You can find further information on my thoughts on my website 

www.logicalparadoxes.com/index.html

 

Solution of Hempel’s paradox.

Concisely. An evidence e confirms a hypothesis h iff P(h/e)>P(h). Let us consider the hypothesis h=“all ravens are black” in  the most current context; the value P of its probability calculated on the observation of some ravens, all black, is by far higher than the value P’ calculated on the observation of some non-black individuals, all non-ravens, though h and h’=”all non-black individuals are non-ravens” are logically equivalent. Now (Nicod’s postulate) the new evidence of a non-black non-raven individual (a red apple, say) actually confirms h’, increasing P’ up to P”; but even thus increased, P”<<P, therefore, by definition, h is not at all confirmed by the red apple, because we can already count on its by far higher value P.

Here  is the winning trace for the definitive solution of Hempel’s paradox (in the full respect of classical logic and of Nicod’s postulate). I have similar traces for Goodman’s riddle, McGee’s counter-examples of Modus Ponens, Bértrand’s  polyvalent geometrical probabilities, up to the general solution of logical paradoxes; in fact all these troubles arise from a wrong approach to the various problems.

If some reader is interested in the matter, write to me at italo@italogandolfi.com.

 

GENERAL SOLUTION OF LOGICAL PARADOXES

 

The considerations below sketch an extremely concise exposition of the solution.

Reflexivity is a kind of indexicality. Indexical expressions depend on the context because they are affected by some lack of information: and the context is just the integrative source of information which converts the indexical (open) expression into  an absolute (closed) one. On this ground a basic distinction can be focused between linguistic conversions (Socrates was a great man; he ...) and ostensive conversions (“yesterday”, “down there”). Ostensive conversions can be neglected for the present purpose.

The aforesaid lack of information is symbolizable by a free variable whose conversion is carried out by its substitution with a constant. Of course that conversion cannot be effective if the free variable to substitute continues to appear in the substitutor. Here I do not dwell on the improvement that the introduction of a reflexive variable can bring  to a formalization of the entire theme.

The current criterion for the classification of open sentences (Skolem) is the number of different  free variables occurring in the sentence under examination.  We are interested in monovariable sentences. Skolem criterion does not account for the basic difference between sentences where the free variable occurs either in the subject or in the predicate, and sentences where the same free variable occurs both in the subject and the predicate (reflexive sentences). Logical paradoxes arise with self-conversions. Whatever sentence resulting from a self-comversion cannot be closed  (the conversion cannot be effective), as the free variable to substitute continues occurring in the substitutor. Paradoxical dilemmas are defective and no defective dilemma can admit a well-founded answer; the usual argument (if it is were so, then it ought to be the contrary) is nothing but the intrinsically vain attempt to fill through an arbitrary hypothesis the lack of information created  by our same definitions.

This general solution of logical paradoxes entails very wide consequences.

 

 

SOLUTION OF GOODMAN’S RIDDLE

 

 

ABSTRACT

 Goodman’s riddle arises from a flagrant misapplication of the truth table for a basic propositional connective. I mean that the truth of  a binary inclusive disjuntion does not at all require the truth of both disjuncts; so, provided that an examined emerald is green, it is grue regardless of its future color.

The point seems to me manifest; yet  the vast bibliography about the riddle instils the doubt that the same point so manifest is not. Anyhow I assume that the next few pages will make my opinion unquestionable.

 

CURRENT APPROACH

A premise. The occurrence of asterisks and inverted commas as diacritical symbols is not  a typographical carelessness, but the consequence of  theoretical reasons (here left out).

 

 

DEFINITION:

(i)        something is grue iff (it is examined before t° and is green) or (it is examined after t° and is blue)

(superfluous brackets facilitate the analysis below).

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 incompatible results.

 

SOLUTION

Goodman’s example is potentially misleading because, so to say, its institutional (then unobjectionable) oddity contributes to mask the logical mistake affecting the argument on which the riddle stands; then I prefer to propose another example .

 

Our new universe is constituted by the twenty bear cubs (briefly: twenty cubs) of a breeding. They are classified on the basis of two oppositions: in fact the attitude of every cub depends on the moment it is examined (either day or night) and can be either sulky or aimable.  For the sake of concision I will speak also directly of day examinations, of aimable cubs  and so on. Furthermore I integrate the exposition by a representation R where our universe  is visualized in a circle whose horizontal diameter opposes day examinations (upper semicircle) to night examinations (lower semicircle) and whose vertical diameter opposes sulky examinations (left semicircle)  to aimable ones (right semicircle). Thus the circle is partitioned in four quadrants: Q1 for the day-aimable, Q2 for the night-aimable, Q3 for the night-sulky and Q4 for the day-sulky examinations. Each examination is represented by a dot (which obviously falls within the respective quadrant).

Since the first sequence of n  examinations has been performed during the day and did not give any sulky outcome it tells us that

(ii)                                           all cubs are day-aimable

and is represented by the corresponding sequence of n dots falling within Q1. Indeed the greater is n  the more reliable is the inductive generalization, until, for n=20, (ii) becomes a deductive statement etcetera; anyhow here I do not dwell upon this theme because, once the reliability of (ii), inductively or deductively supported, is achieved, the reasoning follows identically. In fact our very problem is: can (ii) be the basis for predicting the night-attitude of a cub? Intuition suggests us a negative response, anyhow, in order to strengthen it let me agree that  by definition

(iii)                              a cub is domestic        iff        he is (day-aimable or night-aimable)

and

(iv)                               a cub is tempered      iff        he is (day-aimable or night-sulky)

 (where “or” continues meaning an inclusive disjunction). Now, can  the contradictory conclusions of an argument like

as all cubs are day-aimable, by definition they all are domestic, therefore night-aimable, but by definition they all are tempered too, therefore night-sulky and not night-aimable

be seriously accepted? Claiming that a given evidence enables us to predict something means simply that the information provided by that evidence contains the information of which the prediction is the bearer. And, until we have no information about the night attitudes of our cubs, we cannot reasonably infer any specific information about such attitudes. In other words: since  the information that all cubs are day-aimable is compatible with both their possible night-attitudes, it does not allow us to infer specifically one of them. All what we can reasonably infer is just that, at night, a day-aimable cub will be either aimable or sulky. In R, while the dots representing day examinations fell within the upper semicircle (more precisely, within Q1), the dots representing  night examinations will fall within the lower semicircle. A cub is domestic iff AT LEAST ONE of his two dots (less concisely: iff AT LEAST ONE of the two dots representing the respective day-examination and night-examination) falls within Q1 or (inclusive) within Q2: analogously a cub is tempered iff AT LEAST ONE of his two dots falls within Q1 or within Q3. Thus, since the first dot both of domestic and tempered cubs falls within Q1, both conditions are already satisfied, within any quadrant of the lower semicircle the second dot will fall.  That is: owing to the inclusive import of the disjunctions occurring in (iii) and in (iv), the day-aimableness implies the domesticity, but the the domesticity does not at all imply the night-aimableness (analogously: the day-aimableness implies the temperedness but the temperedness does not at all imply the night-sulkiness). In order to impose these latter implications we should betray Goodman’s definition, replacing the inclusive disjunction with a conjunction, as, for instance,

(v)                               a cub is capricious iff he is (day-aimable AND night-sulky)

yet  it would a sterile betrayal because, under (v), the capriciousness is sufficient to imply the night-sulkiness, but the day-aimableness is no more sufficient to imply the capriciousness: the conclusion is that, in any case, the information obtained by the performed day examinations is insufficient to infer any night-attitude of a cub. So, going back to the original example, we see that if an emerald examined before t° is green, by definition it is already grue with full rights, therefore its being grue does not at all exact its being blue after t°.

At this point the riddle is overcome, because the argument according to which we should predict a blue emerald has been confuted. Anyway, for the sake of completeness, I wish also to confute the argument according to which we should predict a green emerald. Thus the claim is achieved that Goodman’s procedure, far from entailing  two incompatible expectations, does not entail either.

Concisely. As soon as a definition gives a discriminating value to a predicative opposition concerning the universe we deal with, no generalization can disregard this opposition. Now (i), besides the discriminating chromatic factor, gives a discriminating value to a chronological factor too (actually the brackets in (i) show that the two disjunct predicates are not “green”/“blue” but “green-and-examined-before-t°”/ “blue-and-examined-after-t°”). Therefore, since the green emeralds a, b, c etc. have been examined before t°

(vi)                                          all emeralds EXAMINED BEFORE t° are green.

and not

(vii)                                                    all emeralds are green

is the right inductive generalization. The difference between (vi) and (vii) is indeed momentous since (vi) does not provide any information about the color AFTER t° of an emeralds exactly as (ii), where the chronological factor si meant by “day”, does not provide any information about the night-attitude of a cub. In other words: the chronological specification occurring in (vi)  (respectively: occurring in (ii)) reduces the domain of “all”, leaving out of  quantification every emerald examined after-t° (respectively: every cub examined at night).

Frankly there is little need to point out that in our daily practice we all are convinced that a green-before-t° emerald is also a green-after-t° emerald; in fact  this conviction does not involve the analysis above because, rather than being derived from the mentioned theoretical assumptions, it is derived from collateral and contingent peculiarities as for instance the high implausibility of the hypothesis that an emerald changes its color at t°. Here is the reason why I think that Goodman’s example is misleading. Would the hypothesis that a day-aimable cub changes its attitude at night be equally implausible? It is just the plausibility of both the incompatible alternatives about his night attitude, that is the absence of some collateral and contingent notion privileging one of them, which shows their theoretical unpredictability.

To summarize: a definition is not an evidence , then, regardless of any definition we propose, on the only ground of the evidence that all the emeralds examined before t° are green neither we can predict that an emerald examined after t° ought to be blue nor that it ought to be green.