1880.] expressing the common propositions of Logic. 39 



terminateness by describing S as the #-part of what is not P 

 instead of directly reminding us that it is an unknown part. It 

 also abbreviates by substituting a single letter p for the fuller 

 equivalent 1 — P. No. (5) is employed by Wundt (Logik) in his 

 account of symbolic procedure. It only differs from the third and 

 fourth by making use of X as the universe-symbol, instead of unity. 

 As regards the representation of the class not-P by p, in (6), there 

 is one serious defect; for the purposes of symbolic logic, a fatal 

 defect. We cannot thus represent the negation of a composite 

 class. The other schemes meet the difficulty. " What is not both 

 B and G" can be readily represented by 1— PC, and "what is 

 neither B nor C" by (1 — B) (1 — G) ; and the similar devices of a 

 bar over two or more letters, or an accent put outside a bracket 

 Avhich contains them (as in specimens 12 and 13) will subserve the 

 same purpose. But on the plan of employing small letters to 

 mark negations, ab would stand for " What is neither A nor B," 

 and there seems no ready mode of simply expressing the negation 

 of AB as a tuhole. We have to break it up into detail and write 

 it a + Ab, or in some equivalent form. This is a serious symbolic 

 blemish. 



7. This form is employed by Prof. Delboeuf (Logique Algo- 

 rithmique, 1877), and seems to me identical with one proposed by 

 Mr J. J. Murphy (Relation of Logic to Language, 1875; Mind, v. 52). 

 It only differs from the preceding ones by adopting the subtractive 

 instead of the multiplicative symbol. Whereas those four say, 

 Make an indetermiuate selection from not-P and we obtain 8; 

 this says, Make an indeterminate rejection by omission, and we 

 obtain S. It is not incorrect, but seems to me to suffer from the 

 drawback of demanding a tacit condition, viz. that y shall be 

 included in 1 — P. Wlien this condition is expressed, it coincides 

 with Boole's form (If B is included in A, so that B = vA, then 

 A — B = (1 — v)A : but, 1 — v having the same limits of uncertainty 

 as v, this may be written vA), and becomes exactly equivalent to 

 S = v(l — P). It should be noticed that Delboeuf divides this 

 general form into several distinct cases according to the extent of 

 the whole universe covered by S and not-P, and so forth (Log. 

 Alg. p. 58). 



8. This was a scheme projDOsed by Holland, a friend and cor- 

 respondent of J. H. Lambert. Though not sound in this particular 

 application it deserves notice, both for its ingenuity, and historically 

 as an anticipation of some more modern results. (It is given in 



Lambert's Deutscher gelehrter Briefwechsel, I. 17.) The general 



a p 



propositional form which he proposed was — = — , which is really 



