The Logical Syntax of Inductive Support-gradings
Abstract and Keywords
According to Karl von Frisch's method of reasoning, the conjunction of two generalizations must have the same grade of support as has the less well supported of the two or as both have if they are equally well supported. Also, any substitution-instance of a generalization must have the same grade of support as the generalization, since it is equally resistant to falsification by manipulations of relevant variables. So substitution-instances conform to the same conjunction principle as generalizations. The assumption of evidential replicability is crucial here and bars Carnap's confirmation-measures, or any form of enumerative induction, from applying to experimental reasoning like von Frisch's. Once the correct negation principle for inductive support has been established, it becomes clear how the emergence of mutually contradictory support-assessments can function as a reductio ad absurdum argument for the revision of a list of relevant variables. A support-function for generalizations of a certain category applies not only to those propositions that are constructed out of the basic vocabulary of the category but also to those that have been constructed out of this category when it has been enriched by the addition of terms describing the variants of relevant variables.
§50. According to von Frisch's method of reasoning the conjunction of two generalizations must have the same grade of support as has the less well supported of the two or as both have if they are equally well supported. §51. Also, any substitution-instance of a generalization must have the same grade of support as the generalization, since it is equally resistant to falsification by manipulations of relevant variables. So substitution-instances conform to the same conjunction principle as generalizations. §52. The assumption of evidential replicability is crucial here and bars Carnap's confirmation-measures, or any form of enumerative induction, from applying to experimental reasoning like von Frisch's. §53. Once the correct negation principle for inductive support has been established it becomes clear how the emergence of mutually contradictory support-assessments can function as a reductio ad absurdum argument for the revision of a list of relevant variables. §54. A support-function for generalizations of a certain category applies not only to those propositions that are constructed out of the basic vocabulary of the category but also to those that have been constructed out of this category when it has been enriched by the addition of terms describing the variants of relevant variables. Top-grade support can than be maintained for a generalization, in the face of adverse evidence, by qualifying its antecedent appropriately. But the consequence principle must be correspondingly restricted in its application to qualified generalizations.
§50. The conjunction principle for inductive support
To show that inductive probability has an independent standing, and is not to be derived in any way from mathematical probabilities, it is necessary to show first that inductive support, from which inductive probability does derive, is not itself a function of mathematical probabilities. Two arguments to (p. 168 ) that effect will be given below (in §§55–7). But the second of the two arguments depends on a certain feature of inductive support-gradings which has not yet been established. So §§50–3 are devoted to the discussion of this feature and of some of its corollaries.
The feature in question is that in any particular field of inquiry the conjunction of two propositions of certain kinds turns out to have the same grade of inductive support, on the evidence of any third proposition, as has the less well supported of the two conjuncts or as both have if they are equally well supported. More specifically, where H and H′ are first-order universally quantified conditionals belonging to the same category as one another, or substitution-instances of such, and E is any proposition whatever, the principle asserts
(1) If s[H′,E] ≥s[H,E], then s[H & H′,E] = s[H,E]
There is a quite general argument in favour of (1) if only a finite number, n, of grades of support are possible. This argument runs as follows. If s[H′,E] =s[H,E], there are just three possible kinds of value for s[H & H′,E], viz.
(i): s[H & H′,E] 〉s[H′,E]
(ii): s[H & H′,E] 〈s[H,E]
(iii): s[H & H′,E] = s[H′,E]
But the above argument, plausible as it may seem, rests on the assumption that only a finite number of grades of support are possible, and there is no a priori reason why this should in fact be true in every field of inquiry (cf. §45). The argument (p. 169 ) therefore needs to be fortified by some further consideration of what is actually involved in grading inductive support for a consistent conjunction on the basis of experimental evidence, irrespective of whether a finite or an infinite number of such grades is possible.
Let H and H′ be testable first-order generalizations. Since H and H′ are hypotheses in the same field of inquiry, they are open to the same series of tests as one another. For example, H might be
(2) Anything, if it is a bee-population, is taste-discriminatory
while H′ was
(3) Anything, if it is a butterfly-population, is smell-discriminatory.
Of course, the conjunction of H and H′ is not a testable generalization, according to the definition of testability given above (in §44, p. 137). But what an inductive test establishes is the circumstances under which a generalization may be relied on, and this is what the separate tests on H and H′ establish for the conjunction H & H′. So it would hardly be reasonable to prohibit the support-function which grades the reliability of H and H′ separately from also grading the reliability of their conjunction, as in the case of von Hess's hypothesis that both fishes and invertebrates are colour-blind.1
Moreover, when H and H′ conflict with one another it can easily be shown that, according to the method of relevant variables, either s[H,E] or s[H′,E] or both must be equal to zero, so that by (1) s[H & H′,E] is also zero.2
(p. 170 ) §51. The uniformity principle for inductive support
But what about the cases in which H and H′ are substitution-instances of one or two such generalizations? To deal with these cases I shall need two other principles as premisses.
The first of these is the equivalence principle
(4) If H is necessarily equivalent to H′ and E to E′, according to some non-contingent assumptions, such as laws of logic or mathematics, then s[H,E] = s[H′,E′] and s[H] = s[H′].
The second principle needed here is a principle of uniformity:
(5) For any U, P, and E, if U is a first-order universally quantified conditional that contains no reference to specific individuals and P is just a substitution-instance of U, then s[U,E] = s[P,E] and s[U] = s[P].
to count as substitution-instances of generalizations of the form
α ≠ a → (Raa → Saa)
we obtain some logically true substitution-instances for contingent generalizations. Such a substitution-instance will, in virtue of its logical truth, either have maximum support or at least be assigned always the same support-grade, while there may be different grades of support for different contingent generalizations of the form in question.4
(x)(y)(x ≠ y → (Rxy → Sxy) )
Inductive support is derived ultimately from the results of experimental or observational tests on universally quantified conditionals. The problem now is to determine the way in which such support may belong to singular propositions that are the substitution instances of those conditionals. Clearly, if U were an appropriately formulated generalization about some causal connection in experimental science and P a substitution-instance of it, the equivalence principle (4) would entitle us to assert the uniformity principle (5), since P would be necessarily equivalent to U. For example, if removal of a bee's antennae genuinely sufficed to produce loss of olfactory sense in one case, it must also do so in any other case.5 But this mode of argument from assumptions about causal uniformity is not open to us in relation to hypotheses like (2) and (3), which describe abilities rather than causal connections. If U and P in (5) could be any first-order universally quantified conditional and one of its substitution-instances, respectively, there is no reason to suppose that P is necessarily equivalent to U or to any other substitution-instance of U.
Nevertheless assumptions about causal uniformity can help (p. 172 ) us indirectly here. For, though U itself may not be a hypothesis about a causal connection, yet every experimental test on U investigates certain possible causal connections. It investigates, by the variations typical of causal enquiry (cf. §§42–6 above), whether certain combinations of circumstances suffice to cause the falsification of U. And to say this is equivalent, in virtue of appropriate assumptions about causal uniformity, to saying that if an experiment is to count as a valid piece of evidence it must be replicable: compare what was said about replicability in §43, p. 133, above. If this or that combination of relevant circumstances really suffices to cause the falsification of one substitution-instance of a particular generalization, it should suffice to falsify any other, and if it really fails to falsify one it should also fail to falsify any other. Accordingly, if inductive support is to be assessed in the light of resistance to falsification by combinations of relevant circumstances and if P is as resistant as U but no more resistant than U—i.e. if P resists in just the same combinations of relevant circumstances as U does—then it is reasonable to assign P just the same grade of inductive support as U in the light of any reports of experimental evidence. So the uniformity principle (5) follows in a relatively unrestricted form from the bearing of causal uniformity on the experimental testing of any hypotheses. We do not need to suppose that (5) is defensible only if U is itself a hypothesis about a causal connection. (Similarly where the tests are observational rather than experimental, (5) will follow from the fact that the variants of relevant variables are reliably regular signs of certain outsomes in appropriate circumstances.)
But once (5) is accepted we also have to accept (1) where H and H′ are substitution-instances of first-order quantified conditionals.6 Consider first the case where H and H′ are (p. 173 ) substitution-instances of two different generalizations, U and U′, respectively. There will always be a further generalization U″ that stands in the same relation to the conjunction U & U′ as does the generalization
to the conjunction of (2) and (3). Also there will always be a substitution-instance, H″, of U″ that stands in a corresponding relation to the conjunction H & H′. Now it has already been established (in §50) that if s[U′,E] ≥s[U,E], then s[U & U′,E] = s[U,E]. So, since according to the uniformity principle (5), where H and H′ are substitution-instances of U and U′ respectively, we have s[U′,E] = s[H′,E], s[U,E] = s[H,E] and s[U″,E] = s[H″,E]; and since, according to the equivalence principle (4), s[U & U′,E] = s[U″,E] and s[H & H′,E] = s[H″,E]; we can conclude that, if s[H′,E] ≥s[H,E], then s[H & H′,E] = s[H,E]. Replacing U′ by U we get the same result for the case where H and H′ are substitution-instances of the same generalization.
For any x and any y, if either x is a bee population or y is a butterfly population, then x is a bee population only if taste-discriminatory and y is a butterfly population only if smell-discriminatory.
In order for the above argument to go through, it is not necessary to premise that all causal connections between individual events should be treated as instances of uniformities. It is necessary to suppose only that such connections are sometimes so treated in, say, experimental entomology—which is adequately warranted. However, the case against a uniformity analysis of causation is not as strong as it is sometimes taken to be. As the topic is marginal to my present concern I shall discuss just two arguments that are sometimes put forward against such an analysis.
First, it is sometimes suggested that the uniformity analysis is an outcome of excessive reverence for Newtonian mechanics, and that in modern science
One event is the cause of another if the appearance of the first event is followed with a high probability by the appearance of the second, and there is no third event that we can use to factor out the probability relationship between the first and second events.7
(p. 174 ) On this view causal uniformities are merely the limiting case of an essentially probabilistic and Pascalian relationship. But it would be historically more correct to say that the long reign of Newtonian mechanics merely strengthened a pre-existing belief in uniform causation, since such a belief is clearly essential to Francis Bacon's inductive logic and its search for ‘forms’. Of course, no one ever doubted that causal uniformities were hard to come by, in a fully determinate formulation. Hence the common use of prefixes like ‘in normal circumstances …’. ‘under laboratory conditions …’, etc. The question at issue is therefore whether one should suppose that there is always a causal uniformity to be found, if only one searches long enough, or whether instead, at a certain stage of refinement in one's investigations of a problem, one should be prepared to give up the search for uniformities altogether and rest finally content with mathematical probabilities. Short of a theory that compels one to do this in a particular area of inquiry, as in quantum physics, it is not easy to see why any philosopher of science should want to say or imply that this is in principle the right way to proceed, if adequate time and resources are available for research. No doubt statistical approximations, or generalizations restricted to ‘normal circumstances’, are often all that may in practice emerge. But to surrender the thesis that causal connections are, in rerum natura, always instances of determinate, unrestricted uniformities is to surrender the most powerful intellectual impetus to further inquiry that has ever existed in human history.
Secondly, it is sometimes claimed that certain causal connections between individual events in nature are instances of neither determinate nor approximate uniformities. For example, it may be suggested that this claim is borne out by statements like
(6) That woman's death was caused by the contraceptive pill she took
§52. Some consequences of evidential replicability
Thus, so far as the conjunction principle (1) depends on the uniformity principle (5), it is a consequence of the assumption that any valid piece of experimental evidence—any result that is to count as evidence at all—is replicable. And that assumption also has another important consequence. Inductive support, as here described, cannot be obtained by what Bacon and Mill called enumerative induction.
Specifically what is assumed in the normal discourse of experimental science is that any statement of the form ‘The experiment described in E is evidence supporting H’, or ‘The experiment described in E is evidence against H’, already implies that the result of the experiment described in E will recur whenever the initial conditions described in E are reproduced. Hence so far as getting more evidence is concerned there is no point in just repeating the same experiment. If the replicated outcomes are again favourable to the hypothesis concerned, this cannot be regarded as increasing the grade of support for that hypothesis. If they are again unfavourable, this cannot be regarded as decreasing its grade of support. Instead, if you accept the normal assumption about replicability, you must regard any attempt at the actual replication of such an experiment as being merely a check-up on its evidential validity. You must regard such an attempt as being merely a crude test—at best a t1-type test—of whether the reported outcomes occurred as a result of the reported experimental conditions. There may conceivably be some contexts of human enquiry in which enumerative induction is legitimate—i.e. in which the replicability of experimental results (or the repeatability of observational ones) is not assumed and more evidence of the same favourable kind as before is properly taken to constitute an increase (p. 176 ) of support. But in the analysis of experimental reasoning like von Frisch's nothing is gained by confusing the question whether an alleged experimental result is replicable and therefore evidentially valid, with the question how much it would favour or disfavour a particular hypothesis if it were valid.
To avoid confusion here is particularly important if one wants to come to some conclusion about the applicability of Carnap's inductive logic. In his system it can be proved8 (with the help of Reichenbach's convergence axiom) that the confirmation of a singular statement H1 is increased when, in addition to original evidence stated in E, which is consistent with H1, we accept a new observational statement H2 which reports an instance of the same attribute as H1,: i.e. c[H1,H2 & E] 〉c[H1,E]. But the discovery of a proof for this principle, though welcomed by Carnap, was in fact a heavy blow to his claim to be explicating natural-scientific modes of discourse about inductive support. The discovery showed that this ‘principle of instantial relevance’, as Carnap called it, is rather deep-rooted in his confirmation-theory; and the harsh truth is that functions for which the principle is provable are cut off therewith from grading the support given by experimental evidence in any context in which an assumption of replicability operates. For in such a context, if E already reports an instance of the attribute described by hypothesis H1, H2 cannot be regarded as stating evidence that increases the support for H1 merely because it describes yet another instance of the same attribute. In relation to von Frisch's type of experimental reasoning Carnapian confirmation-functions could at best provide us with crude measures for the evidential validity of experimental results, in so far as this validity depends on replicability.
Similarly it is easy to construct supposed counterarguments to inductive principles like (1) and (5) if the error is made of taking these principles to be intended to apply to enumerative induction. For example, if the evidence reports nine observed swans to be white and one black, then the generalization that all swans are white might seem much less well supported than (p. 177 ) some single substitution-instance of that generalization. The substitution-instance seems to have, as it were, some chance of being true, while the generalization has none. So the uniformity principle (5) seems quite untenable if support may be assessed by reference to the numbers of favourable or unfavourable instances. But once you recognize that replicability is a condition of evidential validity in relation to (5), you cannot consistently pay any regard to numbers of instances in assessing the degree of support that some piece of valid evidence affords a given hypothesis. And if enumerative induction is thus clearly distinguished from the familiar, Baconian form of inductive reasoning at present under discussion, the supposed counterarguments all collapse.9
§53. The negation principle for inductive support
A pattern of axiomatization for the logical syntax of statements about inductive support is given below in §66. The axiom-schemata themselves and most of the main theorem-schemata have been provided with informal justifications elsewhere,10 though further arguments for the conjunction and uniformity principles have been given in §§50–1 above. We shall find, for example, that a familiar form of consequence principle is derivable, viz.
(7) For any E, H and H′, if H′ is a consequence of H according to some non-contingent assumptions, such as laws of logic of mathematics, then s[H′,E] ≥s[H,E].
(8) For any E and H, if E reports a physically possible event or conjunction or events, then, if s[H,E] 〉 0/n, s[not-H, E] = 0/n.
(9) (x) (Rx → Sx)
(10) (∃x) (Rx & -Sx).
(11) (x) (Px → Qx)
(12) (x) ((Px → Qx) → (Rx & –Sx))
If s[H] 〉 0/n, then s[not-H] = 0/n.
(p. 179 ) However, the above argument rests on the assumption that the definition of testability (given in §44 p. 137) may be safely extended in a way that would allow testability to (12). To avoid having to make that assumption one could argue instead that one has to adopt the above negation principles if one is not to be forced by the conjunction principle into admitting the existence of inductive support, on occasions, for the truth of a self-contradictory statement. If we did have both s[H,E] 〉 o/n and s[not-H,E] 〉o/n, the conjunction principle would give us s[H & not-H,E] 〉o/n. And then, if E were true, we could detach s[H & not-H] 〉o/n. To avoid that undesirable result from the consequence principle we have to suppose that for any E describing an encounterable set of events either s[H,E] or s[not-H,E] must be zero. The negation principle (8) states just this, and thus ensures, via (1), that for such an E we shall always have s[H & not-H,E] = 0/n.
Someone may perhaps object at this point that a difficulty now emerges for the proposed treatment of anomalies in theoretical science (§49 above). ‘Let H1’, the objection would run, ‘be a theory that has greater than zero support on E, where E includes the report of evidence of an anomaly in relation to H1. Let H2 be a theory that explains this anomaly, together with all the other laws explained by H1, so that H2 has even higher support on E than H1 has. Then H1 and H2 are apparently inconsistent in their consequences, so that H2 implies not-H1. By the consequence principle therefore we have s[not-H1,E] ≥s[H2,E] 〉 0/n. But we also have s[H1,E]〉 0/n. So we now have both s[H1,E] 〉 0 and also s[not-H1,E] 〉 0 even where E states perfectly respectable test-results; and that contradicts your negation principle.’
This line of reasoning is quite impeccable. But the conclusion to which it leads is not that the present philosophical account of inductive support is untenable. The conclusion should instead be that the particular method of assessing support which generates such a contradiction has to be rejected. All assessments of inductive support that are actually made, and the particular criteria by which they are made, are empirically corrigible. What has happened here is just that the emergence of apparent support for a contradiction forces us to regard one such method of support-assessment as requiring readjustment. If E indeed (p. 180 ) states a physically possible combination of circumstances, then it contains evidence arbitrating effectively between H1 and H2. The relevant variable thus arbitrating—call it ‘υ’—may therefore be taken to be the most important relevant variable so far as judgement on H1 and H2 are concerned; and in accordance with an appropriately readjusted list of relevant variables with υ now as first member of the series, we shall have s′[H1,E] = o/n. In other words the regularity that was previously regarded as an anomaly, while H1 was the only available theory, need no longer be so regarded, because a new theory H2 is now available to explain it. In relation to the two theories υ is now to be considered the relevant variable with greatest falsificatory potential and put first in the appropriate list of relevant variables. Accordingly there is no longer any need to suppose that H1, despite the existence of evidence refuting it, has greater than zero support, and thus no longer any apparent support for the contradiction H1 & not-H1.
The negation principle may therefore be viewed as enabling us to construct a reductio ad absurdum argument here for revising the ordering of our set of relevant variables. By the old ordering we had s[H1,E] 〉o/n, from which the negation principle entitled us to infer s[not-H1,E] = o/n for the E in question. But we could also infer s[not-H1,E] 〉o/n. Now these two conclusions cannot both be true, if E is true, since s[not-H1 cannot be both equal to and greater than zero. It follows that there must be something wrong with our method of assessing support. And if we have no reason to suspect the correctness and adequacy of our set of relevant variables it must be the ordering of this set that is at fault.
We must be particularly careful to keep apart here our ontological conception of inductive support from our epistemological one. Support may actually exist for a theory, though it is as yet unknown to us and unrepresented in our judgements. So major advances in science, as already remarked (§48, pp. 159 f.), proceed on at least two levels. Such advances involve not only the discovery of better theories about the world but also the discovery of better methods for assessing the reliability of these theories. The point emerged earlier in relation to the superiority of a theory that predicts hitherto unknown facts about the world, since this extends our list of relevant variables. (p. 181 ) And the same point emerges now in relation to the superiority of a theory that eliminates anomalies. That kind of theory enables us to reorder our list of relevant variables.
Yet a third way in which the point emerges is also worth mentioning. Two theories H1 and H2 may have incompatible implications about which no evidence is at present available. Popper11 cites the example of Einstein's and Newton's theories of gravitation, which lead to incompatible results for strong gravitational fields and fast-moving bodies. In such a case, where the actual evidence does not discriminate at all between the two theories and supports both equally, we again have a result, with the help of (7), that contradicts the negation principle. But here too the proper conclusion to draw is that there is something wrong with the particular method of assessing support which generates the contradiction. Specifically, the method is at fault in not including some of the relevant variables, viz. the ones that would in fact arbitrate between the two theories. The situation is that tests with these variables are at present impracticable, so they cannot be listed in our current criteria of assessment. But then we must be content to find the incompleteness of our current criteria reflected in the contradictions which are generated by the claim to adequacy that is implicit in these criteria. The evidence shows that claim not to be fully supported. That is, we have to act on a hypothesis about the proper method of assessment for our scientific theories which, because it encounters its own special kind of anomaly, has itself to be assessed—at the appropriate, higher level—as falling short of full reliability.
In short, the emergence of such contradictions is very far from being an objection to the proposed logical syntax for statements about inductive support. Rather, it enables us to elucidate how the actual progress of science in a particular field of inquiry imposes a continuing local readjustment in our criteria of evidential support. Inductive appraisal has a vitally important internal dynamic that must not be ignored. Any globally applicable analysis, like Carnap's, which supposes each derivable support-assessment to be a priori true for all time, inevitably obscures this dynamic.
(p. 182 ) §54. Some consequences of modifying a generalization in order to ensure reliability
I shall assume that the logical syntax of statements about inductive support is the same for non-testable generalizations and their substitution-instances as for testable ones. This assumption is underwritten partly by the equivalence principle (4), and partly by considerations of systematic analogy. But it should be noticed that a support-function for hypotheses of a certain category applies also to certain other propositions besides those testable and non-testable ones that are constructed out of the basic vocabulary of the category. Specifically, it applies also to those propositions that are constructed out of this vocabulary when it has been enriched by the addition of one or more predicables describing variants of υ2, υ3, … or υn in the series of relevant variables. And the reason is that any member of the series of relevant variables must also be relevant to some generalizations constructed out of the enriched vocabulary.
Suppose that the series includes variable υj as well as variable υi. Then the mutual independence of relevant variables implies that each variant of υj falsifies some generalization which is not falsified by any variant of υi. So, a fortiori, each variant of υj would falsify some of the generalizations it does falsify even if they were qualified in their antecedents so as to apply only to this or that variant of υi—provided that the qualification did not in fact protect the generalizations against such falsification. And, where it did protect them, each variant of υj would falsify some of the generalizations it does falsify when they were qualified in their antecedents so as to apply only when this or that variant of υi was absent. The variable υj is therefore relevant to propositions formulated in the basic vocabulary enriched by the terms of υi, for just the same kind of reason as it is relevant to propositions formulated in the basic vocabulary alone.
It follows that, whenever a testable generalization passes all tests up to t1 but fails ti + 1, we can always modify it so as to produce a generalization implicit in the original one which will in fact pass ti+1 and therefore has just as good a claim to be called ‘testable’. We can do this in either of two ways, and (p. 183 ) such modified propositions will hereafter also be regarded as testable.
One way is to specify some particular circumstance in the antecedent that excludes the falsifying variant. For example, if the original generalization were
(13) Anything, if it is R, is S
(14) Anything, if it is R and V2i+1, is S.
Essentially this was von Frisch's move when he discovered that bees did not discriminate all colours.
The other way of buying support for a generalization at the cost of weakening it is to restrict its range of application. This may be achieved by specifying in the antecedent that no variant at all of a certain relevant variable is present. For example, we might replace (13) not by (14) but by
where V1i+1 and V2i+1 are all the variants of υi+1. In other words we would replace (13) by
Anything, if it is R and not V1+1 and not V2i+1 is S
(15) Anything, if it is R and not characterized by any variant of υi+1, is S.
The qualification of causal generalizations proceeds analogously. To assert that anything's being R is a cause or a sign of its being S is tantamount, as we have seen (§46, p. 150), to asserting that the conjunction of (13) with an appropriate statement about a control for R is maximally reliable. So if a (p. 184 ) causal generalization is qualified by adding one or more specified variants of relevant variables to the causal conditions, then the conjunction to which maximal reliability is ascribed must include not only a qualified conditional generalization, like (14), but also appropriate statements about controls for each of these specified variants as well as for R itself. But if instead—as is also very common—a causal generalization is qualified negatively, like (15), the situation is somewhat different. To assert that, when not characterized by any variant of υi+1, anything's being R is a cause or a sign of its being S, is tantamount to asserting just that the conjunction of (15) with a similarly qualified statement about a control for R is maximally reliable.
Indeed the expression ‘in normal circumstances’, which is so often used in practice to qualify hypothesised generalizations, may be interpreted as a comprehensive, collective restriction of the type present in (15) or in an analogously qualified causal generalization. So may ‘ceteris paribus’, ‘under laboratory conditions’, and similar phrases. Accordingly the introduction of non-exhaustive relevant variables into a theory of inductive support is not just an ad hoc dodge to get round the objection mentioned above (in §44, p. 135). Rather, it is essential if we want to be able to precisify the common proviso ‘in normal circumstances’ and elucidate its role in inductive reasoning. A non-relevant circumstance that is co-ordinate but incompatible with each variant of a particular relevant variable is a safe or ‘normal’ circumstance because, qua non-relevant, it does not exclude any generalisation (that belongs to the category concerned) from operating. Hence this concept of ‘normal circumstances’ also plays an important role in hypothetical or counterfactual inference. When we discuss what would have happened if the match had been struck in the oil refinery, we have to make some assumption about the other circumstances in this hypothetical or counterfactual situation. And the most appropriate assumption is that, so far as they are unstated, they are normal. That ensures that the generalization which covers our inference would have operated.12
(p. 185 ) The value of operating with relevant variables that are not logically exhaustive is thus that however low-grade the test our hypothesis has actually passed we can always claim that something has been fully established. If a hypothesis like (13) is reported to have passed test ti but failed ti+1, for example, we can legitimately claim that the hypothesis
(16) Anything, if it is R and not characterized by any variant of υi+1, υi+2 … or υn, is S
(17) In normal circumstances, anything if it is R is S
(18) Anything, if it is R and Vki+1 and Vki+2 and … and Vkn, is S
(19) Anything, if it is R and Vki+1 and Vki+2 and … and Vkj and not characterized by any variant of υj+1, … or υn, is S
But what should be said where a hypothesis like (13) is modified by a variant of υi+1 as in (14) and the modification is disadvantageous rather than advantageous? That is, what should be said when the circumstances described in the antecedent of the modified version suffice to exclude the truth of the consequent, and the generalization is therefore false? Clearly no other support-grading than zero is appropriate for the modified version: this version does not resist falsification under any test.13 But the unmodified version, (13), may have greater than zero reliability because it manages to pass some of the simpler tests. It follows that, since the unmodified version logically implies the modified one, the consequence principle mentioned in §53 cannot apply unrestrictedly to hypotheses like (14) or (18).14 We cannot say that such a hypothesis will always have at least as much inductive support as any hypothesis of which it is a logical consequence. So what principle is operative here?
Obviously the essence of the consequence principle is that a proposition is exposed to any source and extent of falsification to which any consequence of it is exposed. Hence the principle applies only where whatever exposes the consequence to falsification also exposes the premiss. So the restricted consequence principle, which can apply not only to propositions in the basic terminology of a particular category but also to appropriately modified versions of such propositions, will be
For any propositions E, H, and H′, such that H′ is a (p. 187 ) consequence of H according to some non-contingent assumptions such as laws of logic or mathematics, if either
(i) no variant of a relevant variable is mentioned in H′, or
(ii) each of H and H′ is a testable generalization or a substitution-instance of one, and for any variant V of a relevant variable, if H′ states its consequent to be conditional on the presence of V, H also states its consequent to be conditional on the presence of V,
then s[H′,E] ≥s[H,E] and s[H′] ≥s[H].
It is worth noting, finally, that a testable generalization could be modified not only by addition but also by subtraction. Instead of adding the presence of further variants to the circumstances upon which the consequent of (13) is conditional—as in (14), (18) or (19)—we might also subtract the existing antecedent R, which is a variant of υ1 in the series of relevant variables. This kind of modification would produce from (13) the generalization
(20) Anything is S
(1) Bees, p. 4.
(2) Cf. TIOI, pp. 64 ff., and also § 53, p. 178, below.
(3) Cf. TIOI, pp. 18 f. and 101.
(4) In TIOI (pp. 20 ff.) this restriction was unfortunately omitted from the informal statement of the principles that required it; and the formal versions of the principles in TIOI (pp. 228 ff.) need a corresponding qualification, which is allowed for in §66, p. 236, below (in connection with criterion of theoremhood 112). The assignment of maximum inductive support to logical truths may be justified by systematic considerations of analogy: cf. TIOI, §§20–2, and especially p. 200. But see also p. 137, n. 14 above.
(5) Bees, pp. 29 ff.
(6) It is perhaps worth pointing out that the argument given here for this conclusion does not have as one of its premisses the principle referred to in TIOI, pp. 19 f., as the instantial comparability principle, viz. the principle that inequality of inductive support for two first-order generalizations entails, and is entailed by, a similar inequality of support for their substitution-instances. In TIOI that principle was a premiss for the conclusion in question—a premiss which some critics found counter-intuitive. Here, however, the principle is itself easily derivable from (5). (In neither case can the conclusion be established for second-order generalizations which correlate variables with one another. This is because the same combination of relevant circumstances may not have the same causal impact on each variant of the independent variable. Cf. TIOI, pp. 82 f.)
(7) P. Suppes, A Probability Theory of Causality (1970), p. 10.
(8) R. Carnap, ‘A Basic System of Inductive Logic’, in R. Carnap and R. C. Jeffrey (eds.) Studies in Inductive Logic and Probability, vol. i (1971), pp. 161 ff., and proofs by Jürgen Humburg and Haim Gaifman, Ibid. pp. 227 ff.
(10) Cf. TIOI, p. 221.
(11) K. R. Popper, Logic of Scientific Discovery (1959), p. 374.
(12) This problem is discussed at greater length by N. Goodman, Fact, Fiction and Forecast (1954), pp. 17 ff., and by N. Rescher, Hypothetical Reasoning (1964). The solution proposed here meets all the difficulties discussed by Goodman and Rescher.
(13) This is because tests t1, t2, … ti would be inapplicable, since a variant of υi+1 would be present in each (cf. p. 134), and tests ti+2, ti+3, … tn would falsify the generalization, since the series of relevant variables is so ordered that a more important relevant variable can never have its falsificatory potential wholly nullified by a less important one (cf. p. 139).
(14) The need to restrict the consequence principle was not recognized in TIOI, pp. 148–9, because the only modifications considered there were advantageous ones and the concept of an ith grade version was applied not only to modifications of the type exemplified by (14) and (18) but also to those of the type exemplified by (15) and (16)—which in any case create no special problem.
This restriction of the consequence principle for inductive support turns out to be rather important for the possibility of developing a concept of inductive probability: cf. §63, p. 218 f., below.