Note 489 — Chapter II

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.”