AB is C, or that No B is AC. For to say that No AB is AC is the same thing as to deny that anything is ABAC ; but, as shewn in section [429], the repetition of the determinant A is superfluous, and the statement may therefore be reduced to the denial that anything is ABC. And this may equally well be expressed by saying No AB is C, or No B is AC.[490]

[490] See also the [sections] in the following chapter relating to the conversion of propositions.

Again, Some AB is AC may be shewn to be equivalent to Some AB is C, or to Some B is AC ; for it simply affirms that something is ABAC, and the proof follows as above.

(2) In a universal affirmative or a particular negative proposition any determinant of the subject may be indifferently introduced or omitted as a determinant of any alternant of the predicate.