Just to add to the whining ... Vacuous has a pretty rigorous meaning in mathematics, and it derives from one way the conditional (implication, only if, etc.) can be true. The conditional p -> q is true whenever p is false or q is true (false only when p is true and q is false). In the case that p is false, we say that the conditional is *vacuously* true. One example in mathematics is the subset operator. Given two arbitrary sets S and T, S is a subset of T if and only if every element of S is also an element of T. Symbolically, we write (x)(x an element of S -> x an element of T). Now, let 0 denote the empty set and A denote some arbitrary set. Then 0 is a subset of A because it vacuously fulfills the requirements of the subset. That is, there is no x such that x is an element of 0, so the antecedent of the conditional is false and the conditional itself is true. Trivial gets one into much more trouble as it is bandied about more often and less technically in mathematical circles. Sometimes it is taken to mean a result that is apparent or obvious. Sometimes it also refers to the conditional in the case where a problem reduces to p -> p. In abstract algebra, it is used to refer to a one element set, group, ring, etc. As in, the subgroup {e} is the trivial subgroup of a group G, where e is the identity element. Anyway, this probably *is* a bit more off the tack of the discussion, but oh well. Trivially, jlive
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