Is God a Mathematician_ - Mario Livio [85]
Boole’s calculus could be interpreted either as applying to relations among classes (collections of objects or members) or within the logic of propositions. For instance, if x and y were classes, then a relation such as x y meant that the two classes had precisely the same members, even if the classes were defined differently. As an example, if all the children in a certain school are shorter than seven feet, then the two classes defined as x “all the children in the school” and y “all the children in the school that are shorter than seven feet” are equal. If x and y represented propositions, then x y meant that the two propositions were equivalent (that one was true if and only if the other was also true). For example, the propositions x “John Barrymore was Ethel Barrymore’s brother” and y “Ethel Barrymore was John Barrymore’s sister” are equal. The symbol “x · y” represented the common part of the two classes x and y (those members belonging to both x and y), or the conjunction of the propositions x and y (i.e., “x and y”). For instance, if x was the class of all village idiots and y was the class of all things with black hair, then x · y was the class of all black-haired village idiots. For propositions x and y, the conjunction x · y (or the word “and”) meant that both propositions had to hold. For example, when the Motor Vehicle Administration says that “you must pass a peripheral vision test and a driving test,” this means that both requirements must be met. For Boole, for two classes having no members in common, the symbol “x y” represented the class consisting of both the members of x and the members of y. In the case of propositions, “x y” corresponded to “either x or y but not both.” For instance, if x is the proposition “pegs are square” and y is “pegs are round,” then x y is “pegs are either square or round.” Similarly, “x y” represented the class of those members of x that were not members of y, or the proposition “x but not y.” Boole denoted the universal class (containing all possible members under discussion) by 1 and the empty or null class (having no members whatsoever) by 0. Note that the null class (or set) is definitely not the same as the number 0—the latter is simply the number of members in the null class. Note also that the null class is not the same as nothing, because a class with nothing in it is still a class. For instance, if all the newspapers in Albania are written in Albanian, then the class of all Albanian-language newspapers in Albania would be denoted by 1 in Boole’s notation, while the class of all Spanish-language newspapers in Albania would be denoted by 0. For propositions, 1 represented the standard true (e.g., humans are mortal) and 0 the standard false (e.g., humans are immortal) propositions, respectively.
With these conventions, Boole was able to formulate a set of axioms defining an algebra of logic. For instance, you can check that using the above definitions, the obviously true proposition “everything is either x or not x” could be written in Boole’s algebra as x (1 x) 1, which also holds in ordinary algebra. Similarly, the statement that the common part between any class and the empty class is an empty class was represented by 0 · x 0, which also meant that the conjunction of any proposition with a false proposition is false. For instance, the conjunction “sugar is sweet and humans are immortal” produces a false proposition in spite of the fact that the first part is true. Note that again, this “equality” in Boolean algebra holds true also with normal algebraic numbers.
To show the power of his methods, Boole attempted to use his logical symbols for everything he deemed important. For instance,