sabato 12 ottobre 2019


A premise. The occurrence of asterisks and inverted commas as diacritical symbols  is not  a typographical negligence,  but the consequence of well intentional reasons (here left out). Furthermore the inclusive disjunction, from Latin, is meant by “vel” and not by a potentially equivocal “or”.
The genesis of Goodman’s riddle can be outlined in three steps. First step. Blending two incompatibile current predicates (green/blue) in a just defined one (grue). Second step. Avoiding an open conflict between the two current predicates by a diachronical distinction (before/after a future moment t°). Third step. Violating the institutive assumptions by a mismanaged application of inductive generalization. The solution consists simply in restoring the coherence.
Now let me enter into details.
As for the definition of  *grue* I draw my track from Stanford Enciclopedia of Philosophy, voice “Nelson Goodman, §5.3. It runs as
(i)      something is grue iff (it is examined before t° and is green)
vel (it is examined after t° and is blue) (superfluous brackets for the sake of clearness).
THE RIDDLE. Owing to the above definition, before t° for each evidence statement asserting that a given emerald is green, there is a parallel statement asserting that the very emerald is grue. Thus the two inductive hypotheses *all emeralds are green* and *all emeralds are
grue* result equally confirmed. But as soon as we survey an emerald after t°, we knock against a contradiction, for the two hypotheses lead to opposite results.
And as far as I know, the worst aspect of the situation has not even been focalized. I mean that if we in (i) substitute “yellow” to “blue”
thus introducing by definition *grellow*, the same evidences above leading to *all eneralds are grue*, lead to *all emeralds are grellow* and so on for whatever color. The inductive inference from the observation of some green emeralds to the conclusion that all emeralds are grue (or grellow …)  precipitates us into a  chaos, more than into a riddle. It is absolutely wrong. Fortunately a sound appeal to logic overcomes the impasse.
CORRECT  REASONING. An observation represents an inductive evidence if and only if it sets up a particular instance of the generalization under scrutiny. The predicate  of the first inductive generalization, as (i) establishes explicitly, is  not “green” (simbolically “G”), but “examined before t° and green” (simbolically “G°”). This remark is not a
pedantry: the temporal specification is essential, since by definition the meaning of  “grue” depends strictly on it. Therefore, symbolically,
             G°(a) & G°(b) & G°(c) & …. → G°(x)
  is all what the inductive generalizations allows us to state; all emeralds examined before t° are green, therefore all emeralds examined before t° are grue. The extrapolation of this inductively legitimate conclusion  until
(ii)                    all emeralds are grue
smuggles a piece of information (all emeralds examined after t° are grue, then blue) quite arbitrary for supported by no evidence. In other words, (ii) states an absolutely ungrounded enlargement of the range “all” refers to.
Thus the riddle vanishes. The lesson is not the incompatibility between induction and (I quote SEP) “weird predicates” but the respect of the weird connotations imposed by definition to those predicates.
If a super-skeptic reader still had some doubt about the soundness of the argument claiming that the temporal conditions cannot be omitted, he ought to realize what follows. Let us remove the temporal conditions from (i); then the definition becomes
  (iii)                      something is grue iff (it is green) vel (it is blue)
and the inductive generalization (ii) follows unobjectionably from the greenness of the performed observations. Yet we are thus facing an obviousness, not a riddle. In fact from (ii) and (iii) we can infer that all emeralds are green vel blue, but such a conclusion, on the ground of the truth table for inclusive disjunction, agrees perfectly with the greenness of the performed observations: a green emerald, obviously, is also a green-vel-blue emerald.
The solution applies a fortiori to an apocryphal version of Goodman’s riddle. My strong purpose of concision induces me to neglect some other perhaps interesting considerations. Anyhow I am  ready to debate the theme.

11 commenti:

  1. I summarize below a couple of criticisms submitted by email to I prefer to reply from this blog for my words might interest other readers too.
    FIRST CRITICISM. The solution neglects the problem of interdefinability.
    REPLY. From the viewpoint of a ‘green speaker’, “grue” (symbolically “R”) and “bleen” (symbolically “L”) are predicates defined on the ground of the primitives “green” (symbolically “G”) and “blue” (symbolically “B”). Reciprocally, from the viewpoint of a ‘grue speaker’, “green” and “blue” are predicates defined on the ground of the primitives “grue” and “bleen”. The following definitions (“or” inclusive, single inverted comma for “after t°”) are valid.
    Green speaker.
    R iff G° or B’ an emerald is grue iff (it is examined before t° and green) or (it is not so examined and blue
    L iff B° or G’ an emerald is bleen iff (it is examined before t° and blue) or (it is not so examined and green
    Grue speaker
    G iff R° or L’ an emerald is green iff (it si examined before t° and grue) or (it is not so examined and bleen)
    B iff L° or R’ an emerald is blue iff (it si examined before t° and bleen) or (it is not so examined and grue)

    My solution applies identically to each of the four items; it only exacts the respect of coherence. The predicate of an inductive generalization must be the same predicate of the various evidences the generalization is grounded on. An inductive generalization like
    P°a & P°b & P°c & …. → (P°&P’)x
    is a quite unreasonable rule. So, since all the four defnitions above depend on diachronical specifications, all the four inductive generalizations must make reference to such specifications. And since the four defined predicates cover the whole time of reference (in order to be green without temporal restrictions, an emerald must be green both before and after t°) G iff G°&G’ and so on. The simple respect of these nearly obvious exigences assures us that no riddle can arise.
    SECOND CRITICISMS. What about the grueness of a blue emerald observed at t°?
    REPLY. It is obviously non-grue. As the definition above exacts, so that an emerald observed at t° be grue, it must be green, and being blue is being non-green. Analogously, if at t° an unfavorable light forbids us to ascertain the color of the emerald we are examining, the lack of chromatic information forbids us to decide about its grueness.

  2. As far as I understand, the (“well intentional”) reasons (“left out”) on whose ground a distinction is introduced between asterisks and double inverted commas are a sterile perfectism. As a matter of fact in the text above both symbols are used as graphic means for making an expression the object of our discourse, instead of using it in order to speak of other referents.

    1. Diacritical symbology is a rather neglected yet very wide and important topic, since it involves the same logical bases of developed languages.
      I dare to say that it represents the first step toward a very refoundation of logic. More than eighty years ago Morris, genially, denounced the poorness of the means by which natural languages can speak of themselves, but till now no improvement has been realized. Actually the means at our disposal are insufficent. They are still inspired by the scholastic distinction between suppositio formalis (which just mentions expressions, thus making them the object of our discourse) and suppositio materialis (which just uses expressions for speaking of other referents). The crucial point is that there is not one only kind of mention, but two strictly different kinds of mentions which, therefore, must be distinguished. In fact sometimes we mention the expression as linguistic (grammatical) entity, and sometimes we mention the meaning (the piece of information) it adduces. Two easy examples. First. The poet stating that “poltroon” rhymes with “platoon” refers to the words; to claim that he refers to the meanings is unsustainable, since by substitution of identity we could infer that “coward” too rhymes with “platoon”, for the meaning is the same. Second. On the contrary the editor who exhorts the young reporter to moderate the too inflammed tones of his article telling him that “poltroon” is offensive, refers to the meaning; to claim that he refers to the word is unsustaineble since we could infer the derisive proposal of substituting “ poltroon” with “coward” (derisive just because no change in the meaning would correspond to the change of the word), Coming back to the specific case, definitions concern meanings, not words. Defining “grue” could only be something like “g” concatenated with “r” et cetera. On the contrary defining *grue* is explaining what piece of information is adduced by the word in question, that is by “grue”.

  3. The very refoundation of logic” is not too heavy a burden?

    1. Undoubtedly the very refoundation of logic is a heavy task. If it is furthermore a too heavy task, in my opinion, depends on the results. A strict informational approach does miracles

  4. My interlocutor who prefers to contact me in Italian through email is
    disconcerted: Goodman’s riddle, he writes (in Italian), allows to forecast whatever we want. I try to improve as follows the argument disconcerting him. Let us call “astolfian” those objects which have been observed before t° and are not hippogryphs, or are not observed before t° and are hippogryphs. The definition is unobjectionable, for it is coherent and establishes unequivocally the requisites an object must possess in order to be astolfian. Since till now no hippogryph has been observed, all objects till now observed are astolfian, therefore the inductive generalization “all objects are astolfian” results chiefly supported; and since all objects are astolfian, all objects observed after t° have to be hippogryphs.
    Of course the absurdity of the conclusion tells us that a fallacy hides in the same argument. And the fallacy is exactly the one already denounced above, that is the arbitrary (better: the incoherent) enlargement of the range “all” refers to. The correct application of the definition states explicitly that the predicate of the inductive generalization promoted by our observations, far from being “not to be a hippogryph” is “to be observed before t° and not to be a hippogryph”. In a future I hope not very remote, dealing with the general solution of logical paradoxes, we’ll ascertain that they too depend on a disregarded lack of information. In general, the plausibility of an inductive generalization bases itself on the coherent plenitude of relative particular evidences. Therefore the predicate occurring in the generalization must be the predicate of the same particular evidences the generalization is supported by. The rule
    P°a & P°b & P°c & …. → (P°&P’)x
    has been above refused just becasue it states a totally unjustified increment of information. So, coming back to hippogryphs, what our evidences support is not the generalization that all objects are astolfian, but that all objects examined before t° are astolfian, i.e.
    that all objects, so to write, are astolfian° (I emphasize the apex).
    But by what evidences the extrapolation from *astolfian°* to the stronger *astolfian* ought to be supported? Have we observed some hippogryph after t°? In other words. “astolfian” joins in its meaning two opposite connotations (being or not being an hippogryph) discriminated by a temporal criterion, therefore when we generalize inductively on astolfianity we must introduce a temporal specification telling us which sort of astolfianity we are referring to. Let me
    insist: as the opposite definitory clauses of “astolfian” (being or not being a hippogryph) depend on the moment of the observation, we cannot leave the temporal specification out of consideration. And as all our evidences concern objects observed before t° the generalization must stop at t°. No evidence supports the extrapolation beyond that temporal border, therefore the predicate subject to the inductive procedure is not “astolfian” but “astolfian°”, which cannot tell us anything about the objects after t°:
    As soon as the remarks above are acknowledged, the whole matter agrees with the requirements of our common sense.

  5. May I know at least, one "miraculous" result this strict informational approach leads to?

  6. In my opinion, Italo Gandolfi didn’t understand the real puzzle, which
    (adopting his symbology) can be resumed as follows.

    R iff G° or B’ By definition an emerald is grue iff it is green before t° or blue after t° then, since the disjunction is inclusive one only of the two disjuncts is enough to entail the grueness

    G°→R Green before t° → Grue

    and so

    R→B’ Grue → Blue after t°

    for the grueness entails the blueness after t°. Thus the riddle holds.

  7. I thank G.V. Chiapponi. My regret is not to misunderstand the riddle, but not to succeed in explaining a solution which appears clear and unobjectionable to me. I try again. Exactly as (cfr. above)
    R iff G°˅B’ an emerald is grue iff (it is examined before t° and green) or (it is not so examined and blue)
    defines *grue*,
    L iff G°˅ Y’ an emerald is grellow iff (it is examined before t° and green) or (it is not so examined and yellow)
    defines *grellow*,
    K iff G°˅ P’ an emerald is grink iff (it is examined before t° and green) or (it is not so examined and pink)
    defines *grink*, and so on for millions of chromatic nuances. Therefore, if we accept that the observations before t° of some green emerald represents an inductive evidence for the generalization that all emeralds are grue, we must accept also that such observations represents an inductive evidence for the generalization that all emeralds are grellow, and grink and so on. Since these conclusions are manifestly incompatibile, the only way to avoid the incoherence is to refuse that the observation of green emeralds before t° does represent an inductive evidence for the generalization according to which all emeralds are grue. The irrefragable solution, I repeat, is to acknowledge that in *grue* (or in *grellow*, *grink* and so on) we join two incompatibile connotations, and that all the observations of green emeralds before t° concern the first connotation, so that, as soon as we distinguish, so to write, a grueness° (or a grellowness° et cetera) from a grueness’ (…), the coherent frame of the whole topic agrees perfectly with our common sense. This conclusion re-proposes what has been formally remarked in the original note above: the inductive generalization does not comcern the predicate “green” (“G”) but “green before t°” ("G°”). Thus the fact that the observation of green emeralds before t° represents an inductive evidence for grueness°, grellowness°, grinkness° and so on, far from representing a puzzling incoherence, becomes a nearly obvious consequence since it concerns only the first connotation, that is the connotation all the defined properties have in common. Roughly: the observation of green emeralds before t° cannot be an inductive evidence as for their color after t°. In other words: the range of the *all* by which we generalize must be limited to the temporal interval the generalized predicate refers to.

  8. I thank Alberta, too.
    The informational approach open wide horizons and concern many (all?) logic topics. For instance it allows an actually exhaustive analysis of *variable* (what is a variable?), a theorization of reflexive variables, the general solution of logical paradoxes and so on. Such an approach is a way to face the problems. With reference to Goodmanìs riddle, the conclusion says: in the specific context, the information supplied by the observation of green emeralds before t° is not sufficient to support an inductive generalization involving occurrences after t°.

  9. Questo commento è stato eliminato dall'autore.