− y) = zx − zy [distributive law] Lastly is a notion of "identity" symbolized by "=". This allows for two axioms: (axiom 1): equals added to equals results in equals, (axiom 2): equals subtracted from equals results in equals. (7) Identity ("is", "are") e.g. x = y + z, "stars" = "suns" and "the planets" Nothing "0" and Universe "1": He observes that the only two numbers that satisfy xx = x are 0 and 1. He then observes that 0 represents "Nothing" while "1" represents the "Universe" (of discourse). The logical NOT: Boole defines the contrary (logical NOT) as follows (his Proposition III): "If x represent any class of objects, then will 1 − x represent the contrary or supplementary class of objects, i.e. the class including all objects which are not comprehended in the class x" (Boole 1854:48) If x = "men" then "1 − x" represents the "universe" less "men", i.e. "not-men". The notion of a particular as opposed to a universal: To represent the notion of "some men", Boole writes the small letter "v" before the predicate-symbol "vx" some men. Exclusive- and inclusive-OR: Boole does not use these modern names, but he defines these as follows x(1-y) + y(1-x) and x + y(1-x), respectively; these agree with the formulas derived by means of the modern Boolean algebra. Boole derives the law of contradiction Armed with his "system" he derives the "principle of [non]contradiction" starting with his law of identity: x2 = x. He subtracts x from both sides (his axiom 2), yielding x2 − x = 0. He then factors out the x: x(x − 1) = 0. For example, if x = "men" then 1 − x represents NOT-men. So we have an example of the "Law of Contradiction": "Hence: x(1 − x) will represent the class whose members are at once "men," and" not men," and the equation [x(1 − x)=0] thus express the principle, that a class whose members are at the same time men and not men does not exist. In other words, that it is impossible for the same individual to be at the same time a man and not a man. ... this is identically that "principle of contradiction" which Aristotle has described as the fundamental axiom of all philosophy. ... what has been commonly regarded as the fundamental axiom of metaphysics is but the consequence of a law