a collection of individuals, to each of which a particular name or description may be applied; but in this work the meaning of the term will be extended so as to include the case in which but a single individual exists, answering to the required name or description, as well as the cases denoted by the terms "nothing" and "universe," which as "classes" should be understood to comprise respectively 'no beings,' 'all beings.'" (Boole 1854:28) He then defines what the string of symbols e.g. xy means [modern logical &, conjunction]: "Let it further be agreed, that by the combination xy shall be represented that class of things to which the names or descriptions represented by x and y are simultaneously, applicable. Thus, if x alone stands for "white things," and y for "sheep," let xy stand for 'white Sheep;'" (Boole 1854:28) Given these definitions he now lists his laws with their justification plus examples (derived from Boole): (1) xy = yx [commutative law] "x represents 'estuaries,' and y 'rivers,' the expressions xy and yx will indifferently represent" 'rivers that are estuaries,' or 'estuaries that are rivers,'" (2) xx = x, alternately x2 = x [Absolute identity of meaning, Boole's "fundamental law of thought" cf page 49] "Thus 'good, good' men, is equivalent to 'good' men". Logical OR: Boole defines the "collecting of parts into a whole or separate a whole into its parts" (Boole 1854:32). Here the connective "and" is used disjunctively, as is "or"; he presents a commutative law (3) and a distributive law (4) for the notion of "collecting". The notion of separating a part from the whole he symbolizes with the "-" operation; he defines a commutative (5) and distributive law (6) for this notion: (3) y + x = x + y [commutative law] "Thus the expression 'men and women' is ... equivalent with the expression" women and men. Let x represent 'men,' y, 'women' and let + stand for 'and' and 'or' ..." (4) z(x + y) = zx + zy [distributive law] z = European, (x = "men, y = women): European men and women = European men and European women (5) x − y = −y + x [commutation law: separating a part from the whole] "All men (x) except Asiatics (y)" is represented by x − y. "All states (x) except monarchical states (y)" is represented by x − y (6) z(x