OF THE LAWS OF THOUGHT. 18l 



(«f) consist of all invertebrate animals having red blood (zV) and not 

 naked (1 — w), and an indefinite remainder {%') of invertebrate ani- 

 mals (i) not having red blood (i — r) and not naked (1 — 7i) and of 

 vertebrate animals (1 — i) which are not naked (1 — n). 



We have entered so fully into the explanation of Professor Boole's 

 system in its bearing on what he terms Primary (virtually equivalent 

 to Categorical) Propositions, that we cannot follow him into the field 

 of Secondary (virtually equivalent to Conditional, that is, Disjunctive 

 and Hypothetical) Propositions. Nor is it necessary that we should 

 do so ; for our object is not to give a synopsis of the " Investigation," 

 but simply to make the nature of the work understood ; and, for 

 that purpose, what has been said is sufficient. The application 

 of the Calculus to Secondary Propositions is exceedingly similar, in 

 respect not only of the general method followed, but even of the 

 particular formulee obtained, to its application to Primary. All that 

 is peculiar in the treatment of Secondary Propositions arises from 

 the introduction of the idea of Time. For instance, the proposition, 

 "If Xis T, ^ is J5," is held to be not substantially difierent in 

 meaning from this: "the time in -which X is T, is time in which 

 A is B." Such being the fundamental view taken, symbols like x and 

 y are used to represent the portions of time in which certain pro- 

 positions (e.g., Xis T, A is S) are true. Then, the symbol 1 denot- 

 ing the universe of Time, or Eternity, the expressions, 1 — oc, 1 — y, 

 will denote those portions of time respectively in which the propo- 

 sitions, X is Y, A is B, are not true ; and so on. 



The extension of his method, by Professor Boole, to the theory of 

 Probabilities, is a splendid efibrt of genius on the part of the author, 

 and furnishes a most convincing illustration of the capabilities of 

 the method. The part of the " Investigation " which is devoted to 

 this subject, is much too abstruse to admit of being here more par- 

 ticularly considered ; but, to show what the method can accomplish 

 — though the bow of Ulysses perhaps needs the arm of Ulysses to 

 bend it — we may simply state one of the problems of which Pro- 

 fessor Boole gives the solutioji. " If an event can only happen as a 



consequence of one or more of certain causes, A^, A^, , A„, 



and if generally G^ represents the probability of the cause A^, and 

 Pi the probability that, if the cause A^ exist, the event JS will occur, 

 then the series of Cj and^^ being given, required the probability of 

 the event ^" 



