Here below I resume my definitive reflections on what firmly seems to me the complete and exhaustive solution of Goodman’s riddle.
(i) something is grue iff (it is examined before t° and is green) or (it
is examined after t° and is blue)
(superfluous brackets are intended to facilitate the argument below).
BASIC POINTS: Inclusive disjunction and range of quantification.
THEOREM. Claiming that *being green if examined before t°* ought to entail *being blue if examined after t°* is inconsistent.
CONSISTENCY CONDITION. No evidence can support at the same time incompatibile inductive generalizations.
PROOF . By reductio ad absurdum. Exactly as (i) defines *grue*, something is grink iff (it is examined before t° and green) or (it is examined after t°and pink) defines *grink*, and so on for every chromatic nuance we like (*grellow*, *grack* , *grurple*…). Therefore, if we claim that the observation before t° of some green emerald is an inductive evidence for the generalization that all emeralds are grue, we must also claim that the same observation is an inductive evidence for the generalization that all emeralds are grink and grellow and so on.
Consequently if we claim that being grue entails being blue after t°, we must also claim that being grink entails being pink after t°, that being grellow entails being yellow after t° and so on. But as *being blue after t°*, *being pink after t°, *being yellow after t° * and so on are all incompatible predicates, we can conclude that our initial claim leads to inconsistency.
The consistency is restored by a less superficial analysis of the matter. Here it is.
ANALYSIS. The definition of *grue* depends on four predicates, The adopted symbology is:
° examined before t°
’ examined after t°
G° green and examined before t° (I call it “gruebef”)
B’ blue and examined after t° (I call it “grueaft”)
The couple of brackets occurring in (i) tells us that by definition the two predicates inclusively disjunct are not “green” and “blue” but “green and examined before t°” and “blue and examined after t°”. The various evidences we get by examining emeralds before t° are
(ii) G°(a) & G°(b) & G°(c) & …
and then, by induction, the respective generalization is
(iii) (x) G°(x)
(all emeralds are gruebef).
Now, since a temporal opposition occurs in (i), a basic question imposes itself: what is the range of the “all” (iii) speaks of? Is it the range of all emeralds examined before t° or the range of all emeralds independently on the moment of their examination? Of course, as every evidence testified in (ii) refers necessarily to emeralds examined before t°, the inductive generalization cannot but refer to emeralds examined before t°; otherwise the definitory import of the temporal specification would be neglected. Our same intuition corroborates such a
position: the evidences we acquire by examining before t° the various emeralds cannot tell us anything about the color of emeralds examined after t° (“B” not even occurs in (ii)). Actually, owing to the inclusiveness of the disjunction occurring in (i), *being gruebef * is sufficient to derive *being grue*, that is
(iv) If (x) G°(x) then (x)R(x)
(if all emeralds are gruebef, all emeralds are grue, and since all emeralds are gruebef, all emeralds are grue). But once more a question imposes itself: what is the range of “all”? An easy answwer, indeed: as the protasis of (iv) refers to all emeralds examined before t°, the apodosis too must refer to all emeralds examined before t°. The evidences we got do not allow any extrapolation to emeralds examined after t°. Thus the various definitions above (*grink*, *grellow*
etcetera) result perfectly compatible, and the riddle vanishes for, so to write, the ascertained gruebefness of emeralds does not allow us to expect any grueaftness (that is: does not allow us to state that emeralds examined after t° ought to be blue).
P.S. An apocryphal version of the riddle replaces the inclusive disjuncion with a conjunction. In this case, as actually being grue requires being blue after t°, the range of quantification in All emeralds are grue should concerns emeralds examined both before or after t°. Yet being green before t° is no more a sufficient evidence for an emerald in order to conclude its grueness, so that no riddle arises.