In lieu of an abstract, here is a brief excerpt of the content:

210 32 Algebra of the Copula [Version 1] Spring 1891 Houghton Library Logical quantity has but two values, t the true and f the false. But one operation is necessary, defined as follows: a. If b is true, a R b is true. b. Either a or a R b is true. g. If a and a R b are true, b is true. This gives the following: f R t  t f R f  t t R t  t t R f  f. Any proposition written is supposed to be true. In writing propositions parentheses are employed to enclose compounds to be treated as single letters in combining them with letters or other such compounds. These may be called clauses. Parentheses ending clauses or propositions are omitted, and the clauses they would have included are not commonly regarded as such. The last letter of a proposition or clause is called its consequent. Its other immediate parts, letters or clauses, are called antecedents. Thus in the proposition a R [(b R c) R d R e] R f the antecedents are a and (b R c) R d R e, and the antecedents of the latter clause are (b R c) and d. ALGEBRAIC RULES Rule 1. The order of antecedents is immaterial. Thus, a R b R c is identical with b R a R c. Rule 2. Any antecedent may be inserted under an even number of parentheses, or none, and omitted under an odd number. Thus if we have given (a R b) R c we can write d R (a R b) R c 32. Algebra of Copula. Version 1, 1891 211 or [(d R a) R b] R c or b R c. Rule 3. The insertion of an antecedent of a proposition a second time as antecedent of the proposition or of a clause of it has no effect. Thus a R b R c is identical with a R (a R b) R c, and a R (b R c) R d is identical with a R (a R b R c) R d and with a R [(a R b) R c] R d. Rule 4. The insertion of a clause having an antecedent identical with the consequent of the proposition has no effect; and a proposition having an antecedent identical with its consequent is identically true. Thus, (a R b) R c R a is identical with c R a. And so is [(a R b) R c] R a. Rule 5. Any proposition having a clause within a clause may be changed by expunging any of the antecedents of the inner clause and making each of these the consequent of an antecedent clause of the proposition, such clause having for one antecedent a clause with the consequent of the original outer clause as antecedent and the consequent of the proposition as consequent and for other antecedents whatever antecedents the original outer clause may have had beside the inner clause. Thus [a R (b R c R d R e) R f R g] R h is identical with [a R (g R h) R f R b] R [a R (g R h) R f R d] R [a R (c R e) R f R g] R h. ...

Share