sábado, 12 de maio de 2012


This unpublished draft is a failure. Nevertheless, I think that this is the only plausible way to deal with the problem. I think that there is something more adequate in the paper "O problema humiano da indução", also published in this blog.

                            THE METAPHYSICS OF INDUCTIVE  REASONING*

                                 It would be impossible to say truly that the universe
                                 is a chaos, because if the universe where  genuinely
                                 chaotic there could be no language in which  to  say
                                 Jenny Teichman and K. C. Evans

                                                                                     Claudio F. Costa

This is an attempt to solve Hume’s problem of induction. Its approach is not very far from the ordinary language solution, which proposes to dissolve the problem. But the dissolution of the problem, in this case, contains a theoretical side, in the acceptance of the principles of induction grounded in the conditions of possibility of our understanding of a world or of any objective reality.
Esse artigo contém o esboço de uma solução para o problema humiano da indução. Trata-se de uma solução analítica: a indução é racional porque para que possamos ter acesso cognitivo a uma realidade objetiva, a um mundo, seja ele qual for, é conceitualmente necessário que procedimentos indutivos sejam aplicáveis com sucesso.
In this attempt to deal with the resilient problem of the justification of induction, a strategy with two steps will be followed. In the first one, a justification for enumerative and generalizing inductive inferences will be suggested recurring to major rules of induction, which play the role of inferential rules for inductive reasoning. In the second step, a justification for these rules will be attempted. This is the hard philosophical task. The basic insight I want to pursue – which has a light Kantian flavor – is the idea that it is constitutive of concepts like those of world or nature or of any epistemic field that we can’t conceive them as existing without being warranted the conditions for the application of inductive rules to them. Since there is nothing belonging to the realm of existence that does not already belong to the realm of conceivable, and since we can’t conceive a world refractory to induction, we conclude that the hypothesis that induction can fail to be applicable to some existing world is incoherent: if there is a world, it must be open to induction.

1. Proposing a strategy
Suppose that in antiquity a mathematician perceives for the first time that all even numbers he has reviewed are the sum of two prime numbers. He does not have any mathematical proof that all even numbers are the sum of two prime numbers, but after some further examination, he suspects that it may always be so. Moreover, when before a new even number, he turns to expect that it will be shown to be a sum of two prime numbers, attributing a positive probability to this, and the more instances confirming the hypothesis he finds, the more probable it will appear to him that the next even number will be a sum of two prime numbers...  
     This is an example of real inductive reasoning in mathematics, which has nothing to do with the so-called mathematical induction(1). Indeed, even in a formal epistemic field, it is possible to formulate the problem of induction: how to justify the created expectation that, for example, the next even number considered will be the sum of two prime numbers, or the further expectation that maybe all even numbers could be the sum of two primes? A seemingly natural answer would be that this is simply so, namely, that the inductive reasoning is so natural here that any attempt at justification would be unreasonable.
     I think that this answer would be over-hasty. It may look correct, maybe because we don’t yet have an adequate system of inductive logic. To see this, consider a parallel case concerning deductive reasoning. How do we justify an argument like ‘If there is fire, there is light; there is fire, hence there is light’? A natural reaction – from someone completely ignorant of propositional logic – would be the same as in the case of inductive reasoning. One would say: ‘It is natural to think so; there is no sense in attempting to justify a reasoning like that.’ However, as we all know, propositional logic has shown that this inference can be fully justified in two steps. The first consists in the application of the inferential rule of modus ponens (MP), as follows:

1   If there is fire, there is light.
2 There is fire.
3   Hence: There is light.            (1, 2 MP)

     Many think that justification necessarily ends here(2). We don’t think so. For we can add a second justificatory step showing why the modus ponens is a permissible rule of inference. This can be shown by reflection on the fact that valid rules of deductive inference must at least preserve truth, that conditionals expressing tautologies – that is, something that remains true whatever turns to be the situation of the world – are able to preserve truth, and that the modus ponens is a conditional expressing a tautology, as we all know by its truth-table:

         MP: A B    A & (A → B) ├ B
                 v v         v        v        v
                 v f          f         f        v
                 f v          f         v       v
                 f f           f         v       v

Consequently, the modus ponens must serve as a rule of inference. It is true that in making this inference we applied the rule of modus ponens to reach its conclusion. But this doesn’t make the argument circular, since the modus ponens does not enter in it in the premises as an inferential rule. Furthermore, we could avoid this complication simply by giving an example of deductive inference in which we should justify another rule of inference, for example, the rule of addition (A, B ├ A & B).
     Now, our working-hypothesis is that we could find a parallel case in the justification of inductive inferences. Here we also must first turn to rules of inductive inference, and them validate these rules by showing that they fit in our conceptual framework as conditionals expressing tautologies, remaining true whatever turns out to be the situation of the world, though this might not be evident at first view.

2. A general rule of inductionWe begin with the task of formulating the general rule or principle applicable to all conceivable cases of enumerative or generalizing inductive inference. Calling elements all kinds of properties, events, objects, and states of affairs, we can formulate the idea that the greater the number of instances in which an element A is found associated with an element B, the greater will be the probability that the same association will be found again in unconsidered cases. This fundamental idea is used in the formulation of the following rule justifying inductive inferences in general:
RI:  Considering a sufficiently large number of instances, if elements of kind A are found associated in a certain way with elements of kind B in a frequency of n%, then this makes probable (or reasonable to expect) that the next unconsidered element of kind A will be found to be associated with an element of kind B in the same way in a frequency of ~n% (or that all elements of kind A are associated in the same way with elements of kind B in such a frequency), increasing the probability with the increase of the considered instances.
     As presented, RI can justify enumerative as much as generalizing induction, having as important limit-cases those in which n = 100%. This is the case of our initial example. Putting the property ‘even number’ in the place of A, and the property ‘sum of two prime numbers’ in the place of B, we can apply RI to 1 in order to justify our expectation that probably the next A will be B:

1 All even numbers considered until now were
sums of two prime numbers.
2 Hence: the next even number will be a sum         (1 RI)
of two prime numbers (or: all numbers are
sums of two prime numbers).

     By means of RI, the conclusion of 3 turns to be justified as probable, namely, as having a positive probability (higher than 0,5) of being successful, which implies that our expectation that this conclusion is true, being the premise true, must be a reasonable one.

3. The main sub-fields of inductionAs formulated, RI is a general rule, applicable not only to inductive inferences with abstract elements in formal domains, but also to empirical enumerative or generalizing inductive inferences, namely, those ampliative inferences dealing with spatio-temporally located elements, directly or indirectly accessible by means of sensible experience. Consider, for example, the argument: ‘Fire has been always warm; hence (probably) the next fire will also be warm’. Taking ‘fire’ as element A and ‘warm’ as element B, we may apply the same RI as an inferential rule in order to justify a similar empirical inference, concluding more carefully that the next fire will have a positive probability of being warm or that it is reasonable to expect that the next fire will be warm. In fact, most inductive inferences are empirical, handling spatio-temporal elements, like physical (and eventually also mental) properties, events, objects, and states of affairs.
     In the last example, we made a move from what we have empirically observed in the past to what remains unobserved in the future. But the inductive move can also be made from past events to other past events, when one says, for example, that the fire made by the Neanderthals was probably warm; and it can also be made from here to there, when someone infers, for example, that a fire made in Burma now is as warm as fires made in England(3)
     In order to catch inductive inferences in a finer net, we can formulate versions of RI that are restricted to a limited epistemic field. Theoretically, we can split RI in RI-a, restricted to inductive inferences in the abstract field of formal sciences, and RI-e, restricted to inductive inferences in the whole empirical field constituted by spatio-temporal elements. RI-e can be split in two more restricted rules, RI-et, concerning empirical inductive inferences to the future or to the past, but in a given spatial extension (for example, on the earth), and RI-es, concerning inductive empirical inferences to another region of the space, but in a given temporal period (for example, in the last few years). Finally, RI-et can be split in RI-etf, concerning only inductive empirical inferences to the future, and RI-etp, concerning only inductive empirical inferences to the past. These divisions can be schematically represented in this way:
                                RI-a                                            RI-etf
          RI                                           RI-et
                                RI-e                                            RI-etp
    Representing space and time respectively by a horizontal and a vertical axis, we can produce some graphic representations, only in order to give a visual illustration of the different parts of the empirical epistemic field reached by each sub rule of RI-e:

RI-e:                 ______________    space 

RI-et:               RI-etf:                RI-etp:               RI-es     

     A detailed examination of these restricted forms of RI isn’t our aim here. But we will explain RI-e, RI-etf and RI-ee, since this will help us to scrutinize separately successive and distinct dimensions of the Humean problem of induction.

4. Formulating some restricted rules of induction
We begin with RI-e. Taking for granted that the elements are not Goodman’s predicates, that the instances are chosen with sufficiently variable background circumstances, and that there is no available collateral information(4), this empirically restricted version of RI can be presented as follows:
RI-e:  Considering a sufficiently large number of observed instances, if the elements of kind A are found to be associated in a certain way with the elements of kind B in a frequency of n % in already experienced spatio-temporal extensions, then it makes probable (or reasonable to expect) that a next element of kind A in a sufficiently near non-experienced spatio-temporal extension will be found to be similarly associated with an element of kind B in a frequency of ~n% (or: that all elements of kind A are similarly associated with elements of kind B in a sufficiently near spatio-temporal extension in such a frequency), increasing the probability with the increase of the observed instances.
     This restricted version of RI should justify all enumerative or generalizing empirical inferences from the observed to the unobserved, no matter if the unobserved is here or there, in the future or in the past.
     It is important to see, however, that not only the induction by enumeration but, particularly, the inductive generalization, should be usually restricted to sufficiently near spatio-temporal regions. This is to make justice to the fact that usually it does not make sense to apply induction to elements occurring in very distant spatio-temporal regions. Suppose, for example, that based on the fact that the sun has always risen, someone infers that the sun will rise in one billion years. This would be far from reasonable. And the same can be said about the inductive generalization ‘The sun always rises’, if it means that the sun will rise and has risen for all eternity. Taken literally, this generalization would be non-reasonable and probably (in the case, completely) false. The inductive generalization tends to lose any reasonability when applied to an extension of time or of space that is too great or infinite. Hence, the ‘all’ used in inductive generalizations is in most cases non-literal and should be better understood in the sense of ‘all spatio-temporally reasonably near cases’. It is inductively probable, for example, that the sun will always rise in the next years or centuries or millennia…(5) 
     An interesting case of a more restricted version of RI-e is RI-etf, which says that considering a certain extension of space, elements in the future tend to repeat associations already observed in the past. More precisely:

RI-etf:  Considering a sufficiently large number of observed instances, if the elements of kind A are found to be associated in a certain way with the elements of kind B in a frequency of n%, in already experienced proximal time-periods, then it makes probable (or more reasonable to expect) that a next element of kind A, in the future and in the same given spatial extension, will be found to be similarly associated with an element of kind B with a frequency of ~n%  (or that all elements of kind A in the same given spatial extension are similarly associated with elements of kind B in such a frequency), increasing the probability with the increase of the observed instances.

     Returning to our first example with empirical elements, now we can justify the passage from the
premise to the conclusion with the help of a more specific rule:

     1  Fire was always warm.
     2  Hence: The next fire will      (1 RI-etf)
         be warm.

     Finally, we can formulate a restricted form of RI-e, valid from observed to unobserved spatial extensions only, as RI-es:
RI-es:  Considering a sufficiently large number of observed instances, if elements of kind A are found to be associated in a certain way with the elements of kind B in a frequency of n% in experienced extensions of space, then it is probable (more reasonable to expect) that a next element of kind A, in the next non-experienced spatial extensions and in the same given temporal period, will be found to be similarly associated with an element of kind B in a frequency of ~n% (or that “all” elements of kind A in the same restrictive temporal period are similarly associated with elements of kind B in such a frequency), increasing the probability with the increase of the considered instances.
One can use RI-es as a specific rule for giving a positive probability to inductive inferences like the following:

1. All apple trees in this triangular area have apples.
2. Hence: the first apple tree outside the north edge     (1 RI-es)
of this triangular area will have apples (or: all
apple trees in the region outside this area have

Since space, unlike time, has no singular direction, we don’t need to go further in the specification of RI-es.
     Finally, why RI alone, ranging over all these cases, would not be enough? The answer is that, though RI is all-inclusive, its formulation doesn’t make explicit what is alternatively meant in its application within different epistemic fields. This is what the formulation of rules like RI-e, RI-etf and RI-es do. They make explicit what should be alternatively meant by the applications of RI to particular cases occurring in the corresponding fields, being in this way more informative.
     This is the outline of what I believe to be a first and easier step in the attempt to justify induction. The second step of justification should not be this kind of ‘vertical’ justification, appealing to some evidential truth (the rule of induction), but a more ‘horizontal’ justification or legitimation, showing that the rules of induction are constitutive of the conceptual framework we need in order to have cognitive access to any epistemic field. I believe that we can make this idea plausible by showing that the rules of induction are conditionals that – as much as the tautologies of logic – must express something that is true whatever turns to be the situation or course of the world. And that this can be done by showing that these rules are supported by metaphysical principles that are also conceptual truths, necessarily reflecting the nature of their corresponding epistemic fields, as far as they can be conceived and cognitively grasped by us. This is what we will try to do in the next sections. 

5.  A general principle of regularity
We will begin considering RI, the most general rule, telling us that the consideration of a sufficient number of instances of a certain association makes probable that these elements will be found associated in the same way in unconsidered instances. The only way I can see that could justify this epistemic rule is by suggesting that it is supported by what we call a metaphysical principle of regularity or PR, telling us that any epistemic field must have some order, some regularity, some uniformity, some repetition. Calling an epistemic field any cognitively recognizable domain of elements in all its possible combinations, and calling regularity a repeated or preserved relation between elements of the epistemic field, this principle can be roughly stated as follows:

PR: Any epistemic field must have some amount of regularity, so that at least some regularities belonging to a part of the epistemic field will be preserved in the next ones.

     It is interesting to consider PR as PR-a, namely, as applied to epistemic fields constituted by formal elements, like mathematics, since they display the principle in a form that lacks further complications given by spatio-temporality. It is intuitively clear that an epistemic field combining formal elements, as the arithmetic, must have some order, some repetition, some shared regularity, some homogeneously distributed patterns of association between its elements, in order to be cognitively grasped as a necessary knowledge (for example, natural numbers must have successors, they must be subject of arithmetic operations, etc.). Consider, for example, some sub-fields of mathematics like the theory of progressions, the theory of matrixes, or trigonometry. Each sub-field can be seen as an epistemic field needing to preserve a necessary degree of homogeneity in the distribution of its regularities, in order that problems appearing in the field are identified as belonging to it. At least for the case of formal sciences, the idea that PR is a conceptual truth, constitutive of each conceivable epistemic field, seems to be plausible.
     The plausibility of PR as a logically necessary truth is essential for us, since this principle seems to provide a metaphysical foundation for RI: if any epistemic field must have some regularity, some kinds of associations of elements spreading in its extension, this explains why finding a sufficient number of associations between the kinds A and B of elements in a part of an epistemic field allows us to make the inductive inference that this association will probably be preserved in the next parts of the same field.
     Against this would be possibly objected that, even allowing that the necessary existence of regularities is a constitutive condition of any conceivable epistemic field, the number of regularities of an epistemic field could be in some cases too small to justify the application of induction. However, it also seems that we could always surround this charge by requiring more extensive experience, a larger inductive basis, that is, a greater number of considered instances, and less ambitious inductive inferences, what would always ultimately lead us to discover the existing regularities.

6. Nature must have some uniformity (…)
If RI is supported by PR, it seems that RI-e must be supported by PR-e, a principle constitutive of any empirical epistemic field, that is, of any empirical world or world-part, telling us that they must have some amount of distributed regularity, order, uniformity, continuity. Calling an empirical world (the nature, the universe) a multiplicity of elements (properties, events, objects, states of affairs…) in all their possible spatial and temporal association, and calling regularity a repeated or preserved association between elements, so that a regularity can be usually divided into a sequence of regularities, a rough statement of PR-e could be the following:

PR-e: Any empirical world must have a certain amount of regularity, so that the greater the number of regularities belonging to a spatio-temporal region of the world, the greater will be the number of preserved regularities at least in the next similarly dimensioned region and proportionally to their proximity with those of the first region.

     We believe that this principle could be shown to be a conceptual truth(6). As for its first sentence, it seems simply to belong to our concept of empirical world that it must have at least some spreading regularity, some recognizable pattern, some continuity in the contiguity, in order to instantiate a conceptual framework that allows it to be epistemically accessible, what means, ultimately, to make it inductively accessible (an existing world must be conceivable and a world, to be conceivable, must be inductively accessible). Indeed, as in the case of analytical statements, the denial that an empirical world contains some amount of regularity seems to be inconsistent, because irregularities are thought against some background of order and it makes no sense to say that the empirical world may lose all its regularities and continue to be a world. Maybe the idea that an empirical world must have some amount of regularity spreading in its spatio-temporal extension could be contested by someone thinking on regularities only in diachronic terms, as repeated successions of events, like the causal regularities focused by Hume in his sceptical treatment of the problem of induction. For a world without causal changes, a frozen world, would indeed lack this kind of regularity. However, we should not forget that some synchronic permanence of the states of affairs in time is constitutive of any spatio-temporal world, being also object of our inductive knowledge(7). Thus, for example, I know that I’m now sitting in front of a lap-top, which is on a table, in this room, in this house, street, city, country, planet… and all the elements constituting these structures are contiguously associated in synchronic and more or less permanent ways, being my expectation of the permanence of these states of affairs for some time in the future also a question of inductive inference (in this case the elements of kind A and B in the application of RI-etf would be elements a and b in t1… elements a and b in tn, so that a and b in tn would be the same elements as a and b in t1). Only an instantaneous world would escape the assumption of some synchronic permanence of its states of affairs, but since existence is ‘being in time’, requiring some permanence, an instantaneous world would not be able to exist.
     To show the necessity of the second clause of PR-e is a more difficult task, which will be developed later, when we attempt to justify the principles governing RI-etf and RI-es, which are partial forms of PR-e.
     If any conceivable world must satisfy PR-e, it seems that they all must ultimately satisfy RI-e. The link is the following: if an empirical world is so that it must have regularities spreading to its contiguous regions, then, considering a regularity as an enduring or reiterating association between elements, it seems understandable that, when a sufficient number of elements of kind A is found related in a certain way with elements of kind B, then these elements would probably be found associated in the same way in the next still non-experienced spatio-temporal region in a similar proportion, or even in the whole not excessively extended spatio-temporal region, which tells us the same as RI-e. It is true that we can’t know a priori how much regularity must have our world. But if it is an a priori truth that it must have some amount of spreading regularity, this already gives us a sufficient reason to try the inductive reasoning based on RI-e. In this way the rule of empirical induction seems to be justified by the nature of any conceivable empirical reality and must be always applicable, given, of course, the adequate amount of experience, enough inductive basis, and a sufficiently humble inductive projection. Consequently, any conceivable empirical world must satisfy RI-e.
     Considering what we have said until now, and considering that we can only attribute existence to something that is conceivable, resulting from this that for us any existent world, as our own, must also be conceivable, we can state the following argument, which goes from the conceivable to the real, in the attempt to give a general justification for the application of induction to the empirical world:

1. Any conceivable empirical world must be epistemically accessible.
2. Any epistemically accessible world must be inductively accessible.
3. (1,2) Any conceivable empirical world must be inductively accessible.
4. If induction must be applicable to any conceivable world, then RI-e must be applicable in any conceivable world.
5. Where RI-e is applicable, it must be supported by the truth of a PR-e.
6. PR-e is a conceptually necessary truth.
7. (3, 4, 5, 6) Any conceivable empirical world must satisfy PR-e.
8. Where PR-e is true, it must support the application of RI-e.
9. (7, 8) Any conceivable empirical world must satisfy RI-e.
10.  If an empirical world exists, then this world must be at least conceivable.
11.  (9, 10) RI-e must be applicable to any existent empirical world.
12.  Empirical induction is always possible when RI-e can be applied.
13.  (11, 12) Any existing empirical world must be open to induction.

     If this argument has a force, it has to do more with coherence than with anything else: it comes from saving the greatest number of seemingly central conceptual truths in a consistent set. However, this is unavoidable when we try to validate a concept as central as that of induction, since what we really have is a cluster of central interdependent concepts (inductive knowledge, empirical world, time, space…) and what we need is to defeat our inductive worries by explaining their interdependence.

7. The future must have some similarity with its past (…)
This general argument can be reinforced when we consider RI-etf, the rule allowing us to make inductive projections to the future. The metaphysical basis for this epistemic rule of inference should be PR-etf, telling us that the future must hold some amount of similarity with its past in proportion to its proximity to it or, more precisely:

PR-etf: Any empirical world, being extended in time, must preserve a certain amount of regularities (some continuity or repetition) in its future, so that the greater the amount of past regularities, the greater must be the regularities preserved at least in its similarly dimensioned future period and proportionally to the proximity to its past.

     This principle can be also seen as stating a conceptual truth. Concerning its first sentence, it seems to belong to our concept of future that, at least when it is sufficiently near, it must hold similarities to its past in order to be the future of the same past, and that this can’t be denied without incoherence. It doesn’t make sense to say that a sufficiently near future may hold no similarity to its past, yet continuing to be the near future of the same past; for what would be the future, if not the future of a certain past? Indeed, even in a one-dimensional world, a purely acoustic world constituted by a casual sequence of sounds randomly chosen, there would remains the regularity of the casual sequence of sounds randomly chosen being preserved in the future for the time this world remains.
     A more perspicuous argument to the same point is the following. The concept of future involves the concept of time, which depends upon change; but changes are only possible against a background of rest: if the event E2 occurs after event E1, there must be some unchanged state of affairs S1 relating both events; hence, the future must have something in common with its past(8).
     This argument can be extended in order to justify the second clause of PR-etf, since we can show that there is a hierarchy in which smaller changes rest on changeable states of affairs that rest on less changeable states of affairs, and so on. Suppose that in the time-interval t1–t2 someone changes the form of a piece of wax from a stick to a ball. Other properties, like color, texture, solidity, remain the same, as much as the material. Now, suppose that in the next time interval, in t2–t3, the wax is submitted to heat, so that it changes into a liquid. Now only the matter remains the same, while color, texture, and solidity, have changed or disappeared. By the time-interval t1–t2, while the first change occurs, much more things are presupposed than in the whole time interval that comprises t1–t3. It seems also that the shorter the time, the greater the amount or regularities that might be expected to remain.
     This isn’t an isolated case. The same pattern can be found overall. I will give a more detailed historical example. Consider the historical change characterized by the hardening of repression in the USSR after 1928 (E1  E2). This change was dependent on a common state of affairs, namely, the permanence of Stalin in power (S). But the change of power from Stalin to Kruschev (S  S1) was dependent on the permanence of Soviet communism (S’). And the change from Soviet communism to democracy (S’  S1’) was dependent on the permanence of the state (S’’). We can represent the structure of a change like this by means of the following diagram:

     t1.........t2..…........t3…..……………...t 4…      
     E1  E2
     (----S----)  (----S1----)
     (-------------S’------------)  (----------S1’----------)
     (------------------------------S’’------------------------)  …

     Now, if time is dependent of change, then the nearer is the future period from its similarly dimensioned past period, the greater must be the preserved similarity between both, increasing this similarity with the increase of the amount of regularities. So, for example, if t1 is the present, then the near future t2 must have the elements S, S’ and S’’ in common with t1, while the more distant future t4 will have only the element S’’ in common with t1. But this seems to validate the second clause of PR-etf as a conceptual truth, telling us that the preservation of regularities in the future is directly dependent of the proximity of the future to its past and of the amount of regularities constituting its past.
     Now, if PR-etf is a conceptually necessary truth, it seems to provide a metaphysical support for the truth of PR-etf. For the terms ‘future’ and ‘past’ in PR-etf must refer here to segments of time in ways that forcefully include all the associations of elements constituting these periods. Expressed in a more detailed way, PR-etf applies the idea of necessary regularity to the preservation of associations in the future, telling us that the near enough future will hold at least some regularities which are the same as those taking place in its past, and suggesting that the more similar were the past regularities to the regularities of their own past pasts, the greater will be the similarity of the regularities of the future of the present with those of its past. But this principle can be used as a metaphysical basis to the epistemic rule PR-etf, which tells us that next observation of an element of kind A in a near enough future will probably hold a certain association with an element of kind B, when elements of kinds A and B were found associated in the same way in a sufficiently large number of instances in already experienced periods of time, increasing the probability with the increase of the number of observed instances.

8. World-CoursesA more effective way to make evident that RI-etf expresses a tautological conditional is by means of a thought-experiment in which we can show that this rule remains applicable, whatever the situation of the world turns to be. In order to do this thought-experiment, we call a ‘world-course’ the becoming of a world with regard to its uniformity or regularity. We may conceive of two opposite world situations. One is of a highly regular world, full of enduring and constant associations between its elements, which might be called an ‘ordered world’ or OW. The other is a highly irregular world, lacking enduring or regular associations between its elements, which might be called the ‘chaotic world’ or CW. This allows us to conceive the following four basic world-courses:

1. The course of an ordered world: The amount of regularity remains very high; it goes from OW to OW.
2. The course of a world in formation: The amount of regularity increases; it goes from CW to OW.
3. The course of a chaotic world: The amount of regularity remains very low; it goes from CW to CW.
4. The course of a world in disaggregation: the amount of regularity decreases; it goes from OW to CW.

      Surely there are other world-courses; but since they can be seen as combinations of these four basic ones, we don’t need to consider them here. The question we can pose now is this: how well can the RI-etf be used in the process of sustaining inductive procedures in each world-course? If we can show that RI-etf does well in all world-courses, since PR-etf is preserved, then we have a good reason to believe that we are dealing with conceptual truths, with tautologies that remain true whatever the empirical changes may be.
     Certainly, RI-etf is applicable to a world-course going from OW to OW, maintaining its high regularity. Since PR-etf is true to this world, because the regularities constituting the world remain fast always the same, spreading homogeneously in the future, RI-etf, telling us that the most common regularities of the past will probably be preserved in a sufficiently near future, remains applicable.
     PR-etf is also surely applicable to a world in formation, going from CW to OW. Since this is a world in which the already existent associations between its elements tend to persist, while new preserved regularities are being added to them, we can expect that PR-etf, telling us that the future holds at least some similarities to the past, being the amount of preserved regularities directly proportional to their temporal proximity, remains obviously true, warranting the application of the corresponding inductive rule.
     A more interesting question is whether RI-etf applies to a chaotic world, to CW. This question must be subdivided in two others. The first is: can RI-etf be applied to a relatively chaotic world, to what we could call an anarchical world? The second is: can RI-etf be applied to an absolutely chaotic world?
     Beginning with the second question: indeed, it doesn’t seem possible that RI-etf is applicable to an absolutely chaotic world or that PI-etf is true to it. But this doesn’t need to worry us, for the concept of an absolutely chaotic world (as expected) is incoherent. Such a world should lack not only diachronic regularities, like the causal ones, but also enduring synchronic regularities (states of affairs) warranting some structural permanence, which would make it impossible to be re-identified and experienced as existent.
     Now, what about the applicability of RI-etf to a world that is and remains only relatively chaotic, to an anarchical world? Here I would still hold a positive answer: a relatively chaotic world, preserving some regularity, still satisfies PI-etf, still has regularities tending to spread in the future. Consequently, RI-etf still finds opportunities of application, even if less often, since the inductive basis required would be greater, demanding much more experience, and since our forecasts would need to be limited to a nearer future, near enough to assure some similarity with the past. Cognitive subjects surviving in an anarchical world should cope with the lack of regularities by adjusting application of RI-etf, in ways that can’t be a priori. In fact, since by means of inductive experience we can plausibly measure the level of regularity of our world, and then, applying PR-etf to the known amount of regularity, adapt the required amount of experience and inductive basis, and the extent of our inductive projections to it, our inductive procedures and expectations seems to be inductively self-adjustable. Moreover, it is worth remarking that most of our world is, in reality, fairly chaotic, a fact of which natural scientists are well aware. And it is also true that parts of our world are less regular than others, as the sociological world, the psychological world, the meteorological world, the world of quantum-physics... what does not hinder the application of inductive procedures even to them.
     One can ask, however, what guarantee we have that our self-adjusted expectations concerning inductive experience will remain reliable in the future. Suppose, for example, that our empirical world has always been fairly regular until now, but that in the next minute it begins to lose regularities taking the regretable course of a world in disaggregation… Our answer is that PR-etf warrants that the accumulation of past regularities simply makes probable that this disaggregation would not occur, what makes rational to expect that our world would not change its course, even if this expectation turns out to be contradicted by the facts. Since ‘probable’ is not the same as ‘certain’, the fact that the disaggregation really takes place does not change the correctness of the inference. (Consider an example: adults have no fear of darkness, not because they’ve got used to sleeping in the darkness, as Hume would think, but because the accumulation of knowledge and inductive experience makes probable for them that to be alone in the darkness isn’t usually dangerous. And the wonder we sometimes have about the future persistence of our world as it is – we are aware that all things could suddenly change – isn’t rooted in lack of probability, but simply in lack of certainty.)
     The further question is whether RI-etf applies to a world in disaggregation. This is the worst scenario we can imagine. Indeed, nothing warrants that our not very regular world will not begin to lose its very reliable regularities, becoming more and more chaotic. Would induction work in this case? Following Hume, many have concluded that in such world RI-etf would not be applicable and inductive reasoning would collapse. Consider the following enumerative kind of induction:

1. The element F was associated in a certain way
with the element G in all the sufficiently large
          number of instances observed in the past.
2. Hence: F will remain associated with G in the      (1 RI-etf )
same way the next time.
     Some have said that F could always be found associated not with G, but with H, or with I, or with J… which lowers the probability that the next F be found associated with G near to null, and they would agree that this must be particularly true in a world in disaggregation. Or, to give a concrete example: a person throws a coin in a world in disaggregation… Even when the coin has always fallen it could instead move in any other different direction or stay suspended in mid-air or explode or disappear or change into a big stone or a camel…(9) What reason do we have to believe that this will not occur with the whole world, or at least in isolated cases, showing that induction doesn’t really makes its conclusion probable?
     Nevertheless, here we must disagree. The objection seems to work only because people are considering one case alone, without connecting it with the huge context of other surrounding cases against which it should be evaluated. I will explain. First, suppose that you throw a coin and it stays floating in the air, but that this is only an isolated case. This would not prove that there is something wrong with the inductive inference, since this would be the exception that proves the rule: since if RI-etf makes its conclusions only probable, it is to expect that it will sometimes fail to give us the expected result. Only a massive failure of induction would eventually show that RI-etf isn’t warranted. We already saw that this would not be possible in an absolutely chaotic world, since this world couldn’t exist for us. But would this be possible in a world in disaggregation? Here too, we think that the answer must be ‘no!’. Certainly, a world in disaggregation would be frightening, making our weaker inductive inferences made in the past completely unreliable. For it would elude the self-adjustment of our inductive expectations and of the required amount of experience and inductive basis that has been achieved by means of our past inductive experience of the level of regularity of the world. However, if a world in disaggregation is still accessible to experience or even conceivable, then it must satisfy PR-etf, holding enough distributed regularities in the associations of its elements to make RI-etf still applicable. This must be so because in a world where no inductive basis would be enough to warrant inductive inferences, there would not be enough regularity for the temporal permanence of any physical object or state of affairs or even of complex properties. In this world we would also lack enough regularity to warrant the continuation of our perceptual experience of the objects around us and even of ourselves, and our mental mechanisms of cognition and perceptual experience would collapse. Consequently, we would be back to a non-cognoscible chaotic non-world again. The expression ‘sufficiently large number of instances’ in our first premise of 6 means ‘sufficiently large to warrant that F will probably remain associated with G in the same way in the next time in a future that is near enough’. It is true that in a disaggregating world the past inductive knowledge would lose much of its reliability, but since we know a priori that there must be at least enough regularity to allow us to experience the world, we would still trust those past inductions based on the largest number of instances, the inductive knowledge concerning the most fundamental regularities, necessary for the existence of many others. For our inductively based knowledge of the world can be seen as having the form of a pyramid, and the better entrenched inductive beliefs, those constituting the basis of this pyramid, must be at least preserved, if there is to be a world.
     To see this clearly, we need a concrete example. The following passage from Sartre’s book Nausea gives us an idea of what would be a disaggregating world-course:
That may happen at any time, straight away perhaps: the omens are there. For example, the father of a family may go for a walk, and he will see a red rag coming towards him across the street, as if the wind were blowing it. And then the rag gets close to him, he will see that it is a quarter of rotten meat, covered with dust, crawling and hopping along a piece of tortured flesh rolling in the gutters and spasmodically shooting out jets of blood. Or else a mother may look at her child’s cheek and ask him: “What’s that – a pimple? And she will see the flesh puff up slightly, crack and split open, and at the bottom of the split a third eye, a laughing eye, will appear. Or else they will feel something gently brushing against their bodies, like the caresses reeds give swimmers in a river. And they will realize that their clothes have become living things. And somebody else will feel something scratching inside his mouth. And he will go to a mirror, open his mouth and his tongue will have become a huge living centipede, rubbing its legs together and scraping his palate. He will try to spit it out, but the centipede will be part of himself and he will have to tear it out with his hands. And hosts of things will appear for which people will have to find new names – as stone-eye, a big three-cornered arm, a toe-crutch, a spider-jaw…(10)
     Although losing regularities in a hopeless way, the world in disaggregation described by Sartre still remains to a great extent uniform enough to be inductively accessible. The elements, the usual physical objects (the street, the rag, the eye, the centipede…), the persons (the men, the mother, the child), the properties belonging to these things (the eye see, the centipede behaves as a living being, the limbs move following the will…), remain the same and similarly recognizable, though in some cases not in their usual combinations. Indeed, the permanence of a very great number of re-identifiable elements and their combinations is a necessary background for making a world in disaggregation experientially and cognitively accessible and describable, and it is because there is still a considerable amount of preserved regularity that this world must remain to some extent inductively trustful. Even a coin thrown in the air, one could bet, would more probably fall and not float like the rag, since the persons and all other bodies are placed on the ground and presumably preserving their normal weight. Moreover, certain parts of the world, the physical objects, the background of physical states of affairs, the cognitive experiences, remain as they always have been, suggesting that we still can trust the inductive beliefs based on past regularities of these world-parts.
     Another possible objection is that in non-absolutely regular worlds, particularly in the chaotic or disaggregating worlds, a sufficiently long period of time, all associations of elements could eventually change completely. This charge, however, would be below the point, for, as we already saw, the inductive probability should not be extended without limits to the future, and inductive generalization for all the eternity is something deeply flawed. Suppose, first, that the considered time interval between past and future is short enough to make impossible for the world to free itself from all associations of elements belonging to it, without ceasing to be a world (we already saw that future involves change, which involves permanence, which involves similarity in time…). In this case, our sufficiently humble inductive inferences with enough inductive basis would preserve a positive chance of being well succeeded. Now, suppose that the time interval is were really large enough to allow for all the associations of elements constituting the world to change completely. In this case, the inductive inferences using RI-etf would preserve a positive chance of succeeding only when applied to associations of elements within periods of time belonging to this larger time interval. Nonetheless, this result is fair enough: consider the inductive inferences about something highly irregular like the weather: we really don’t expect to do it for more than a very near future. Why should we expect something different for a very distant future of the world? Our conclusion is that RI-etf remains applicable to all conceivable world-courses, since the world can’t conceivably change in a way that contradicts PR-etf. The denial of RI-etf, as of the principle governing the rule, is inconsistent, what makes this inferential rule a tautological conditional, just as the modus ponens, though warranting only positive degrees of probability instead of certainty.

9. The spreading of spatial regularities
Let us consider, finally, RI-es, which must be supported by PR-es, a principle telling that a spatially extended world must hold some amount of spreading regularities in order to be conceived. This principle can be roughly stated as:
PR-es: Any spatially extended empirical world must have some amount of regularity in its spatial extension, so that the more regular patterns belonging to an extension of space, the greater will be the extent of them being preserved at least in the next similarly dimensioned spatial extensions, being this proportional to their proximity.
     We suggest that this principle is a conceptually necessary truth, because a spatial field, in order to be epistemically recognizable, must be conceivable, and this does not seem possible if space isn’t endowed with any regularity, order, uniformity, homogeneity. Being a conceptual truth, this principle would support the idea that the more associations between certain kind of elements are found in a certain region of space, the greater will be the probability to find this kind of association repeated in the next region, which is the basic intuition belonging to RI-es.
     To show the intuitive plausibility of PR-es, suppose that in all well distributed past observations of a bi-dimensional world constituted only by an homogeneous spatial surface, the same related patterns are found: dots with asterisks, which are equally distant one from the other. The following figures reproduce the first four observations subsequently made in adjacent extensions of space:

                                ●*    ●*                                                  6th              ?
                 3rd      ●*    ●*    ●*
                               ●*    ●*                 4th   
            ●*     ●*              ●*     ●* 
 2nd  ●*    ●*     ●*     ●*     ●*     ●*  
            ●*     ●*              ●*     ●*     
                          ●*    ●*
                     ●*     ●*    ●*            ?              5th
     1st                 ●*    ●*               

     Is there any reason to expect that such sequence of observations makes objectively probable that the same patterns will be found in the 5th observation made in an adjacent extension of space? And that we will found in the 5th observation the two patterns still preserved, instead of only one or none of them? And that we would be less sure about what to find in the more distant 6th observation? And that we would be less and less sure about the answer, the more distant were the extensions of space to be observed from the already observed ones? Against the Humean philosopher, but on the side of the plain common sense, we believe that the answer to all these questions should be in the affirmative. But if the answer is affirmative, it must be because we know for sure that our application of RI-es is being warranted by the truth of PR-es, telling us that any identifiable spatial field must be endowed with some patterns of regularity, which are distributed in direct proportionality with their amount and proximity. Moreover, even if an adjacent extension of space might lose regularities, it seems that it is impossible that this adjacent extension abruptly loses all synchronic associations of elements present in the already observed regions and remains experienced as the next spatial extension.
     Finally, a thought-experiment similar to the proposed in order to justify RI-etf can be imagined. Suppose that Ethel(1) goes from her room to another room in her house, called Rose Cottage. Then she (2) leaves the house and walks through her town, Witney, to the train station, where she goes to Oxford, to walk. The smaller change in space (1), from one room to another, presupposes the permanence of Rose Cottage, but Rose Cottage would still be there, even if the rooms were changed or destroyed, and only the external walls of the house would remain. Rose Cottage is officially a part of Witney, which is part of Oxfordshire. This means that, when Ethel moves from her room to another, she is assuming the permanence of the house and, to a certain extent, at least, the permanence of Witney and Oxfordshire. But her going to Oxford does presuppose less the permanence of her house. This seems to show that a move from a place to another, which is near, presupposes more spatial regularity than a change from one place to another, which is a more distant one. And the reason should be that the nearer the place, the bigger seems the amount of spatial regularities presupposed. If this is true, RI-es must remain always applicable, since it requires only that the next region maintains some similarity with the already experienced regions, a similarity that might always be mirrored by inductive procedures.
     I finish my attempt here. Although I’m aware that the arguments should be more carefully and adequately developed, I hope to have given some plausibility to the idea that inductive procedures are rational because they are able to make their conclusions objectively probable, since the rules promoting the inductive inferences are governed by principles constitutive of any conceivable epistemic field. It is interesting now to compare our line of argument with the argument followed by Hume’s in his classical attempt to show that induction is irrational.

10.  The Humean argument against the justifiability of induction
I begin by reconstructing Hume’s classical argument against the justifiability of inductive inference. By doing this, I’m not worried in matching the historical Hume, but in leading out the rationale for scepticism beneath his argument. Thus, beginning with the search for a justification of inductive reasoning, Hume first considered (in a conflating way) major principles like the following principle of the uniformity of nature:

      PU: Nature must be uniform,

And what we could call the principle of future’s likeness:

      PF: The future must be like the past,

     Now, principles like these could be added to inductive arguments as major premises, in order to make them justifiable. This can be respectively shown regarding the following two examples: (i) ‘Water has always quenched the thirst; hence water always quenches the thirst’; (ii) ‘The sun has always risen in the past; hence, the sun will rise tomorrow’. Using respectively the two major principles as major premises, these inductive arguments may be transformed into the following deductive arguments:

1 Nature must be uniform.
2 Water has always quenched the thirst.
3 Hence: water always quenches
the thirst.

1 The future must be like the past.
2 The sun has always risen in the past.
3 Hence: The sun will rise tomorrow.

     These arguments are deductive because the truth of their conclusions should follow necessarily from the truth of their premises(11).
     Nonetheless, Hume seems to have also shown in a compelling way the paradoxical consequences of this strategy. For him, principles like PU and PF need to be either analytical (relations of ideas) or synthetic (matters of fact). Certainly, these principles are not analytical, for they can be denied without contradiction: ‘The nature isn’t uniform’ and ‘The future will be different from the past’ are perfectly intelligible and potentially true sentences. As he writes, there is no contradiction that the course of nature may change and that an object like those we have experienced turns to have different or contrary effects; it is perfectly intelligible, he says, the idea of snow falling from the sky with the taste of salt and the feeling of fire, or of trees flourishing in December and January and decaying in May and June(12).
     Since the major principles are certainly non-analytic (and ignoring the possibility of their being synthetic a priori), the only way to justify their truth must be through experience. Indeed, experience shows us that the world around us has a huge number of uniformities, supporting as probable the conclusion that to a certain measure nature is uniform and that the future will have similarities to the past. However, the inferences leading to these conclusions are typically inductive, what means that they demand the truth of the major principles in order to be justified, which also means that any empirical justification of the major principles would be circular, presupposing what they are intended to justify. The well known sceptical conclusion that Hume draws from this argument is that induction can’t be rationally justified: we believe in its results only because of our psychological dispositions to achieve, by means of repetition, habits of expectance, being unavoidably compelled to follow these habits, like insects flying towards the light. It is no wonder that only few philosophers have been satisfied with his radically sceptical conclusion.

11.  Misleading aspects of the Humean reasoning
     To my mind, Hume’s paradoxical conclusion that induction is something non-rational only arises because his argument, though deeply prescient, is fallacious. And it is fallacious mainly because he confuses the level of epistemic rules with the level of metaphysical principles, using the last ones as the epistemic tools for a deductive attempt to justify induction that is then prematurely shown to be hopelessly flawed.
     The too strong demands of Hume’s strategy are reflected in his excessively demanding formulation of PU and PF, which are metaphysical principles requiring the preservation of all or almost all uniformities and allowing them to be falsified by isolated exceptions, like the snow with the taste of salt, and the feeling of fire and the trees flourishing in the winter. This request was made in order to warrant the place of the principles as major principles of inductive arguments able to turn them into deductive ones (even if in a more sophisticated reading such deductive arguments might lead to a probabilistic conclusion of the form ‘It is probable that p’). However, by proceeding in this way, Hume brought us into serious trouble. For in these too demanding formulations the principles are synthetic, since they are deniable without contradiction: it is clearly conceivable that nature could lose unexpectedly many of its regularities, or that the future could turn to be very different from its past. Nonetheless, if the major principles can be denied, they need to be empirically justified, which inevitably ends up in some kind of circular appeal to induction.
     Differently from Hume, we accept two metaphysical principles that correspond roughly to PU and PF, which are respectively PR-e (‘The world must have some uniformity’) and PR-etf (‘The future must have some similarity with its own past’). We can say that PR-e must be true because an empirical world, to exist, must be conceivable, and we can’t conceive a world without regularities; moreover, in order to be conceivable, a world must be epistemically accessible, that is, it must be open to induction, which is only possible if it has regularities. Hence, a world, to be a world, must have enough uniformity to sustain induction. And PR-etf must also be true because a future, to be a future, must be conceivable, and we can’t conceive the future without supposing that it preserves some regularities; moreover, in order to be conceivable, a future must be cognitively accessible as the future of its own past, which is only possible if it is open to induction by preserving some similarity with its past. Hence, a future, to be a future of its own past, must preserve enough similarity with its past to sustain induction. Thus, differently from Hume, we would formulate the examples 7 and 8 respectively as:

     1. Water has always quenched the thirst.
     2. Hence: water always quenches the thirst.  (RI-e)    (based on PI-e)

      1 The sun has always risen in the past.
      2  Hence: the sun will rise tomorrow. (RI-etf)           (based on PI-etf)

     Summarizing: it is not true that there is no path between the Cillas of an empirical but circular principle of induction and the Caribdes of a useless a priori principle, as Hume tried to show. We have reasons to believe that there is always a tiny trafficable path between them, since our inductive ship seems to have a self-adjusting drive mechanism enabling it to conduct us always safely through this path.

12. Misleading aspects of the ordinary language  attempt to dissolve the problem
It is interesting to compare our view with the very influential ordinary language attempt of dissolution of the problem of induction, since they have important similarities and some noteworthy differences. According to P. F. Strawson’s version of the ordinary language answer, inductive procedures don’t need further justification, since they are themselves the ultimate grounds of justification and rationality(13). When someone justifies his belief that the sun will rise tomorrow by telling us that it has always risen, we ask no more; when someone believes that he will not die when falling from the twentieth floor of a building, we do not hesitate to consider this person totally unreasonable. The rationality of induction is confirmed by the commonsensical intuitions reflected by our ordinary language.
     Strawson also thinks that there must be a dissociation between the rationality and the success of induction(14). With Hume, he believes that it is possible that our inductions turn to be massively unsuccessful, for it is conceivable that all our inductive anticipations might prove unsuccessful in the future. Nonetheless, Strawson holds that these inductive attempts should still be called ‘rational’, since ordinary language allows us to say that induction is rational. Unfortunately, the admission of a discrepancy between rationality and success contributes only to discredit the ordinary-language sense of the word ‘rationality’. For to say that induction is rational in a world in which it does not work is only to make a gesture, meaning as much as, to quote Wesley Salmon’s remark, ‘if you use inductive procedures you can call yourself “reasonable” – and isn’t that nice!’(15). Moreover, it seems that there is in fact a Humean sense of rationality, a sense in which the massive non-applicability of inductive procedures would make them irrational. For, as Lawrence Bonjour writes, to say that we are rationally justified is to say that we have reasons to think that something is ‘likely to be true’(16), that is, to attribute a positive probability to a state of affairs. Since the ordinary language defender of the induction is unable to give any justification for this last claim, his appeal to the ordinary language use of the concepts of rationality and induction turns out to be ineffective.
     I think that allowing dissociation between rationality and success, Strawson makes things too easy for the critics of his view. If our attempt to legitimate inductive rules shows anything, it is that there should be no gap between rationality and success in the case of induction. Inductive procedures are rational in the Humean sense that they really make their conclusions likely to be true, and we can’t conceive any circumstance in which induction would be massively unsuccessful. This would occur only in unconceivable worlds, defying any cognitive access, like the absolutely chaotic world. As a whole, the inductive procedures must be always successful, since this is warranted by the ways we can conceive of an objective world epistemically accessible to us. Indeed, it is because the inductive procedure as a whole must always be successful that we call this procedure rational.
     The problem with the ordinary language solution is that it tends to cut the branches of justification and rationality too short. As proposed by Strawson, this kind of solution recalls the naïve response to the question of the justifiability of deductive arguments, by setting aside the necessity of explicitly considering the rules of deduction, or by denying the necessity of their further justification by showing that they are conditionals expressing tautological truths applicable to all situations and changes of the world. In a similar way, the ordinary language solution overlooks the justificatory importance of making explicit the rules of induction and, in particular, the task of making explicit their epistemic status by means of further justification. As a result, the alleged reasonability of inductive procedures might become uncomfortably mysterious when we oppose them to Hume’s argument.
     Things look quite different when we see them in the light of the conceptual justification of induction that we are proposing. From this perspective, ordinary language calls inductive procedures ‘justified’ and ‘rational’ only because they are based on inductive rules with could be shown to be conceptual truths. Moreover, there is no need for dissociation between rationality and success, since the conceptual nature of our inductive procedures warrants their rationality by warranting enough success if there is to be a world. Of course, this rationality is weak and unable to give us any warrant that the sun will rise tomorrow, or that the floor will not sink under our feet. This is a sad fact about the misery of human cognition. Even so, this limbo of reason will be always far better than the Humean inferno.

Notes:* This paper was written during my post-doctoral stay at the University of Oxford with a grant from CNPq. My best thanks to professor Richard Swinburne for his careful critical reading of the last draft version.
1 Mathematical induction consists in the recursive application of a modus ponens to accepted premises. Thus, given that 0 has the property Ф, and that for each natural number x that has Ф, x + 1 has Ф, one can conclude that 0 + 1 has Ф, and therefore that 1 has Φ, that 1 + 1 has Ф, and so on infinitely. This process is obviously deductive. Our reasons for the inductive hypothesis considered here (the still unproved Goldbach’s conjecture) are merely inductive in the proper sense of this word. However, if this hypothesis comes to be proved true, it is to expect that the proof will proceed by means of mathematical induction.
2 Wesley C. Salmon, “The Concept of Inductive Evidence”, in Richard Swinburne (ed.): The Justification of Induction (Oxford: Oxford University Press 1971) , p. 53.
3 See J. S. Mill, A System of Logic (1843; 8th edn, 1872), book III, ch. 3, sec. 1,2, in J. M. Robson (ed.) Collected Works of J. S. Mill vol. VII (London: Routledge 1973).
4 Here I’m following Laurence Bonjour, who in his formulation of the rule of induction also abstracted these things as unimportant for our present concerns. See his, In Defence of Pure Reason (Cambridge: Cambridge University Press, 1998), pp. 188-9.
5 The case of the laws of nature isn’t necessarily better. Consider the most famous physical law, the law of gravity. It applies to our region of the universe and many others. But astronomers like Fred Hoyle have considered the hypothesis that some regions of the universe would have a negative gravitational force.
6 One could argue against our use of the concept of analyticity appealing to Quine’s objection that this concept can be defined only vaguely and with help of neighbor concepts, like those of identity, necessity, and meaning, what makes this definition ‘semi-circular’. However, this is clearly a fallacious argument, first because of its excessive demand of precision; the usual definition of analytical proposition as being true because of the meanings of its components might require a better analytical understanding, but isn’t itself imprecise. And second, because far from being a disadvantage, it is a necessary characteristic of definitions that the terms of the definiens must belong to the same semantic field of the terms of the definiendum. It would be senseless, for example, try to define a concept of economy using concepts of ornithology or biochemistry. See W. V. O. Quine, ‘Two Dogmas of Empiricism’, in From a Logical Point of View (Cambridge: Cambridge University Press 1953). For the opposite view, see H. P. Grice and P. F. Strawson: ‘In Defense of a  Dogma’, Philosophical Review 65, 1956, pp. 141-58; see also Richard Swinburne, ‘Analyticity, Necessity and Apriority’, in P. K. Moser (ed.) A Priori Knowledge (Oxford University Press: Oxford 1987).
7 Keith Campbell has noted that the complete denial of induction is incompatible with language and thought, since language and thought demand generalization, which demands re-identification of enduring associations by means of inductive inference. See his paper, ‘One Form of Scepticism about Induction’, in R. Swinburne (ed.), The Justification of Induction.
8 Nicholas Maxwell: ‘Can there be Necessary Connections Between Successive Events?’ in, R. Swinburne (ed.), The Justification of Induction, ibid. p. 154.
9  Example taken from John Foster, The Divine Lawmaker (Oxford: Oxford University Press 2003), pp. 2-3. See also W. Salmon: ‘The Uniformity of Nature’, Philosophy and Phenomenal Research, 14, 1953.
10  Jean Paul Sartre: Nausea (London: Penguin Books 2000),  pp. 225-226. 11 This understanding of the argument was defended, for example, in Max Black’s reconstruction in his, Problems of Analysis (London: Routledge 1974), chap. 11.
12 J. Hume: An Inquiry Concerning Human Understanding, ed. L. A. Selby-Bigge, Oxford 1989, sec. IV.
13 See P. F. Strawson: Introduction to Logical Theory (London: Methuen 1977), p. 260-263. See also Paul Edwards, ‘Russell’s Doubts About Induction’ in, R. Swinburne (ed.), The Justification of Induction.
14 Strawson, Introduction to Logical Theory, p. 254. Brian Skyrms has emphasized this point in Choice and Chance (Dickson: California 1962), chap. 2.
15 Wesley Salmon: ‘Should we attempt to justify induction?’ in, Feigl, Sellars, Lehrer (ed.): New Readings in Philosophical Analysis, New York 1972, p. 506. See also the comments from Lawrence Bonjour’s book In Defense of Pure Reason, p. 199.
16 Laurence Bonjour: ‘A Reconsideration of the Problem of Induction,’ Philosophical Topics, vol. XIV, n. 1, 1986, p. 102.

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