grue paradox a paradox in the theory of induction, according to which every intuitively acceptable inductive argument, A, may be mimicked by indefinitely many other inductive arguments – each seemingly quite analogous to A and therefore seemingly as acceptable, yet each nonetheless intuitively unacceptable, and each yielding a conclusion contradictory to that of A, given the assumption that sufficiently many and varied of the sort of things induced upon exist as yet unexamined (which is the only circumstance in which A is of interest). Suppose the following is an intuitively acceptable inductive argument: (A1) All hitherto observed emeralds are green; therefore, all emeralds are green. Now introduce the colorpredicate ‘grue’, where (for some given, as yet wholly future, temporal interval T) an object is grue provided it has the property of being either green and first examined before T, or blue and not first examined before T. Then consider the following inductive argument: (A2) All hitherto observed emeralds are grue; therefore, all emeralds are grue. The premise is true, and A2 is formally analogous to A1. But A2 is intuitively unacceptable; if there are emeralds unexamined before T, then the conclusion of A2 says that these emeralds are blue, whereas the conclusion of A1 says that they are green. Other counterintuitive competing arguments could be given, e.g.: (A3) All hitherto observed emeralds are grellow; therefore, all emeralds are grellow (where an object is grellow provided it is green and located on the earth, or yellow otherwise). It would seem, therefore, that some restriction on induction is required. The new riddle of induction offers two challenges. First, state the restriction – i.e., demarcate the intuitively acceptable inductions from the unacceptable ones, in some general way, without constant appeal to intuition. Second, justify our preference for the one group of inductions over the other. (These two parts of the new riddle are often conflated. But it is at least conceivable that one might solve the analytical, demarcative part without solving the justificatory part, and, perhaps, vice versa.)
It will not do to rule out, a priori, ‘gruelike’ (now commonly called ‘gruesome’) variances in nature. Water (pure H2O) varies in its physical state along the parameter of temperature. If so, why might not emeralds vary in color along the parameter of time of first examination?
One approach to the problem of restriction is to focus on the conclusions of inductive arguments (e.g., All emeralds are green, All emeralds are grue) and to distinguish those which may legitimately so serve (called ‘projectible hypotheses’) from those which may not. The question then arises whether only non-gruesome hypotheses (those which do not contain gruesome predicates) are projectible. Aside from the task of defining ‘gruesome predicate’ (which could be done structurally relative to a preferred language), the answer is no. The English predicate ‘solid and less than 0;C, or liquid and more than 0;C but less than 100;C, or gaseous and more than 100;C’ is gruesome on any plausible structural account of gruesomeness (note the similarity to the English ‘grue’ equivalent: green and first examined before T, or blue and not first examined before T). Nevertheless, where nontransitional water is pure H2O at one atmosphere of pressure (save that which is in a transitional state, i.e., melting/freezing or boiling/condensing, i.e., at 0°C or 100;C), we happily project the hypothesis that all non-transitional water falls under the above gruesome predicate.
Perhaps this is because, if we rewrite the projection about non-transitional water as a conjunction of non-gruesome hypotheses – (i) All water at less than 0;C is solid, (ii) All water at more than 0;C but less than 100;C is liquid, and (iii) All water at more than 100;C is gaseous – we note that (i)–(iii) are all supported (there are known positive instances); whereas if we rewrite the gruesome projection about emeralds as a conjunction of non-gruesome hypotheses – (i*) All emeralds first examined before T are green, and (ii*) All emeralds not first examined before T are blue – we note that (ii*) is as yet unsupported.
It would seem that, whereas a non-gruesome hypothesis is projectible provided it is unviolated and supported, a gruesome hypothesis is projectible provided it is unviolated and equivalent to a conjunction of non-gruesome hypotheses, each of which is supported. The grue paradox was discovered by Nelson Goodman. It is most fully stated in his Fact, Fiction and Forecast (1955). See also PROBLEM OF INDUCTION , QUALI- TATIVE PREDICAT. D.A.J.