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# #header_text h1#site_heading a, #header_text h1#site_heading{ } @media (min-width: 650px) { #header_text h1#site_heading a, #header_text h1#site_heading{ } } Paradoxes and Their Resolutions

## #header_text h2#site_subheading a, #header_text h2#site_subheading{ } @media (min-width: 650px) { #header_text h2#site_subheading a, #header_text h2#site_subheading{ } } A Thematic Compilation by Avi Sion

### 1.    Traditional analysis

Carl Gustav Hempel[1] in the 1940s exposed an alleged “paradox of confirmation”, which suggested that a fully consistent formal inductive logic is impossible. This is commonly called “the raven paradox”, and may be described as follows:

1. The observation that Some ravens are black (Some A are B) confirms the hypothesis that All ravens are black (All A are B).

The latter proposition may be contraposed to All non-black things are non-ravens (All nonB are nonA).

1. Next, consider the observation that Some apples are green (Some C are D). This is convertible to Some green things are apples (Some D are C).

It follows from this proposition that Some non-black things are non-ravens (Some nonB are nonA), since green things are not black and apples are not ravens.

Now, just as Some ravens are black (Some A are B) confirms the hypothesis that All ravens are black (All A are B), so Some non-black things are non-ravens (Some nonB are nonA) confirms the hypothesis that All non-black things are non-ravens (All nonB are nonA).

This induced proposition may in turn be contraposed to All ravens are black (All A are B), and here lies the difficulty, for it appears that the mere observation of some green apples is enough to confirm the hypothesis that All ravens are black! Note well that to achieve this result we did not even need to observe any black ravens.

1. It follows from the preceding that we can equally well, using the same logical process, given Some apples are green, confirm the hypothesis that All ravens are pink, or any other color (except green) for that matter.
2. This is in itself a mystery: how can apples tell us about ravens? Intuitively, this has to be viewed as a non sequitur.

Moreover, in the case of black ravens, the existence of black ravens has empirical backing, as already indicated; so the ‘inference’ from green apples to All ravens are black still seems somewhat reasonable. But in the case of pink ravens, we have never observed any such animals; so the ‘inference’ from green apples to All ravens are pink seems quite unjustifiable.

Moreover, knowing by observation that Some ravens are black, how can we ‘conclude’ that All ravens are pink? Even if we do not claim all ravens black, but only claim all ravens pink, we would in such circumstances be upholding contrary propositions, namely the particular one that some ravens are black and the general one that all ravens are pink.

Moreover, even if we have never observed the color of any ravens, we can according to the above inductive process simultaneously conclude many contrary statements such as All ravens are black, All ravens are pink, All ravens are orange, etc. This too is a result that flies in the face of the law of non-contradiction.

Furthermore, the same can be done with reference not only to green apples, but also to apples of other colors (except black or pink, etc. as the case may be), and indeed to things (non-ravens) other than apples. In that event, almost anything goes in knowledge, and we can at will affirm or deny just about anything about just about everything!

This then, according to traditional presentations[2], is Hempel’s paradox. It appears, from such analysis that the inductive processes of confirming hypotheses (such as generalizations directly from experience or indirectly from logical derivatives of experience) are fundamentally flawed. The analysis involved is quite formal, i.e. it can be performed in terms of symbols like A, B, C, D – and so it has universal force.

It follows that induction is bound to result in various absurdities: apparent non-sequiturs, many contradictions, and ultimately imply the arbitrariness of all human knowledge. Clearly, Hempel discovered here a serious challenge to inductive logic and logic in general.

### 2.    Novel analysis

As I will now show in detail, the above analysis is inaccurate in some important respects. I will show that although Hempel did indeed discover an interesting formal problem for logicians to consider and solve, this problem does not result in what we would call a paradox. That is, there are valuable lessons to be learned from Hempel’s paradox (as we may continue to call it conventionally), but it does not present logic with any insurmountable predicament.

1. The first operation described above is the commonly used inductive process of generalization. A particular proposition (Some A are B) is turned into a general one (All A are B). The particular supports the general in the way that positive evidence confirms a hypothesis. Their logical relation is adductive. ‘Some’ here means ‘at least some, possibly all and possibly only some’ – and by generalizing we are opting for the hypothesis ‘all’ in preference to the hypothesis ‘only some’.

However, it would be an error to consider that Some A are B is alone capable of inductively justifying All A are B. Such generalization is an inductively permissible inference provided we have looked for and so far not found any A that are not B. For if we had found (by direct observation or by some reasoning) that Some A are not B, we would certainly not have generalized. Moreover, if we later do come across an A that is not B, we would have to particularize All A are B back to Some A are B.

This condition sine qua non of generalization, viz. to remain on the lookout for contradictory instances and adjust one’s judgment accordingly, is not stressed or even mentioned in the earlier presentation, note well. Yet this is a known and accepted rule of scientific thought at least since the time of Francis Bacon, who emphasized the importance of the “negative instance” in induction. To ignore this condition is bound to lead to contradictions sooner rather than later.

Regarding the contraposition of All A are B to All nonB are nonA, it is of course a deductive act. Even so, we must keep in mind that the conclusion All nonB are nonA is only due to the prior inductive inference of All A are B from Some A are B. No observation is required for the deduction, but we remain bound by the need to keep checking the previous inductive act, i.e. to remain alert for eventual cases of A that are not B.

1. Now, let us grant that Some C are D, as above. Some C are D readily converts to Some D are C. However, Some D are C does not formally imply that Some nonB are nonA – some syllogistic inference is tacitly involved here, which ought to be brought out in the open. Clearly, we tacitly take for granted the premises that green is not black and apples are not ravens, whence: the following two successive syllogisms are constructed:

1st figure, EIO:

No green thing is a black thing (No D are B)

Some apples are green (Some C are D)

Therefore, Some apples are not black (Some C are not B).

3rd figure, AII:

All apples are non-ravens (All C are nonA)

Some apples are non-black (Some C are nonB)

Therefore, Some non-black things are non-ravens (Some nonB are nonA).

Whence, by generalization we obtain: All non-black things are non-ravens (All nonB are nonA); and then by contraposition: All ravens are black (All A are B). Note that the premises that led to this general conclusion do not include Some ravens are black; i.e. this conclusion is based on no empirical observation of black ravens.

Note too that we could have obtained the same result with the premises No ravens are green (No A are D) and No apples are black (No C are B). Note also that, though the syllogisms involved are deductive processes, all such tacit premises require prior observations and generalizations (i.e. inductions) to be adopted.

Moreover, it is significant to note that these syllogisms could not be constructed if the colors of the ravens and apples under consideration were the same (both green or both black), or if ravens and apples were not mutually exclusive classes. We also assume here that a raven cannot have more than one color (e.g. be partly black and partly green or whatever, or sometimes the one and sometimes the other); and similarly for an apple.

The next step was to generalize Some nonB are nonA to All nonB are nonA. But here again, generalization is allowed only provided we have no evidence or inference from any other source that Some nonB are not nonA. That is, in our example, we must remain conscious that it is possible that some non-black things are not non-ravens, i.e. are ravens, which means we might yet find some non-black (albino) ravens out there.

Here too, we must make sure, in accordance with Bacon’s crucial principle of adduction, that there is no conflicting observation that obstructs our expansive élan. This is all the more necessary, since here the premise of generalization Some nonB are nonA was obtained indirectly by deduction from previous products of induction, whereas previously our premise Some A are B was (supposedly) directly observed.

Note further that these two generalizations have the regulatory conditions that Some A are not B or Some nonB are not nonA, respectively, not be found true – and these conditions are one and the same since these two propositions are logically equivalent by contraposition. This means that in either case, whether we reason directly from black ravens or indirectly from green apples, there is the same implicit condition for generalization – that in our experience or reasoning to date no non-black ravens have appeared.

Thus, whichever of these two generalizations we opt for, the condition that there be no known instances of A which are not B is unaffected, and the dependence of the truth of All A are B on this condition is unchanged. Note too, the same condition holds before and after such generalizations. That is, even after such inductive process, if we discover new evidence to the contrary, we logically may and indeed must retract.

As previously stated in c) and d): using the same logical process, given Some apples are green, we can equally confirm the hypothesis that All ravens are pink, and many other wild hypotheses that conflict with each other[3]. Obviously, we are doing something wrong somewhere, and have to take action to either prevent such absurd eventual consequences or correct them when and if they occur. I will now explain the solution to the problem.

Generalization is never an irreversible process. So if any generalization leads to contradictions, we are free and indeed obligated to particularize. The question of course remains: in what precise direction and how far back should we go? Still, what this means is that there is no ‘paradox’ in inductive logic as there is in deductive logic; almost everything (with the exception of logic itself – especially the laws of thought on which it is built) is and ever remains ‘negotiable’.

In deduction, a contradiction is a far more serious event, because the process leading up to it is presumably necessary. But in induction, we know from the outset that the connection between premise(s) and conclusion is conditional – so contradictions are expected to arise and it is precisely the job of inductive logic to determine how to respond to them.

Dealing with contradictions is a branch of inductive logic, called harmonization or conflict resolution. This is not something rare and exceptional – but occurs all the time in the development of knowledge. Sometimes conflicts are resolved before they take shape, sometimes after. If we see them coming, we preempt them; otherwise, we perform the possible and necessary retractions.

Particularization of a general proposition is retraction. More broadly, retraction means rejection or modification of a theory in the light of new evidence. Thus, for example: till now, I have seen only black ravens, and assumed all are black; tomorrow, I may notice a white raven, and change my view about the possible colors of ravens.

Hempel is evidently or apparently unaware of this crucial aspect of inductive reasoning, else he would not have viewed contradictions arising in the course of induction as paradoxical. Nevertheless, the situation described by him is interesting in this context, for reasons he did not (I think) realize.

For after the first generalization, starting from Some ravens are black (Some A are B), if we belatedly discover that Some ravens are not black (Some A are not B), we simply return to our initial observation that Some ravens are black (Some A are B). Whereas after the second generalization, starting from Some non-black things are non-ravens (Some nonB are nonA), if we belatedly discover the same conflicting evidence, we cannot simply deny All ravens are black (All A are B).

Why? Because this would still leave us with part of our generalization, viz. the claim that Some ravens are black (Some A are B). That is to say, we would expect ‘All A are B’ plus ‘Some A are not B’ to yield the harmonizing conclusion ‘Some A are B and some A are not B’. The negative particular does not eliminate the positive particular underlying the positive generality; since we previously (due to said generalization) believed the generality, we now have a leftover to account for.

In the case of All ravens are black, such retraction is not noteworthy, since we know from experience Some ravens are black; but in the case of All ravens are pink, we have a serious problem, for there is no shred of evidence for a claim that Some ravens are pink! In other words, the proposed retraction cannot suffice in the situation presented by Hempel, i.e. when All A are B is induced from Some nonB are nonA.

### 3.    The problem and its solution

In my view, this is the crux of the problem revealed by Hempel’s exploration. The problem is not exactly a paradox, since the validity of generalization formally depends on such process not giving rise to any eventual contradiction.[4]

That from the observation of some green apples we may by generalization infer that All ravens are black and All ravens are pink and many other conflicting conclusions – this is amusing, but not frightening. For in such situation of self-contradiction, we can by retraction find ways to harmonize our knowledge again. The problem is temporary.

On the other hand, what Hempel has here uncovered is that we cannot always retract simply by particularization of the conflicting theses. Particularization seems acceptable in some cases (e.g. with black ravens), but in other cases it yields unacceptable results (e.g. with pink ravens), because the logical remainder of such retraction is devoid of empirical basis.

Suppose, using Hempel’s method, starting from green apples, we induce both the generalities All ravens are pink and All ravens are orange. These two conclusions are in conflict. Let us say we decide to resolve the conflict by denying them both; that still leaves us with two propositions Some ravens are pink and Some ravens are orange.

These two particular propositions are not in conflict – and, let us take for granted, neither of them has any empirical basis, yet they both got somehow cozily ‘established’ in our knowledge! They were introduced by the generalizations from green apples, yet they were not dislodged when we abandoned the corresponding generalities. We are stuck with them, even though the complex processes that led to them have been revoked.

It is unthinkable that such particulars (whether true or untrue) should emerge from the unrelated observation of green apples (or whatever else). This I believe is the significant problem uncovered by Hempel. The problem is not the conflict of generalities or between general and particular propositions, so it is not about paradox. The problem has to do with ‘collateral damage’ to knowledge, through incomplete correction of errors.

I suggest the following solution for it: when we generalize from Some A are B to All A are B, and then discover that Some A are not B, we particularize All A are B back to Some A are B. That is normal procedure, which we all commonly practice.[5]

On the other hand, when we obtain All A are B by generalization from Some nonB are nonA to All nonB are nonA (followed by contraposition of the latter), then when we discover that Some A are not B, we cannot merely particularize All A are B back to Some A are B, but must also retract the intermediate premise of the proposition All A are B, viz. All nonB are nonA, and return to Some nonB are nonA.

In view of the latter retraction, we in fact no longer have a basis for claiming Some A are B (this cannot be deduced from Some nonB are nonA). It would be an error of induction to forget the actual source of our belief in All A are B. The distinction between the inductive grounds Some A are B and Some nonB are nonA must be kept in mind, so that in the event of discovery of contradictory evidence, viz. that Some A are not B, we particularize back to our exact same previous position in each case.

We may thence formulate the following new law of inductive logic, which may be called the law of commensurate retraction: a product of generalization like All A are B cannot be treated without regard to its particular source; when if ever it is denied by new evidence, we must retreat to the same initial particular and not to some other particular that was implied by the generality when it seemed true but is now no longer implied by it since it is no longer true.

In other words, when and if we come upon a contradiction of the sort considered here, we must realize that this does not merely discredit the generality that was previously induced, but more deeply discredits the inductive act that gave rise to it. Thus, we should not retract by mere particularization, but carefully verify whether the remaining particular has any independent basis and if it has not we should return far back enough to the status quo ante to make sure no unconfirmed particular remains.

This seems like a perfectly reasonable instruction – to reverse and clean up all traces of an inductive act that was found illicit, i.e. that led us into a logical impasse.

All this means that, using ordinary procedures of logic, we would never fall into a self-contradictory situation (e.g. claiming paradoxically All ravens are black and All ravens are pink). The fact that generalizations may yield incompatible results is commonplace; we daily deal with such conflicts without difficulty. When such conflicts arise, we are logically required to harmonize. If we cannot find a specific way to resolve the conflict, the conflict is resolved in a generic manner, viz. all the generalizations involved are put in doubt.

In a situation where two or more propositions are put in doubt by mutual conflict, we would naturally give more credence to one that has some direct empirical basis (like All ravens are black) than to one that merely emerged from indirect projection (like All ravens are pink). We need not treat all conflicting propositions with equal doubt, but may be selective with regard to their inductive genesis.

With regard to the evidence for conflicting thesis – obviously, if we have no data on black or pink ravens, we would not know which way to retract, and both generalizations would be problematic. But if we have observed some black ravens and never observed any pink ones, we would naturally opt for the generalization that All ravens are black (All A are B). On the other hand, if we have observed both black ravens and pink ravens, we would make neither generalization and simply conjoin the two particulars.

With regard to the inductive processes used – direct generalization would naturally be favored over the indirect sort envisaged by Hempel. If the conflict at hand can be resolved by ordinary means, e.g. with reference to empirical considerations, we need not bother to backtrack with reference to process. But in cases where we have no other means of decision, process would naturally be the focus of revision.

A possible objection to the law of commensurate retraction would be that in practice we rarely manage to keep track of the exact sources of our generalizations. Such ignorance could conceivably occur and cause some havoc of the type Hempel described in our knowledge.

However, we may also point out that in practice we just about never find ourselves in the situation described by Hempel. How often does anyone generalize from a proposition like Some nonB are nonA? The statistical answer is ‘probably never’ – Hempel’s paradox is just a remote formal possibility that logicians have to consider, but its practical impact is just about nil.

Moreover, we are not likely to arrive at a proposition of the form Some nonB are nonA, except by the sort of reasoning above depicted, i.e. through some other terms like C and D. We cannot directly observe that Some nonB are nonA. Observation relates primarily to positive phenomena; it can be about negative phenomena but only indirectly. This suggests that if we did encounter a situation of Hempel paradox, we would likely be aware of how it arose.

Another remark worth making is that the above solution of the problem raised in Hempel’s paradox can be characterized as heuristic; it is repair work by trial and error. But I have already proposed in my work Future Logic[6] a detailed, systematic, formal treatment of induction, by means of factorization and formula revision. I believe that is free of the Hempel’s problem, since every formal possibility is included in the factorial formulas developed.

With regard to solutions to Hempel’s paradox offered by other logicians, e.g. those by Goodman and by Quine described in the earlier mentioned Wikipedia article:

“Nelson Goodman suggested adding restrictions to our reasoning, such as never considering an instance as support for ‘All P are Q’ if it would also support ‘No P are Q’ … Goodman, and later another philosopher, [W.V.] Quine, used the term projectible predicate to describe those expressions, such as raven and black, which do allow inductive generalization; non-projectible predicates are by contrast those such as non-black and non-raven which apparently do not. Quine suggests that it is an empirical question which, if any, predicates are projectible; and notes that in an infinite domain of objects the complement of a projectible predicate ought always be non-projectible. This would have the consequence that, although “All ravens are black” and “All non-black things are non-ravens” must be equally supported, they both derive all their support from black ravens and not from non-black non-ravens.”

I find these proposals reasonable and not incompatible with my own. However, I think mine is a little more precise in pinpointing the problem at hand and its solution.

Goodman’s suggestion to restrict induction from a proposition if such process yields conflicting conclusions is logically sound. Only his instruction cannot be obeyed preemptively, but only after we discover that the process yields conflicting conclusions. So it is not a preventative, as he seems to consider it, but an after the fact correction. It can therefore be regarded as about the same as the law of commensurate retraction I above propose. The only difference is that he does not seem to have made a distinction between the conflict of generalities and the underlying leftover particulars.

As for “non-projectible predicates”, I would agree that negative terms (complements) present a general problem in induction. Although deductive logic makes no distinction between positive and negative terms, phenomenology does distinguish between the presence of positive phenomena and their absence. Whereas we can observe a positive phenomenon (like a black raven) without regard to its negation, we cannot mention a negative term (like non-black or non-raven) before thinking of and looking for the corresponding positive phenomenon and failing to find it.[7]

Thus, a truly negative term can never be truly empirical. Its content is never ‘I have seen something’, but always ‘I have diligently looked for something and not found it’. A negative is ‘empirical’ in a lesser, more derivative sense than a positive. It already involves a generalization of sorts, from ‘could not be found’ to ‘was not there to be found’.

It follows from this insight that generalization from negative terms, such as Some nonB are nonA, can only proceed with unusual caution and skepticism. Hempel’s scenario further justifies such tentativeness. We are might even be tempted as a radical solution to simply always interdict generalization for a truly negative subject. If any manner of discourse has certain likely illogical consequence, logicians are wise to formulate a preemptive law of logic of this sort.

Another temptation is to deny any meaningful content to propositions of the form Some nonB are nonA. Such a proposition is formally implied by All A are B, and compatible with Some A are B, No A are B and Some A are not B – but does it really tell us anything? Indeed, since nothing can be inferred about A or B (as subjects) from Some nonB are nonA, what information does such a proposition contain? Could one not conceivably assert such a proposition using any almost two terms taken at random? This sort of doubt could be used to further justify interdiction of generalization from such propositions.[8]

However, since a less radical solution, namely the above-proposed law of commensurate retraction is possible, we perhaps need not go so far. Rather than preemptively forbid certain doubtful processes under all conditions, I prefer to allow them in case they occasionally work, and prepare the appropriate corrective mechanism for when they fail to work.

To sum up, I believe we have convincingly shown here that Hempel's so-called paradox does not present the science of logic with any insuperable difficulty; it is made out to be a bit more daunting than it really is. Even so, it is an interesting contribution for logicians to ponder over.

From Logical and Spiritual Reflections 1.8.

[1]              Germany-USA, 1905-97.

[2]              See for instance: the article in Wikipedia at http://en.wikipedia.org/wiki/Raven_paradox.

[3]              To show propositions with different predicates are in conflict, we use syllogism. For instance, All ravens are black and All ravens are pink are incompatible, because knowing that No black things are pink, we obtain, by syllogism (1st figure, EAE): No ravens are pink, which is contrary to All ravens are pink.

[4]              A paradox is a thesis that formally contradicts itself or deductively leads to contradictory propositions. From a single such paradox, we may conclude that the thesis in question is false; logic as such is not put in question, because the contradiction involved is merely conditional. A double paradox, on the other hand, is a serious threat to logic; here, both a thesis and its contradictory are paradoxical, so the contradiction is unconditional. In that case, logic cannot declare either of them true or false – but must among them find either a non-sequitur (as in the Barber paradox) or a meaningless term (as in the Liar paradox). That is, logic must challenge either one or more of the implications involved, and/or one or more of the terms or theses involved. The Hempel scenario does not give rise to an unconditional/double paradox.

[5]              Symbolically, A + O = IO.

[6]              First published in 1990, a few years before Hempel’s death. See part VI.

[7]              See my essay on this topic in Ruminations (part I, chapter 9).

[8]              These questions are made clearer if we consider the eventual negation of Some nonB are nonA, i.e. the form No nonB is nonA, which implies All nonB are A. In the event the latter proposition is true, we would have a negative term (nonB) included in a positive (A). This could be taken to mean that almost all the world (except things that are B) falls under A. For this to happen, A would have to be a very large concept. Such a concept would be very exceptional and almost meaningless. Whence, we can say that Some nonB are nonA is almost always true, and at the same time not very informative.

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