SYMBOLIC EEASONING. 497 (7 = 0) will very well express that a is a certainty, ft an impossibility, and 7 a variable ; but in complicated cases (in order to economise space and dispense as far as possible with those necessary evils called brackets] still simpler symbols were desirable. I therefore chose indices to denote classes of statements, so that, for example, the symbol a u asserts that the statement a belongs to the class of statements denoted by the symbol u. On this interpretation, the three symbols a , ft*>, y 9 respectively assert that a is a certainty, ft an impossibility, and 7 a variable, and are therefore synonymous with the three longer symbols (a = e), (ft = 77), and (7 = 9}. Again, putting T for a true statement (not necessarily a certainty) and t for a false statement (not necessarily an impossibility], a7 will assert that a is true, and a 1 that a is false. Thus a r : ft T in my new notation becomes equivalent to a : ft in my former ; a7 : ft' becomes equivalent to a : ft' ; a T ft r to aft ; a7 + ft' to a + ft' ; and so on ; while a 6 (not a T ) becomes equivalent to the former (a = 1), and a? (not a 1 ) becomes equivalent to the former (a = 0). The symbols 1 and being thus displaced by and ?; were ipso facto set at liberty for other purposes. I use the former in conjunction with its old comrades 2, 3, 4, etc., in such cases as u v u. 2 , u 3 , etc., which respectively denote particular statements of the class u. Thus the equational statement a/3 : a + ft = e t asserts that the im- plication aft : a + ft is a particular statement of the class called certainties, and is the first certainty that has entered into our present argument. Similarly the equational state- ment 0* = ?7 3 asserts that 6* 1 is a particular statement of the class called impossibilities, and is the third impossibility that has entered into our argument. And so on for other particular statements ?; 2 , e 5 , 4 , etc., of their respective classes. On the other hand, I use the symbols a of3 , a, &o(u+,), etc. (chiefly to avoid cumbersome brackets) as con- venient synonyms for the denials (a^)', (a")', (+,)'. etc. For example, the formula (a + ft) m = a ou + ft ou is much more convenient than its equivalent, the formula {(a + ft]uY ()' + (#)'. Other symbols of abbreviation employed by me in my fifth paper (recently published) in the Pro- ceedings of the Mathematical Society are the following : (1) a u is short for (a")*, so that aF, a , (V>, and (a = 77)' are all four synonymous, each denying the truth of a* and asserting that a is a possibility. Similarly, a um> means (a"') w ; and so on. (2) a" 1 " is short for a u -f- a* ; , is short for a u + a v ; (a, ft) u is short for a" + ft". Hence it is evident that a", a 04 , 32
