The Paradoxes of ConfirmationAihepiirit:
Tekijät:
Julkaistu: 1965 Julkaistu filosofia.fi sivustolla: 15.06.2009
Published in/Publicerad i/Julkaistu
Theoria, 31, 1965 (255275).
© The von Wright Heirs/ von Wrights arvingar/von Wrightin perikunta The Georg Henrik von Wright Online Collection, Filosofia.fi (Eurooppalaisen filosofian seura ry) <http://filosofia.fi/vonWright> ed./red./toim. Yrsa Neuman & Lars Hertzberg 2009. Inskanning & transkribering/Skannerointi & litterointi/Scan & transcription: Filosofia.fi Tommi Palosaari  Sonja Vanto The Paradoxes of Confirmation1. The name "Paradoxes of Confirmation", as far as I know, was coined by Hempel.^{1} The locus classicus of the treatment of the paradoxes is Hempel's paper "Studies in the Logic of Confirmation" in Mind of 1945.^{2} The puzzle is sometimes referred to under the name "Hempel's Paradox". I think it is useful to distinguish between several puzzles in the context. I shall here single out for special treatment two such puzzles. One of the puzzles which I have in mind may be said to belong to the bundle of perplexities which are known under the generic name of Paradoxes of Implication. It can be briefly described as follows: Consider a generalization of the form "All A are B". An example could be "All ravens are black". We divide the things, of which A (e. g. ravenhood) and B (e. g. blackness) can be significantly (meaningfully) predicated into four mutually exclusive and jointly exhaustive classes. The first consists of all things which are A and B. The second consists of all things which are A but not B. The third consists of all things which are B but not A. The fourth, finally, consists of all things which are neither A nor B. Things of the second category or class, and such things only, 256 afford disconfirming (falsifying) instances of the generalization that all A are B. Since things of the first and third and fourth category do not afford disconfirming instances one may, on that ground alone, say that they afford confirming instances of the generalization. If we accept this definition of the notion of a confirming instance, it follows that any thing which is not A ipso facto affords a confirming instance of the generalization that all A are B. It also follows that any thing which is B ipso facto affords a confirmation of the generalization. Such consequences as these may strike one as being, somehow, "paradoxical". They would entail, e. g., that a table, since notoriously it is not a raven, affords a confirmation of the generalization that all ravens are black—or that any black thing, whether it is a raven or not, confirms the generalization that all ravens are black. It may now be thought that a way of avoiding this paradox would be to give to the notion of a confirming instance a more restricted definition. One suggestion would be that only things of the first of the four categories, i. e. only things which are both A and B, afford confirmations of the generalization that all A are B. This definition of the notion of a confirming instance is sometimes referred to under the name "Nicod's Criterion".^{3 }According to this criterion, only propositions to the effect that a certain thing is a raven and is black can rightly be said to confirm the generalization that all ravens are black. But if we adopt Nicod's Criterion as our definition of the notion of a confirming instance we at once run into a new difficulty. Consider the generalization that all notB are notA. According to the proposed criterion we should have to say that only things which are notB and notA afford confirmations of this generalization. The things which are notB and notA are the things which are neither A nor B (but of which Aness and
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Bness can be significantly predicated). That is: they are the things of the fourth of the four categories which we distinguished above. But, it is argued, the generalization that all A are B is the same as the generalization that all notB are notA. To say "all A are B" and to say "all notB are notA" are two ways of saying the same thing. It seems highly reasonable, not to say absolutely necessary, to think that any thing which affords a confirmation of the generalization g also affords a confirming instance of the generalization h, if "g" and "h" are but two different expressions for the same generalization. This requirement on the notion of a confirming instance is usually called "The Equivalence Condition".^{4} To accept Nicod's Criterion thus seems to lead to conflict with the Equivalence Condition. This conflict constitutes another Paradox of Confirmation. There has sprung up an extensive literature on the Paradoxes of Confirmation. I shall not attempt to review it here. It seems to me, moreover, that there may exist several interesting and rewarding ways of treating the paradoxes. I do not think there is a unique, correct solution of them.^{5} 2. There is a theorem of confirmation theory, originally proved —in slightly different ways—by Broad, Keynes, and Nicod to 258
the effect, roughly speaking, that the probability of a generalization increases with each new confirmation of the generalization provided that (1) the probability of the generalization prior to confirmation (its socalled probability a priori) is not 0, and (2) the probability of the new confirmation relative to the bulk of previous confirmations is not 1.^{6} I shall refer to this as the Principal Theorem of Confirmation. This theorem may be thought to provide a possibility of handling confirmation paradoxes—in particular that paradox which may be counted as one of the Paradoxes of Implication. For, assume it could be shown that things which afford "paradoxical" confirmations of a generalization also afford confirmations which necessarily are maximally probable relative to the bulk of previous confirmations. This would mean that confirmations afforded by such things cannot possibly contribute to an increase in the probability of the generalization in question. And this would constitute a good ground for saying that they confirm the generalization, not "genuinely", but only "paradoxically". To show that some confirmations, which are intuitively felt to be "paradoxical", are thus futile (irrelevant, unimportant) from the point of view of the probability of the generalizations, is one way and, it seems to me, a rather good one of settling the paradox under discussion. I shall now try to explore this way in some detail. 3. We shall sketch a proof of the Principal Theorem. In order to do this we need an axiomatic basis to start from. This basis will here be introduced with a maximum of what seems to me to be permissible simplifications. We shall assume that a probability is a notnegative real number, uniquely associated with a twoplace functor of the form
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"P(φ/ψ)". "φ" and "ψ" are schematic names of generic characteristics of the same logical type. By generic characteristics we understand attributes and properties of individual things,—and also features of events, processes or states which may or may not take place, go on or obtain on individual occasions. To say that characteristics are of the same logical type shall here mean that they have the same range of significance, i. e. that the individuals of which it makes sense to say that they have or lack those characteristics are the same. The expression "P(φ/ψ)" can be read "the probability that a random individual is φ, given that it is ψ". Instead of "is" we can also say "has the characteristic", and instead of "given" we can say "on the datum" or "relative to". The application of probabilities, which are primarily associated with characteristics, to individuals is connected with notorious difficulties. The application is sometimes even said to be meaningless. This, however, is an unnecessarily restricted view of the matter. If x is an individual in the range of significance of φ and ψ, and if P(φ/ψ) = p is true, then we may, in a secondary sense, say that, as a bearer of the characteristic ψ, x has a probability p of being a bearer also of the characteristic φ. This secondary sense of a "probabilityrelation" which holds primarily between generic characteristics must not be confused with another "probabilityrelation" which holds primarily between propositions. This other type of probability, sometimes also called "logical probability", is not involved in the argument of the present essay at all. For the sake of simplifying our formulae, we here assume that all generic characteristics (and molecular compounds of such characteristics) which appear as arguments in the probability functor belong to at least one individual. (If we did not make this assumption, we should have to add an existential clause to every formula.) The probability functor obeys the axioms: A1. (ψ Ì φ) → P(φ/ψ) = 1. A2. P(φ/ψ) + P(~φ/ψ) = 1. A3. P(φ & ψ/χ) = P(ψ/χ) × P(ψ/χ & φ).
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4. Consider two characteristics φ and ψ and a sequence of characteristics φ_{1},...,φ_{n},… . For the conjunction of the first n members of the sequence we introduce the abbreviation "Φ_{n}". We shall assume that for every individual x and for every natural number n, if this individual has the characteristic φ it also has the characteristic Φ_{n}. Let the value of P(φ/ψ) be p, and let the value of P(φ_{1}/ψ) be p_{1}, the value of P(φ_{2}/ψ & φ_{1}) be p_{2}, and, generally, the value of P(φ_{n+1}/ψ & Φ_{n}) be p_{n}_{+1}. For the product of the first n members of the sequence of numbers p_{1},..., p_{n}, ... we introduce the abbreviation "II_{n}". By repeated applications of A3 we calculate that P(Φ_{n}/ψ) has the value II_{n}. According to A3, P(φ & Φ_{n}/ψ) = P(φ/ψ) ∙ P(Φ_{n}/ψ & φ) = P(Φ_{n}/ψ) ∙ P(φ/ψ & Φ_{n}). The value of P(φ/ψ), we said, is p. The value of P(Φ_{n}/ψ) we calculated to be II_{n}. Since, on our assumption above, everything which is φ is Φ_{n}, it must also be the case that everything which is φ & ψ is Φ_{n}. Therefore, by virtue of Al, P(Φ_{n}/ψ & φ) has the value 1. We next assume that p > 0. Then we can divide and write P(φ/ψ & Φ_{n}) = p/ II_{n}. On the basis of exactly analogous considerations we derive P(φ/ψ & Φ_{n}_{+1}) = p/ II_{n}_{+1}. Since p > 0, p/ II_{n}_{+1} > p/ II_{n}, if and only if p_{n+}_{1} < 1. This is essentially the content of the Confirmation Theorem within our axiomatic setup of probability. So far there has been no mention of generalizations and their instances. As will be seen, however, the abstract and general thoughts underlying the above theorem may become applied for studying the probability of generalizations.
5. Before we go back to discussing the generalization that all A are B, let us first consider a related generalization. This generalization says that B is a necessary condition of A, meaning that the presence of the characteristic (property) B in an individual (thing) is a necessary condition of the presence of the characteristic (property) A in that same individual (thing).
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When calling the two generalizations related, I am in the first place thinking of the fact that, if B is a necessary condition of A, then everything which is A is certainly also B. Let "Nc_{a}" denote the (secondorder) property of being a necessary condition of the (firstorder) property A. The symbol "Nc_{a}(B)" then expresses the generalization that B is a necessary condition of A. A disconfirming instance of the generalization Nc_{a}(B) is afforded by any thing, in which B is absent although A is present, or as I shall also say: in which B is absent in the presence of A. Any thing in which B is not: absent in the presence of A, affords, we shall say, a confirming instance. The classes of disconfirming and confirming instances of the generalization that B is a necessary condition of A thus coincide with the classes of disconfirming and confirming instances of the generalization that all A are B. Let us assume that we can order the individuals in the range of significance of the characteristics A and B in a sequence x_{1},…, x_{n},… . Let "A_{n}" denote the (secondorder) property which a (firstorder) property has, if and only if, it is not absent in the presence of A in the thing x_{n}. The symbol "A_{n}(B)"^{ }then expresses the proposition that B is not absent in the presence of A in the thing x_{n}. The thing x_{n} in other words affords a confirming instance of the generalization Nc_{a}(B). Any characteristic of the same logical type as A and B will be called "an initially possible necessary condition of A". Let "τ" denote the tautological property in the universe of properties of the same type as A and B. The symbol "τ(B)" then expresses the proposition that B is an initially possible necessary condition of A. Consider the characteristics Nc_{a}, τ, A_{1},…, A_{n},… . For the conjunction of the first n members of the sequence A_{1}, ... we introduce the abbreviation "KA_{n}". It follows from the definition of a necessary condition that any characteristic which is a necessary condition of A is not absent in the presence of A in the first n things x_{1}, …, x_{n}, … . For,
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that a characteristic is a necessary condition of A means that it is never, in no thing, absent in the presence of A. Hence, for any n, the inclusionrelation Nc_{a} Ì KA_{n} necessarily holds. We can now apply the theorem, which we proved in the preceding section, to the above set of characteristics. Since the conjunction of τ and any other characteristic is logically identical with that other characteristic, we can omit "τ" from the expressions of such conjunctions. What the application of the theorem says is then the following: If, and only if, P(Nc_{A}/τ) > 0 and P(A_{n+}_{1}/KA_{n}) < 1, then P(Nc_{A}/KA_{n+}_{1}) > P(Nc_{A}/A_{n}). It is convenient to call P(Nc_{A}/τ), i. e. the probability that a random characteristic, such as e. g. B, is a (n actual) necessary condition of a characteristic A, given that it is an initially possible necessary condition of A, the probability a priori of the generalization that this characteristic is a necessary condition of A. If a random characteristic, such as e. g. B, is not absent in the presence of A in a certain individual, then this individual affords a confirming instance of the generalization that this characteristic is a necessary condition of A. P(A_{n+}_{1}/KA_{n}) thus is the probability that the next individual (of a certain Universe of Discourse) affords a confirming instance of the generalization that a certain characteristic, e. g. B, is a necessary condition of A, given that the n first individuals afford confirming instances of this generalization. It is, finally, natural to call P(Nc_{A}/KA_{n}), i. e. the probability that a random characteristic, such as e. g. B, is a necessary condition of a characteristic A, given that it is not absent in the presence of A in the first n individuals (of a certain Universe of Discourse), the probability a posteriori of this same generalization. ("A posteriori" here means: after n confirmations of the generalization.) Adopting the terminology which we just have introduced and applying the theorem which we have just proved to the characteristic B, we can give to our result the following formulation: If, and only if, the probability a priori of the generalization that B is a necessary condition of A is not 0 (minimal) and the
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probability that the thing x_{n}_{+1 }affords a confirming instance of it, given that the things x_{l}, …, x_{n }afford confirming instances, is not 1 (maximal), then the probability a posteriori of this generalization is greater relative to the n + 1 confirmations afforded by x_{l}, …, x_{n}_{+1 }than relative to the n confirmations afforded by x_{l}, …, x_{n}_{ }. 6. Assume that ~A(x_{n}_{+1}), i. e. assume that A is absent in the thing x_{n}_{+1}. Clearly, it must then be the case that, in this thing, the property B is not absent in the presence of the property A. (Since A itself is absent.) Hence x_{n}_{+1} affords a confirming instance of the generalization Nc_{A}(B). This will strike us as "paradoxical". How could any fact concerning a thing which is not A be relevant to a generalization concerning A's necessary conditions? It may, however, be shown that, if A is absent in the thing x _{n} _{+} i, then the confirmation afforded by this thing cannot contribute to the increase in probability of the generalization that B is a necessary condition of A (on the datum that B has not been absent in the presence of A in so and so many things). The proof is as follows: For reasons of logic it is true that (1) ~A(x_{n}_{+1}) → (A(x_{n}_{+1}) → B(x_{n}_{+1})). The consequent of this implication can also be written in the form A_{n}_{+1}(B). Since "B" does not occur in the antecedent, we can universalize the implication in the consequent. This gives us (2) ~A(x_{n}_{+1}) → (X) (A_{n}_{+1 }(X)). For reasons of logic it is, moreover, true that (3) (X) (A_{n}_{+1} (X)) → (X) (KA_{n }(X) → A_{n}_{+1} (X)). The consequent of this implication can also be written in the form KA_{n} Ì A_{n}_{+1}. From Axiom 1 of our probability calculus it follows that (4) (KA_{n} Ì A_{n}_{+1}) → P(A_{n}_{+1}/KA_{n}) = 1. From (2)–(4) and principles of logic we get (5) ~A(x_{n}_{+1}) → P(A_{n+}_{l}/KA_{n}) = 1. Herewith has been proved that the assumption that the thing x_{n}_{+1} is not A, together with principles of pure logic and probability theory, entails that it is maximally probable (= probable 264
to degree 1) that a random characteristic, such as e. g. B, is not absent in the presence of A in the thing x_{n}_{+1}, given that it is not absent in the presence of A in the things x_{1}, ..., x_{n}. Hence the confirmation afforded by x_{n}_{+1 }does not confirm the generalization under discussion "genuinely", but only in a "paradoxical" or "vacuous" manner. Let us next assume that B(x_{n}_{+1}), i. e. assume that B is present in the thing x_{n}_{+1}. Then, clearly, B is not absent in the presence of A in this thing. (Since, B, on the assumption, is present.) Hence x_{n}_{+1} necessarily affords a confirming instance of the generalization Nc_{A}(B). Can any fact concerning a thing which is B be relevant to the generalization (hypothesis) that B is a necessary condition of A? The answer is obviously affirmative. We would wish to know, whether the thing in question is A or not. If it is not A, we should regard the confirmation as worthless—and incapable of contributing to the probability of the generalization. If it is A, we should regard the confirmation as genuine—and capable of contributing to this probability. (The further question may then be raised, under which circumstances the thing which is B and also A will actually contribute to the increase in probability of the generalization that B is a necessary condition of A. This question, as far as I can see, cannot be answered on the basis of purely formal considerations of probability. But our "intuitive" feeling is, I think, that the thing in question will not contribute to the probability of the generalization, unless it eliminates from the class of initially possible necessary conditions of A some property, call it C, which has hitherto, i. e. in the things x_{1}, ..., x_{n}, not been absent in the presence of A, but which is now, i. e. in the thing x_{n}_{+1}, absent. As I have shown elsewhere,^{7} this idea of increasing the probability of a hypothesis through the elimination of concurrent hypotheses from the class of initially possible ones, can be reflected in a frequencymodel of the ConfirmationTheorem.) Thus the confirmation of the generalization that B is a necessary condition of A through a thing which is known to possess
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the property B before it is known whether it has or lacks also the property A, is not to be counted paradoxical on account of the fact alone that it is known to possess the property B. This is reflected on the formal level in the fact that the assumption B(x_{n}_{+1}), unlike the assumption ~A(x_{n}_{+1}), does not entail that P(A_{n}_{+1}/KA_{n}) = 1, i. e. that the conditions for increase in the probability a posteriori of the generalization are not satisfied.^{8} For, although B(x_{n}_{+1}) → (A(x_{n}_{+1}) → B(x_{n}_{+1})) is as good a tautology as ~A(x_{n}_{+1}) → (A(x_{n}_{+1}) → B(x_{n}_{+1}), we cannot universalize the consequent of the first formula in "B" as we can do with the consequent of the second formula. This is so, because "B" occurs in the antecedent of the first formula, but not in the antecedent of the second. Confirming instances, we said, of the generalization that B is a necessary condition of A are afforded by things which are either both A and B or B but not A or neither A nor B. We have found that things of the second and third categories afford confirmations which cannot affect the probability of the generalization in question, on the datum that B is an initially possible necessary condition of A which has so far never been absent in the presence of A in any thing. If by "genuine" confirmations of the generalization we mean its confirmation through instances which can affect this probability, we may conclude that only things of the first category, i. e. things which are both A and B, genuinely confirm the generalization that B is a necessary condition of A.
7. From the considerations in the last two sections nothing follows as regards the question which things confirm the generalization that all A are B "genuinely" and which things confirm it "paradoxically". At most could it be said that, if we "interpret" that all A are B as a generalization to the effect that B, among a number of rival possibilities, is a necessary condition of A, then the generalization is genuinely confirmed only by things which are both A and B. But the generalization that all A are B need
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not be regarded as a hypothesis about the necessary conditions of A (or about the sufficient conditions of B); and when it is not thus regarded, the answer to the question what confirms it genuinely and what only paradoxically will be different. Let us imagine a box or urn which contains a huge number of balls (spheres) and of cubes, but no other objects. Let us further think that every object in the urn is either black or white (all over). We put our hand into the urn and draw an object "at random". We note whether the drawn object is a ball or a cube and whether it is black or white. We repeat this procedure—without replacing the drawn objects—a number of times. We find that some of the cubes which we have drawn are black and some white. But all the balls which we have drawn are, let us assume, black. We now frame the generalization or hypothesis that all spherical objects in the box are black. In order to confirm or refute it we continue our drawings. The drawn object would disconfirm (refute) the generalization if it turned out to be a white ball. If it is a black ball or a white cube or a black cube, it confirms the generalization. Is any of these types of confirming instance to be pronounced worthless? It seems to me "intuitively" clear that all the three types of confirming instance are of value here and that no type of confirmation is not a "genuine" but only a "paradoxical" confirmation. (Whether confirmations of all three types are of equal value for the purpose of confirming the generalization may, however, be debated.) I would support this opinion by the following ("primitive") argument: What we are anxious to establish in this case is that no object in the box is white and spherical. Not knowing, whether there are or are not any white balls in the box, we run a risk each time when we draw an object from the box of drawing an object of the fatal sort, i. e. a white ball. Each time when the risk is successfully stood, we have been "lucky". We have been this, if the object which our hand happened to touch was a cube (and, since we could feel it was a cube, need not be examined for
267 colour at all); and we have been lucky, if the object was a ball which upon examination was found to be black. To touch a ball, one might say, is exciting, since our tension (fear of finding a white ball) is not removed until we have examined its colour. To touch a cube is not exciting at all, since it ipso facto removes the tension we might have felt. But to draw from the box is in any case exciting, since we do not know beforehand, whether we shall, to our relief, touch a cube, or touch a ball and, to our relief, find that it is black, or touch a ball and, to our disappointment, find that it is white. Let "S" be short for "spherical object in the box", "C" for "cubical object in the box", "B" for "black" and "W" for "white". All things in the box can be divided into the four mutually exclusive and jointly exhaustive categories of things which are S and B, S and W, C and B, and C and W. It is not connected with any air of paradoxicality to regard things of all the four types as relevant (positively or negatively) to the generalization that all S are B. All things in the world can be divided into the four mutually exclusive and jointly exhaustive categories of things which are S and B, S but not B, B but not S, and neither S nor B. Things of the first category obviously bear positively and things of the second category negatively on the generalization. But of the things of the third and fourth category some, we "intuitively" feel, do not bear at all on the generalization, have nothing to do with its content—and therefore "confirm" it only in a "paradoxical" way. The categories of things C & B and S & W differ from the categories of things ~S & B and ~S & ~B in this feature: all things of the first two categories are things in the box, but some things (in fact the overwhelming majority of things) of the last two categories are things outside the box. The things which we "intuitively" regard as affording "paradoxical" confirmations of the generalization that all S are B are those things of the 3rd and 4th category which are not things in the box. I shall introduce the term range of relevance (or of application) 268 of a generalization.^{9} I shall say that the range of relevance of the generalization that all spherical things in the box are black is the class of all things in the box. All things in the range of relevance of a generalization constitute genuine confirmations or disconfirmations of the generalization. The things outside the range neither confirm nor disconfirm the generalization genuinely. Since, however, they do not disconfirm it, we may "by courtesy" say that they confirm it, though only "paradoxically".—This sounds good common sense. We must now try to turn it into good logic as well. 8. My program is to connect considerations pertaining to the ranges of relevance of generalizations with considerations pertaining to the Principal Theorem of Confirmation and the probabilifying effect of instances. Let R be the range of relevance of a generalization to the effect that all A are B. R could, e. g., be the property of being an object in a certain box, A could be the property of being spherical, and B the property of being black. The generalization then says that all spheres in the box are black. Or R could be the property of being a bird, A the property of being a raven, and B the property of being black. Then the generalization is that all birds of the species raven are black. Here the genus bird is being regarded
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as the range of relevance of the generalization that all ravens are black. It seems to me that the generalizations which we make, in the sciences and in daily life, are usually not unrestrictedly about all things, but restrictedly about a range of things. Which this range is, however, may far from always be clearly conceived. I shall use the letter "F" as a name of the characteristic which a thing may be said to have by virtue of the fact that it falls under one of the three categories of things A&B or ~A&B or ~A&~B. ("F" is thus an abbreviation for "A&Bv~A&Bv~A&~B".) The proposition that, in the range R, all A are B may be understood as saying that having the characteristic F is a necessary condition of belonging to the range R. For example: The generalization that all spheres in a box are black may be understood as saying that, in order to be a thing in the box, the thing must not be a notblack sphere. Or: The generalization that all birds which are ravens are black may be understood as saying that, in order to be a bird, a thing must not be a notblack raven. This, admittedly, is not how we ordinarily express ourselves. But it is a possible mode of expression with a perfectly clear meaning. Consider now the hypothesis Nc_{R}(F), i. e. the generalization that, in the range R, all A are B. We can apply to Nc_{R}(F) our confirmation theorem and the condition which it lays down for the increase in the probability of Nc_{R}(F) through the accumulation of confirming instances. In particular, we can show that if a thing is not R, then, although this thing will necessarily afford a confirming instance of the generalization, the confirmation which it affords cannot possibly affect the probability of this generalization relative to the datum that in no thing so far has the property B been absent in the presence of the property A. This being so, we can regard the confirmation as not confirming the generalization genuinely but only paradoxically.
9. We are now in a position to answer such questions as this: Is it possible to confirm genuinely the generalization that all ravens are black through the observations, e. g., of black shoes 270
or white swans? The answer is that this is possible or not, depending upon which is the range of relevance of the generalization, upon what the generalization "is about". If, say, shoes are not within the range of relevance of the generalization that all ravens are black, then shoes cannot afford genuine confirmations of this generalization. This is so, because no truth about shoes can then affect the probability of the generalization that, in the range of relevance in question, all things which are ravens are black. So what is then the range of relevance of the generalization that all ravens are black? When we introduced the notion of a range of relevance and gave examples we spoke of the genus bird as a range of relevance of this generalization. But would it not be more correct to say that the range is the species raven rather than the genus bird? Are not ravens (and not birds) what the generalization that all ravens are black eminently "is about"? Here it should be noted that it is not clear "by itself" which is the range of relevance of a given generalization such as, e. g., that all ravens are black. Therefore it is not clear either which things will afford genuine and which only paradoxical confirmations. In order to tell this we shall have to specify the range. Different specifications of the range lead to so many different generalizations, one could say. The generalization that all ravens are black is a different generalization, when it is about ravens and ravens only, and when it is about birds and birds only, and when it is—if it ever is—about all things in the world unrestrictedly. As a generalization about ravens, only ravens are relevant to it, and not, e. g., swans. As a generalization about birds, swans are relevant to it, but not, e. g., shoes. And as a generalization about all things, all things are relevant—and this means: of no thing can it then be proved that the confirmation which it affords is maximally probable relative to the bulk of previous confirmations and therefore incapable of increasing the probability of the generalization. When the range of relevance of a generalization of the type that all A are B is not specified, then the range is usually understood to be the class of things which fall under the antecedent
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term A. And one may introduce a convention to the effect that an unspecified range should always be thus understood. The generalization that all ravens are black, range being unspecified, would then have to be understood to be a generalization about ravens—and not about birds or about animals or about everything there is. I shall call the class of things which are A the natural range of relevance of the generalization that all A are B. It would be a mistake to think that the natural range must be the range of relevance of a generalization. If it strikes one as odd or unplausible to regard the genus bird, rather than the species raven, as the range of relevance of the generalization that all ravens are black, this is probably due to the fact that the identification of birds as belonging to this or that species is comparatively easy. But imagine the case that species of birds were in fact very difficult to distinguish, that it would require careful examination to determine whether an individual bird was a raven or a swan or an eagle. Then the generalization that all birds which are ravens are black might be an interesting hypothesis about birds. Perhaps we can imagine circumstances too under which all things, blankets and shoes and what not, would be considered relevant to the generalization that all ravens are black. But these circumstances would be rather extraordinary. (We should have to think of ourselves as beings, who quasi put their hands into the universe and draw an object at random.)
10. The generalization that, in the range R, all A are B we have identified with the generalization Nc_{R}(F) where F is a property which a thing has, if and only if, it is not A & ~B. If, in particular, R is the same characteristic as A, then Nc_{R}(F) is the same generalization as Nc_{A}(B). For, if no thing is A, unless it is not: A but not B, then no thing is A, unless it is also B. And, if no thing is A, unless it is also B, then no thing is A, unless it is not: A but not B. Thus what we said in Sections 5 and 6 above, concerning genuine and paradoxical confirmations of the generalization that B is a necessary condition of A, is the same as that which we shall have to say concerning genuine and
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paradoxical confirmations of the generalization that all A are B within the latter generalization's natural range of relevance. As will be remembered, genuine confirmations of the generalization that B is a necessary condition of A are afforded only by things which are both A and B. In other words: Within the natural range of relevance of a generalization, the class of genuinely confirming instances is determined by Nicod's Criterion. But is this not in conflict with the Equivalence Condition? This condition, as will be remembered, says that what shall count as a confirming (or disconfirming) instance of a generalization cannot depend upon any particular way of formulating the generalization (of a number of logically equivalent formulations). Do we wish to deny then that the generalization that all A are B is the same generalization as that all notB are notA? We do not wish to deny that "all A are B" as a generalization about things which are A expresses the very same proposition as "all notB are notA" as a generalization about things which are A. Generally speaking: when understood relative to the same range of relevance, to say that all A are B and to say that all notB are notA is to say the same thing. But the generalization that all A are B with range of relevance A is a different generalization from the one that all notB are notA with range of relevance notB. If we agree that, range of relevance not being specified, a generalization should be taken relative to what I have called its "natural range", then we should also have to agree that, the ranges not being specified, the generalization that all A are B and the generalization that all notB are notA are different generalizations. They are different, because their "natural" ranges of relevance are different. This agrees, I believe, with how we "naturally" tend to understand the two formulations "all A are B" and "all notB are notA". Speaking in terms of ravens: The generalization that all ravens are black as a generalization about ravens, is different from the generalization that all things which are not black are things which are not ravens as a generalization about all notblack things. But the generalization that all ravens are black as a generalization about, say, birds is the very same as the generaliza
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tion that all things which are not black are not ravens as a generalization about birds. (For then "thing which is not black" means "bird which is not black".) Within its natural range of relevance, the generalization that all A are B can become genuinely confirmed only through things which are both A and B and is "paradoxically" confirmed through things which are B but not A, or neither A nor B. Within its natural range of relevance, the generalization that all notB are notA can become genuinely confirmed only through things which are neither A nor B and is "paradoxically" confirmed through things which are both A and B, or B but not A. Within the natural range of relevance, Nicod's Criterion of confirmation is necessary and sufficient. Within another specified range of relevance R, the generalization that all A are B may become genuinely confirmed also through things which are B but not A, or neither A nor B. And within the same range of relevance R, the class of things which afford genuine confirmations of the generalization that all A are B is identical with the class of things which afford genuine confirmations of the generalization that all notB are notA. Thus, in particular, if the range of relevance of both generalizations are all things whatsoever, i. e. the whole logical universe of things of which A and B can be significantly predicated, then everything which affords a confirming instance of the one generalization also affords a confirming instance of the other generalization, and vice versa, all confirmations being "genuine" and no "paradoxical". One has sometimes felt doubts, whether the formula (x)(A(x) → B(x)) of the lower functional calculus is adequate as a proposed symbolic expression ("formalization") of the sentence of ordinary language "all A are B". I think that we have found a reason for saying that the formula is not adequate. The reason is that the universal implication (x)(A(x) → B(x)) may be said to specify the range of relevance of the generalization involved (viz. that all A are B) to the whole universe of things of which A and B can be significantly predicated. This is indeed a possible specification of the range of relevance. But it is not the 274
only possible one. And it is different from the specification which may be said to be inherent in the form of words "all A are B". (It would be a gross misunderstanding of my intentions in this paper, if it were assumed that I have wanted to doubt the logical equivalence of the two formulae (x)(A(x) → B(x)) and (x)(~B(x) → ~A(x)) of the predicate calculus. It seems a plausible conjecture that, had the symbolism of the quantifiers not been invented and the formula (x)(A(x) → B(x)) come to be regarded as a formalization of the universal affirmative of Aristotelian logic, one would never have come to think of the Paradoxes of Confirmation as "paradoxes".

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