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Empirical progress and nomic truth approximation revisited
Studies in History and Philosophy of Science xxx (2014) xxx–xxx
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Empirical progress and nomic truth approximation revisited Theo A.F. Kuipers University of Groningen, c/o Platanenlaan 15, 2061 TP Bloemendaal, Netherlands
a r t i c l e
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Article history: Available online xxxx Keywords: Empirical progress Nomic truth approximation Truth content Falsity content Instrumentalism Realism Conceptual possibilities Nomic possibilities
a b s t r a c t In my From Instrumentalism to Constructive Realism (2000) I have shown how an instrumentalist account of empirical progress can be related to nomic truth approximation. However, it was assumed that a strong notion of nomic theories was needed for that analysis. In this paper it is shown, in terms of truth and falsity content, that the analysis already applies when, in line with scientific common sense, nomic theories are merely assumed to exclude certain conceptual possibilities as nomic possibilities. Ó 2014 Elsevier Ltd. All rights reserved.
When citing this paper, please use the full journal title Studies in History and Philosophy of Science
1. Introduction The intuitive idea underlying the notion of truth approximation can be expressed as follows: one theory is closer to the truth than another when the first says more things about the domain under investigation and more of all things said are true. The first formal definition of ‘closer to the truth’, or ‘more truthlike’ or ‘more verisimilar’, was proposed by Popper (1963). According to Popper, a theory is more truthlike than another if the former implies more true sentences and fewer false sentences than the latter. Notwithstanding its intuitive appeal, Popper’s definition was shown to be untenable by Tichy´ (1974) and Miller (1974), who independently proved that, according to this definition, a false theory can never be closer to the truth than another (true or false) theory. The Tichy´-Miller theorem opened the way to the post-Popperian approaches to truthlikeness, which have emerged since 1975. Such approaches escape the strictures pointed out by Tichy´ and Miller, allowing for a comparison of at least some false theories with regard to their closeness to the truth. Excellent surveys of most post-Popperian accounts of verisimilitude can be found in Niiniluoto (1998), Zwart (2001) and Oddie (2008).
In Kuipers (1982, 1984, 2000) I have developed the so-called nomic account. The intuitive idea underlying the notion of nomic truthlikeness can be expressed as follows. Given a domain of inquiry, let U be the set of all relevant conceptual possibilities which might occur within this domain. U may be construed as the conceptual frame of a given scientific inquiry, specifying the relevant kinds of objects, events or states of natural systems or artifacts under investigation. As an example, U may contain four kinds of object: ‘black raven’, ‘black non-raven’, ‘non-black raven’ and ‘non-black non-raven’. One may assume that there is a unique subset T of U, including precisely all nomic possibilities, i.e., all conceptual possibilities which are ‘really’ possible in the domain of inquiry. Here, ‘nomic possibilities’ may assume different meanings, depending on the particular context: e.g., T may concern the physical, chemical, biological, psychological or socio-economical possibilities of the domain. Of course, the set UT is then the set of the nomic impossibilities, i.e., the set of those conceptual possibilities which are impossible as a matter of fact. For example, T might contain the conceptual possibilities ‘black raven’, ‘black non-raven’, and ‘non-black non-raven’, whereas UT might contain the conceptual possibility ‘non-black raven’. Since theory-oriented
E-mail address: [email protected] URL: http://www.rug.nl/staff/t.a.f.kuipers http://dx.doi.org/10.1016/j.shpsa.2014.02.003 0039-3681/Ó 2014 Elsevier Ltd. All rights reserved.
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scientists usually aim at understanding and discovering the nomic features of the world, T can be construed as the target, or ‘the great unknown’, of scientific inquiry. Accordingly, T may be construed as the (whole) nomic truth about the domain and the nomic truthlikeness of a nomic theory may be defined as its similarity or nearness to the nomic truth T. The logical problem of nomic truthlikeness thus amounts to the explication of the idea that a given theory is closer to T than another. As long as T is unknown, solving the logical problem of nomic truthlikeness by an explication does not enable direct applications. The empirical problem of nomic truthlikeness is to relate the explication to empirical evidence in a convincing way, i.e., in such a way that we may have good empirical reasons to conclude, for the time being, that a new theory is closer to the nomic truth than an old one. That is, that we approximate the nomic truth better or more by the new one than by the old. It is plausible to try to link truth approximation in this tentative sense to (an explication of the idea of) empirical progress, a notion that is used by realists and instrumentalists. More specifically, explicating ‘empirical progress’ and ‘(nomic) truth approximation’ should do justice to some basic instrumentalist/empiricist and realist Conditions of Adequacy. CA-instrumentalist: the explication of ‘empirical progress’ (1) should at most make use of inductive steps and (2) should not be laden by realist notions, notably, ‘the truth’ and ‘closer to the truth’. CA-realist: the explication of ‘truth approximation’ and ‘empirical progress’ should be such that (1) ‘truth approximation’ explains ‘empirical progress’ and (2) ‘empirical progress’ supports the ‘truth approximation’-hypothesis. The first condition is important in order to convince instrumentalists that the realist intentions in the second condition pertain to their crucial notion of empirical progress. The notion of ‘estimated progress’ of Niiniluoto (1987, 2011) cannot work in this respect. It evidently does not satisfy the first condition, for it is defined as (the degree of) increase of estimated verisimilitude, a highly plausible, but typical realist notion for it is based on Niiniluoto’s wellknown quantitative measure of verisimilitude. After 30 years, I discovered in 2012 that my qualitative approach to nomic truth approximation and empirical progress (Kuipers, 1982, 1984, 2000) can be presented in a much more general way than I always thought. The definition of ‘closer to the truth’ can already be conceptually motivated by assuming that the claim of a theory only excludes certain conceptual possibilities as nomic possibilities, i.e. the ‘exclusion claim’. I always thought that the ‘inclusion claim’ had to be added that the not excluded possibilities were nomically possible. The new, simplified approach to nomic truth approximation was strongly stimulated by the related work of Gustavo Cevolani, Vincenzo Crupi and Roberto Festa (2011). In Section 2 the simplified story based on the exclusion claim will be presented in its ‘basic’ form, that is, the explication of nomic truth approximation and its relation to empirical progress assuming that there is just one language that generates the conceptual possibilities. In Section 3 the story will be concretized by taking the crucial (theory-relative) distinction between an observational language and a theoretical language into account. In this case, nomic truth approximation is of course explicated in terms of the theoretical language and empirical progress in terms of the observational language. Finally, in Section 4 several perspectives of the simplified account will be presented.
bulbs the language will have elementary propositions that enable to indicate which switches are on and which are off and also to indicate which bulbs give light and which do not. Among the conceptually possible states there will at any moment be one actual state, but several other states are also physically possible. To represent these states by one proposition or theory one will have to design a complex proposition, the nomic truth. All states in which this proposition is true are physically possible, all others are not. All propositions that can be formulated in the indicated language may be considered as candidates for being this nomic truth and, at least intuitively, one proposition may be closer to the truth than another. This may be a toy example for representing theory oriented science, but in present day epigenetics there is even a close analogy: genes are considered as switches that may be on or off. However this may be, only the general tenet of the example is relevant: theory oriented science is ultimately aiming at characterizing what is physically or biologically possible and theories are tested by experiments which are realizations of possibilities. Let U, as above, indicate the set of conceptual possibilities in a given context (e.g. the possible states of a system), generated by a descriptive vocabulary V in which U is, and subsets of U, e.g. X, Y, R, S, can be characterized. Let (bold) T indicate the subset of nomic, e.g. physical, possibilities, and hence cT (the complement of T, U T) the subset of nomic impossibilities. By the bold ‘T’ we indicate that we do not (yet) dispose of a characterization of it in terms of V. See Fig. 1. The target of research is identifying, if possible, T’s boundary in V-terms, indicated by (non-bold) T, hence T = T, assuming such a characterization exists, which I will do throughout in this paper. T will be called ‘the (explicit) truth’, for reasons that will become clear. In a nomic context attempts to characterize T in V-terms are primarily done by theories that exclude certain conceptual possibilities as nomic possibilities. Let theory X, or simply X, indicate a subset X of U, defined in V-terms, with the (exclusion) claim ‘‘T # X’’, or equivalently ‘‘cX # cT’’, i.e., all non-members of X are excluded as nomic possibilities. It is now plausible to define that X is true iff its claim is true, i.e. iff cX # cT, that is, iff cX cT = £, false otherwise. It is easy to see that there is at most one strongest true theory, called the true theory or the (explicit) truth, viz. the characterization of T in V-terms, if it exists, as before indicated by T, with non-bold ‘T’, i.e. the target of research! It is also plausible to define: the truth content of X, TC(X): the largest subset of cX which is also a subset of cT: cX \ cT, that is, the subset of those members of cX for which the claim cX # cT is true, the falsity content of X, FC(X): the largest subset of cX which is also a subset of T: cX \ T = cXcT, that is, the subset of those members of cX for which the claim cX # cT is false.
2. The basic story Let me start with my favorite toy example of theory oriented science. To represent an electric circuit with several switches and
Fig. 1. The set of conceptual possibilities U and the (unknown) subset of nomic possibilities T.
Please cite this article in press as: Kuipers, T. A. F. Empirical progress and nomic truth approximation revisited. Studies in History and Philosophy of Science (2014), http://dx.doi.org/10.1016/j.shpsa.2014.02.003
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Fig. 2. The truth content and the falsity content of theory X.
Note that the truth content of X as well as its falsity content are ‘outside’ X due to the fact that they derive from an exclusion claim. Their union equals the complement of X, cX, which I like to call the Popperian content of theory X, for it contains all conceptual possibilities that are excluded for the status of nomic possibility. Fig. 2 depicts the defined notions. Finally it is plausible to define: Y is at least as close to T, or to the truth, as X, iff: TC-clause:
the truth content of X is a subset of that of Y: cX \ cT # cY \ cT
TC(X) # TC(Y), that is, & FC-clause:
the falsity content of Y is a subset of that of X: cY cT # cX cT FC(Y) # FC(X), that is,
Y is closer to T, or to the truth, than X, iff: Y is at least as close to T as X & one of the clauses may be strengthened to a proper subset claim.
Of course, we speak of nomic truth approximation when a new theory is closer to the truth than an old one. It is easy to check that the TC- and the FC-clause of ‘at least as close’ are equivalent to: c (X [ T) # c (Y [ T) Y[T # X[T ? YT # XT
resp. resp. resp.
cY \ T # cX \ T TY # T–X T\X # T\Y
Hence, the single claim ‘‘cX # cT’’ already generates the two difference clauses (indicated by arrows), and hence the symmetric difference definition, viz. D(Y,T) # D(X,T) (Kuipers, 1982, 2000).1 To motivate the symmetric difference definition of ‘closer to the truth’ I always thought to also have to assume the ‘inclusion’ claim, X # T, hence together the ‘strong claim’: ‘‘X = T’’, but this turns out to be unnecessary. Consequently, the so far added ‘inclusion’ claim ‘‘X # T’’, leading to the claim ‘‘X = T’’, is totally redundant.2 Fig. 3 represents the situation that theory Y is closer to the truth than theory X as follows: shaded areas indicate empty sets and at least one of the two starred areas has to be non-empty. As is easy to check, the ///-shaded area is empty due to the TC-clause, which is equivalent to TC(X) TC(Y) = £, while the n n n-shaded area is empty due to the FC-clause, which is equivalent to FC(Y) FC(X) = £.
Fig. 3. Theory Y is closer to the truth than theory X.
So far, we have been engaged with the logical problem of verisimilitude, that is, explicating ‘being closer to the truth’, assuming that we dispose of the truth in one way or another. Now we turn to the epistemic problem of specifying empirical conditions that support the conclusion, at least for the time being, that one theory is closer to the unknown truth than another. As we will see, empirical progress is such a condition. In the nomic context the empirical data are asymmetric in the following sense. We can establish by experiments nomic possibilities. In fact, every experiment realizes by definition a nomic possibility. However, it is evident that we cannot establish nomic impossibilities in such a direct way. But in the empirical sciences we use to ‘induce’ nomic impossibilities indirectly by inductive (empirical) generalizations that we accept, for the time being, on the basis of ‘sufficient’ experimentation, notably by trying to realize counterexamples. For example, the observation of a black raven is the realization of a nomic possibility, whereas concluding that non-black ravens do not exist is an inductive generalization, viz. all ravens are black. We indicate the (asymmetric) data at a certain moment by R/S, where R indicates the set of realized nomic possibilities (e.g. realized physical possibilities) and S indicates the strongest law induced on the basis of R, that is, the set of conceptual possibilities that are not excluded by any of the accepted inductive generalizations. In this setup cS indicates the set of induced nomic impossibilities. Of course, we may always assume that R # S, for if not any element in R S would represent a realized counterexample to S and hence to at least one of its constituting inductive generalizations. The following definitions of accepted and rejected content are now plausible:
1 The symmetric difference of sets A and B, ADB, is standardly defined as (A B) [ (B A). It is easy to check that the conditions Y T # X T and T Y # T X together can be abbreviated to D(Y,T) # D(X,T). 2 Soon after 1982 I came to know of David Miller’s symmetric difference definition (Miller, 1978), which he however did not put in the nomic context, but in the context of ‘the actual or descriptive truth’.
Please cite this article in press as: Kuipers, T. A. F. Empirical progress and nomic truth approximation revisited. Studies in History and Philosophy of Science (2014), http://dx.doi.org/10.1016/j.shpsa.2014.02.003
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Fig. 4. The accepted content and the rejected content of theory X relative to evidence R/S.
the accepted content of X, AC(X): the set of ‘induced examples’ of X: cX \ cS = c(X [ S), that is, the subset of those members of cX which are also excluded by the accepted evidence, the rejected content of X, RC(X): the set of realized counterexamples of U: cX \ R = R X, that is, the subset of those members of cX for which the exclusion claim is rejected by the accepted evidence. Fig. 4 illustrates these notions, lying of course again outside X. Analogous to the previous truth-oriented comparative definitions, we can now define success-oriented comparative notions: Y is (empirically) at least as successful as X’, relative to R/S: AC-clause: the accepted content of X is a subset of that of Y: AC(X) # AC(Y), that is, cX \ cS # cY \ cS, or equivalently S[Y # S[X all induced examples of X are (induced) examples of Y, or equivalently all induced laws entailed by X are entailed by Y3 & RC-clause: the rejected content of Y is a subset of that of X: RC(Y) # RC(X), that is, R Y # R – X all realized counterexamples of Y are (realized) counterexamples of X
Fig. 5. Theory Y is more successful than theory X relative to evidence R/S.
Success Theorem: Assuming the Correct Data (CD-)hypothesis): (a) if Y is at least as close to the truth as X, Y is at least as successful, (b) if Y is closer to the truth than X, Y is or will become more successful. Proof (a): as suggested it is formally almost trivial to prove this; however, it is conceptually interesting to write out a proof in some detail. We start with a lemma, following from the CD-hypothesis. Lemma: R # T # S i.e., (CD-i) R # T & (CD-j) T # S, entails
Y is more successful than X, relative to R/S: Y is at least as successful as X & one of the clauses may be strengthened to a proper subset claim. Fig. 5 represents ‘theory Y is more successful than theory X’ as follows: shaded areas indicate empty sets, the -shaded area due to the AC-clause, which is equivalent to AC(X) AC(Y) = £, and the -shaded area due to the RC-clause, which is equivalent to RC(Y) RC(X) = £. Moreover, at least one of the two starred areas has to be non-empty. Now we can turn to the connection between ‘closer to the truth’ and ‘more successful’.4 To begin with, it is conceptually trivial that if R is correctly described and if S is correctly induced, R # T # S. This will be called the correct data (CD-) hypothesis, surely a very strong assumption, in particular the inductive component. However, now it is easy to prove the first clause of the following:
(i1)
RC(X) # FC(X):
cX cR # cX cT
(i2)
RC(Y) \ FC(X) # RC(X)
(cY cR) \ (cX cT) # cX cR
(j1)
AC(X) # TC(X):
cX \ cS # cX \ cT
(j2)
AC(X) \ TC(Y) # AC(Y):
(cX \ cS) \ (cY \ cT) # cY \ cS
In words: the accepted/rejected content of a theory is a subset of its truth/falsity content and the accepted/rejected content of one theory belonging to the truth/falsity content of another also belongs to the accepted/rejected content of that other. To prove (a), hence, assuming the CD-hypothesis and the TCand FC-clause, we get: AC(X) AC(X) & RC(Y) RC(Y)
# (j1) TC(X) # # AC(Y) # (i1) FC(Y) # # RC(X)
TC-clause
TC(Y)
and
hence,
by
(j2),
FC-clause
FC(X)
and
hence,
by
(i2),
3 Note first that cX \ cS = c(X [ S) and hence cX \ cS # cY \ cS is equivalent to c(X [ S) # c(Y [ S), hence to S [ Y # S [ X. Now if L is a superset of S [ X (S [ X # L) it is a superset of both S (S # L) and X (X # L). The former implies that L is at least implicitly an induced law, the latter that it is entailed by X. If S [ Y # S [ X, as assumed by the ACclause, any superset L of S [ X will be a superset of S [ Y, which explains the ‘induced law’ paraphrase of the clause. 4 From here on the section presents an adapted and simplified version of Sections 7.3.3 and 7.3.4 of Kuipers (2000).
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Fig. 6. A visualization of the proof of part (a) of the Success Theorem.
Proof (b): here we only give an informal indication. In view of the fact that Y is closer to the truth than X, there must exist a nomic possibility in (Y \ T) X or a nomic impossibility in X (Y [ T). Hence we may assume that by sufficient experimentation sooner or later such a nomic possibility in (Y \ T) X will be realized or such a nomic impossibility in X (Y [ T) will be induced (by accepting a new inductive generalization). To get a straightforward proof more specific conditions about the experimentation have to be assumed. Fig. 6 makes the proof of (a) visible. Assuming R # T # S, the emptiness of the simple shaded areas, representing ‘‘theory Y is at least as close to the truth as theory X’’, guarantees the emptiness of the areas shaded with double arrows, representing ‘‘theory Y is at least as successful as theory X’’. Now we are in the position to explicate the notion of empirical progress. If a new theory, Y, is more successful than the present one, X, relative to the available data R/S, this suggests the Empirical Progress (EP-)hypothesis, Y (is and) remains more successful than X, i.e., all (past) and future induced examples of X are examples of Y, all (past) and future realized counterexamples of Y are counterexamples of X, Y will have at least some extra induced examples, at least in the past, or X will have at least some extra realized counterexamples, at least in the past. Note that this is a comparative hypothesis, which can be tested by designing tests that may confirm or falsify the EP-hypothesis. In the latter case, it is a first illustration of ‘divided success’. The core idea of Empirical Progress now is the following: acceptance of the EP-hypothesis when it has been ‘sufficiently confirmed’ to be accepted as true, at least for the time being. Of course, being ‘sufficiently confirmed’ will always be a matter of debate. Recall that the explication of ‘empirical data’ and of ‘more successful’ only made use of descriptive and inductive steps. Moreover, the definition of Empirical Progress is based on two (comparative) inductive steps, viz. the two inductive generalizations of the EP-hypothesis. Finally, the definition does not depend on that of T, let alone on that of ‘closer to T’. Hence, the instrumentalist condition of adequacy (CA-instrumentalist) is satisfied. This brings us to the crucial point of my analysis: the connection between Truth Approximation (TA) and Empirical Progress (EP), i.e. the satisfaction of the realist condition of adequacy consisting of two requirements, TA should explain EP and EP should justify TA. 5
5
The first is simple, according to the Success Theorem, TA entails EP, hence it explains EP, at least in a basic sense. The immediate consequence of this is that EP suggests the Truth Approximation (TA-) hypothesis: Y is closer to the truth than X. The way of suggesting is straightforwardly abductive in the Peircean sense: given EP and given that TA explains EP, ‘‘hence, there is reason to suspect that [the TA-hypothesis] is true’’, to quote Peirce’s famous conclusion. The TA-hypothesis is to be tested by testing the EP-hypothesis, for it entails it. The EP-hypothesis even specifies all its empirical consequences. Moreover, the reverse consequences of the Success Theorem are such that EP not only suggests the TA-hypothesis, they also justify it to a substantial extent: 1. the Success Theorem not only makes it perfectly possible that is closer to the truth than , it would even explain the greater success, 2. it is impossible that Y is further from the truth than X (and hence X closer to the truth than Y), for otherwise, so shows the Success Theorem, Y could not be more successful, 3. it is also possible that Y is neither closer to nor further from the truth than X, in which case, however, another specific explanation has to be given for the fact that Y has so far proven to be more successful, e.g. by biased choice of experiments. Note that the TA-hypothesis provides a typical default explanation of EP, that is, an adequate explanation unless there turns out to be reason for another diagnosis. The third reverse consequence provides the room for ‘unless’ conditions and hence for future ‘divided success’. Our conclusion from the reverse consequences is that Empirical Progress justifies ‘‘Inference to the Best Theory (among the available theories) as the closest to the truth’’ (IBT), i.e. acceptance of the TA-hypothesis, at least for the time being. IBT is a sophisticated version of IBE (Inference to the Best Explanation), see Kuipers (2004). Note that IBT is not just an inductive generalization step, for it explicitly refers to (closer to) the truth. See Kuipers (2002) for when and how to take non-empirical, notably aesthetic, success criteria into account. In view of the foregoing analysis we may conclude that our explication of TA satisfies the realist condition of adequacy (CArealist). 3. The stratified story The stratified story is based on the assumption that there is a theory-relative distinction between observational and theoretical terms, that is, although the observational terms may well be laden with underlying theories, they are not laden with the theories we are interested in. Like the excluding nature of the claim of theories, this distinction of an observational and a theoretical level (o- and tlevel) is also typical for theory-oriented science.5 I will restrict the attention to the relevant notions of truth and falsity content, ‘closer to the truth’ and ‘more successful’ and the corresponding relations. Apart from some qualifications, the rest of the story remains the same: the EP-hypothesis, Empirical Progress, the TA-hypothesis and ‘inference to the best theory’. So, let there be for the t-level an encompassing vocabulary Vt with a sub-vocabulary Vo on the o-level, with the consequence that there are two universes or levels of conceptual possibilities, Ut and Uo, the first based on Vt, the second only on Vo. Of course, every member of Ut has a unique counterpart in Uo, but not vice
This section is heavily based on Kuipers (2000, Section 9.1.1, 2004, Section 3.1).
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Fig. 7. See text.
versa. Hence, let p indicate the projection function of members and subsets of Ut to Uo. For the projection of subsets X of Ut we will replace the standard notation p(X) by pX. From the realist point of view we assume that the primary target of research is some subset Tt, e.g. of nomic possibilities, with the aim to find a characterization in terms of Vt, i.e. Tt. Whether or not there is such a characterization, the realist will also assume that there is a subset To of Uo, if possible to be characterized in Vo, i.e. by some To. The instrumentalist has at most to assume that the target of research is To. The realist will claim that this must be pTt, i.e. To = pTt, and hence To = pTt. If To has no characterization in Vo, the instrumentalist will also be happy with an indirect characterization pTt, based on (non-bold) Tt, but of course merely as an instrument. Assuming theories are aiming at Tt, hence on the t-level, with the corresponding (weak) exclusion claim, there is again at most one strongest true theory, ‘the truth’, viz. Tt (non-bold). Hence, we can also take over the definitions of truth and falsity content, of ‘closer to the truth’ and of truth approximation; all with the adjective ‘theoretical’, when confusion might arise otherwise. On the o-level we can do all the same, now with the adjective ‘observational’. Moreover, the notions of ‘more successful’ and ‘empirical progress’ have now to be localized on this level. Hence, on the o-level the full basic story of Section 2 applies, including the connection between empirical progress and (observational) truth approximation. The crucial question is: what is the relation between theoretical and observational truth approximation and, hence, indirectly, what is the relation between theoretical truth approximation and empirical progress? To study this we need to focus on the relation between ‘closer to the truth’ on the two levels. It is not difficult to check that the TC-clause on the t-level is projected or preserved on the o-level. That is, if the TC of Y includes the TC of X, the TC of pY includes that of pX. This can be transformed into: ‘‘Y (X [ Tt) = £’’ implies ‘‘pY (pX [ pTt) = £’’, which is easy to check. However, the FC-clause is not simply preserved by projection. That is, if the FC of Y is a subset of the FC of X, the FC of
pY need not be a subset of the FC of pX. This can be transformed into: ‘‘(Tt \ X) Y = £’’ does not imply ‘‘(pTt \ pX) pY = £’’. It means that there exists the possibility of a ‘lucky hit’ of X relative to a counterexample of Y, that is, a counterexample to the claim that Y is at least as close to the truth as X on the observational level, despite the fact that Y is closer to the truth than X on the theoretical level. That is, there may be a theoretical (nomic) impossibility x, that is rightly excluded by Y, but not by X, however, such that there is also a theoretical possibility z, wrongly excluded by both, that has the same observational projection as x. A picture will be illuminating. Fig. 7 represents on the t-level ‘‘theory Y is at least as close to the truth as theory X’’ by two empty shaded areas. Regarding the TC-clause, it is easy to see that if the ///-shaded area on the t-level is empty, the projections are such that the similarly shaded area on the o-level must also be empty. Regarding the FC-clause, however, the emptiness of the n n n-shaded area on the t-level, leaves room for the non-emptiness of the formally corresponding area, witness the indicated possibility that the projections of x and z are the same, viz. a. Such potential lucky hits can be excluded in a useful general way by assuming that theory X is relatively correct, that is, p(X \ Tt) is not only, by the nature of projection, a subset of pX \ pTt, but also vice versa, and hence:
p(X \ Tt) = pX \ pTt
theory X is relatively correct
The name of the assumption is based on the fact that it precisely guarantees that (relative to pTt) correct items of theory pX (i.e., in the sense of correctly not excluded by theory pX) can be extended (relative to Tt, in the corresponding sense) to correct items of theory X. In other words, it guarantees that X has theoretically correct reproductions of all its observationally correct items. How strong is this assumption? It means something like: theory X is on the right track, as far as Vt relative to Vo is concerned. Apart from this, it is not a very strong condition, as argued in some detail on p. 213 of Kuipers (2000).
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Now it is easy to prove the following Projection Theorem: If theory Y is at least as close to the theoretical truth as theory X, and if X is relatively correct, then theory pY is at least as close to the observational truth as theory pY. Together with the Success Theorem we get a somewhat relativized claim about the consequence of theoretical truth approximation for more successfulness. Success Corollary: If theory Y is at least as close to the theoretical truth as theory X, and if X is relatively correct, then theory pY is always at least as successful as theory pY. For the opposite direction, we assume of course that empirical progress has been established in the sense described in Section 2, now on the o-level, hence it remains to satisfy the instrumentalist condition of adequacy (CA-instrumentalist). In the stratified situation empirical progress not only suggests the observational truth approximation (oTA-) hypothesis, but also the theoretical truth approximation (tTA-) hypothesis, having the same overall test implication, viz. the empirical progress (EP-) hypothesis, however now with the proviso that the discarded theory is relatively correct. Hence, the empirical progress conclusion now not only justifies ‘inference to the best theory (among the available theories) as the closest to the observational truth ‘, i.e. acceptance of the oTA-hypothesis, at least for the time being. It also justifies, though evidently in a weaker sense, ‘inference to the best theory as the closest to the theoretical truth’, i.e. acceptance of the tTA-hypothesis, at least for the time being. In which sense ‘weaker’ is difficult to spell out. However, it is at least possible to construe a case of theoretical truth approximation by some revision of the two theories.6 Note that potential lucky hits may wrongly block the theoretical truth approximation conclusion. It may occur that the AC (accepted content) of theory Y exceeds the AC of theory X, but that the former’s RC (rejected content) is not a subset of the latter’s RC, that is, that Y has extra counterexamples due to the fact that these items are lucky hits of X. This possibility gives reason for the realist to introduce the notion of one-sided or explanatory empirical progress: the AC of theory X is a proper subset of the AC of theory Y, that is, all induced laws entailed by X form a proper subset of the induced laws entailed by Y, even if the latter has extra counterexamples. The explanatory progress might be due to theoretical truth approximation, and the extra counterexamples might be cases of lucky hits. However this may be, since the Tt-hypothesis explains EP, at least when assuming relative correctness, and since EP supports the Tt-hypothesis, the realist condition (CA-realist) is also satisfied. So far for the stratified story. There is no reason to assume that the analysis of referential truth approximation given in Kuipers (2000) is influenced by replacing the strong claim of a theory by the exclusion claim. For the analysis of various forms of inference to the best theory and its relation to induction and abduction, see Kuipers (2004).
4. Perspectives In this section I will sketch a number of possible ramifications, extensions and connections with other fields.
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To begin with, it is important to note that ‘closer to the truth’ and ‘more successful’ are partial order relations. Hence, theories will frequently not be comparable in either way. There are at least two plausible way-outs. The standard way is to design a quantitative concretization. In a finite context it is even plausible to just count number of elements, leading to the quantitative symmetric difference definition of ‘closer to the truth’, viz. |D(Y,T)| 6 |D(X,T)|. However, as soon as U is infinite, ad hoc elements, e.g. weighing factors and other parameters, are unavoidable, witness Niiniluoto’s otherwise impressive approach (Niiniluoto, 1987). However, from a methodological point of view it seems more important to have a strategy to deal with cases of ‘divided success’, that is, when the one theory is more successful in some respects and the other in other respects. The qualitative ideal suggests to try to apply the ‘principle of dialectics’ in this situation, that is: try to improve both in one stroke. In other words, try to design a new theory that is and remains more successful than both, that is, try to achieve genuine empirical progress, and hence truth approximation. There are two plausible qualitative concretizations of the basic version presented in Section 2 (cf. Kuipers, 2000). In Section 3 we have already presented stratification with an observational and a theoretical level, which is more or less crucial for the realism/ instrumentalism debate. As we saw, it leads to some substantial weakening of the connection between empirical progress and truth approximation, but the connection remains remarkable. Another concretization is the refinement by an underlying ternary similarity relation, making one counterexample less severe than another. Below we will see that it is interesting to see the refined analysis of 2000 in the light of the present simplified background of the symmetric difference definition. The possibility of the simplified approach opens the way for an alternative. In fact, it is not difficult to check that if we replace in Section 2 the exclusion claim (T # X) by the inclusion claim (X # T) and proceed conceptually otherwise in the same way, we get formally the same definitions of ‘closer to the truth’ and ‘more successful’. However, the formal versions of the TC/ACclauses and the FC/RC-clauses have been interchanged. Hence, in the stratified version, the FC-clause is now projected, and the projection of TC-clause needs qualification. The consequence of these two roads to the symmetric difference definition is that we can distinguish three types of theories and methods of (nomic) truth approximation, leaving room for concretizations, viz. truth approximation by theories based on an exclusion claim, based on an inclusion claim, and combined or ‘two-sided’ theories. The latter are theories consisting of two subsets of U,, X1 being a subset of X2, with the two-sided claim X1 # T # X2. An extreme case of a two-sided theory results when X1 = X2 = X and hence the claim: X = T. This strong claim I wrongly assumed for 30 years (1982–2012) as necessary for conceptually motivating the symmetric difference definition. All three methods of truth approximation fit in the conjunctive approach to truth approximation (Cevolani et al., 2011; Kuipers, 2013), which is based on an idea of Roberto Festa. In (Cevolani, Festa, & Kuipers, 2013) it is already shown that my nomic approach (Kuipers, 2000) can be presented as a special, viz. strong, case of the conjunctive approach. Without stating this clearly, that paper in fact generalizes the nomic approach to two-sided theories, with on the one hand strong theories as extreme special case and on the other hand two kinds of one-sided theories, that is, theories with merely an inclusion or merely an exclusion claim, where the latter claim seems prototypical for a nomic claim. In Kuipers (2013) it is argued that, besides the nomic interpretation, several other interpretations of U/T make sense, notably the
X⁄ = df X-{x 2 X Y/p(x) 2 pY \ To} and Y⁄ = df X \ Y.
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monadic, the propositional, and the dichotomic interpretation. In the monadic interpretation the elements of U are interpreted as ‘Q-predicates’, i.e. predicates indicating (mutually exclusive and together exhaustive) kinds of objects. An exclusion claim now states that certain kinds are not instantiated in the relevant universe, whereas an inclusion claim states that certain (other) kinds are instantiated. As in the nomic interpretation, the evidence in the monadic case is by nature asymmetric: it can be shown directly that certain kinds are instantiated, but it can only indirectly, by inductive generalization, be concluded that certain other kinds are not instantiated. In the propositional interpretation the elements of U represent logically independent elementary propositions and an inclusion claim states that certain propositions are true and an exclusion claim that certain others are false. In this interpretation the evidence is most plausibly of symmetric nature: of certain elementary propositions it has been established that they are true and of others that they are false. Finally, in the dichotomic interpretation, there is a monadic predicate that divides U into two parts, e.g. equilibrium states and non-equilibrium states. Again the evidence is symmetric: by experiment it may be found out that such and such state is an equilibrium state whereas some other state is not. The exclusion method seems the most suitable for the nomic and monadic interpretation. In both cases, the method reflects what might be called ‘Popperian thinking’ and seems in accordance with scientific common sense. The inclusion method is also suitable for both; moreover, it seems suitable for domain revision. To begin with the former, inclusion theories seem to primarily concern theories that claim the physically possible existence of certain kinds of states or entities (Higgs particles) in the nomic context, and the actual existence of them in the monadic context. How and when empirical progress and truth approximation are achieved with these kinds of theories has still to be elaborated. In Kuipers (2006) I have shown that empirical progress and truth approximation is not only possible by theory revision, but also by domain revision. Of course, this was still based on the strong claim, but it is evident that, given a theory, represented by a set X, characterized in terms of V, the crucial claim about its intended domain of application I is of the form, ‘‘I # X’’, where I can be characterized in some sub-vocabulary of V. Hence, the claim that one domain of application (J) is ‘closer to the truth’ than another (I), now amounts, in its basic form, to the claim that the symmetric difference between J and X is a proper subset of that between I and X. As explained in the paper referred to, this is also relevant in the context of proving ‘interesting theorems’, where the ‘validity domain’ is initially unknown. One will start with claims about certain cases or situations for which the theorem can be proved. If successful one will try to prove it also for other, perhaps more realistic, cases. If unsuccessful, one will posit some other set of cases for which it can be proved. The two-sided method is suitable for propositional and dichotomic interpretation, assuming ‘symmetric data’. Again, how and when empirical progress and truth approximation are achieved with these kinds of theories has still to be elaborated. As already alluded to, in Kuipers (2000, chap. 10) I have presented a refined approach to empirical progress and (nomic) truth approximation based on an underlying ternary similarity relation, making one counterexample less severe than another. Of course, the point of departure was then the strong claim, combining the inclusion and the exclusion method in an extreme complementary way. From the present perspective, the refined definitions of ‘closer to the truth’ and ‘more successful’ seem to be primarily concretizations of the TC-/AC- and FC-/RC-clauses corresponding to inclusion theories, although it is possible to translate them, by taking suitable complements, as a concretization of the exclusion method. In the refined inclusion method new theories revise old theories
by changing the models of the theories to some extent, e.g. by taking new factors into account that have been neglected before. The new models are claimed to be more similar to ‘the true ones’ than those before. Hence, although the exclusion method may appeal to Popperian intuitions, revising theories in the way suggested also seems to reflect part of scientific common sense. Therefore, it is plausible to think that an adequate modeling of scientific common sense has to take into account both intuitions. This is perfectly possible from the present perspective: two-sided theories for which the basic TC-/AC- and FC-/RC-clauses are used for the exclusion subtheories, whereas the refined clauses are used for the inclusion subtheories. In terms of Zwart (2001), this is a way of combining the (Popperian) ‘content-approach’, with the ‘similarity approach’ to truth approximation, à la Niiniluoto (1987) and Oddie (1986). One may even speculate that Lakatosian research program thinking can be represented by such two-sided theories: progress is achieved by revising the corresponding inclusion theory by revising auxiliary hypotheses, however within the boundaries of the corresponding exclusion theory, forming the hard core. One of the relieving consequences of the possibility to base nomic truth approximation merely on the exclusion claim, is that my paper (Kuipers, 2011) in the special issue of Erkenntnis (75.2, 2011) on Belief Revision Aiming at Truth Approximation (ed. T. Kuipers and G. Schurz), entitled ‘‘Basic and refined nomic truth approximation by evidence-guided belief revision in AGM-terms’’ is not at all as ad hoc as I remarked at the end of that paper: after the revision of a theory by evidence according to the AGM-rules of belief revision, I wrongly thought to have to add the inclusion claim in order to prove nomic truth approximation. As a matter of fact, by withdrawing ‘the final step’, Kuipers (2011) presents ‘nomic truth approximation by belief revision’ for nomic exclusion theories, in a basic as well as a refined form. However, as has been demonstrated in Cevolani et al. (2011), starting from the conjunctive approach to truth approximation it is more plausible to dovetail it with belief base revision, i.e., belief revision as developed by Hansson (1999). This is presented in a very general way in Kuipers (2013) for two-sided theories and such that it is easy to derive how it can be restricted to one-sided theories. One remaining challenge is to show how these ‘basic’ versions can best be refined. The above indicated proposal to dovetail refined nomic truth approximation and belief revision was based on a refined form of AGM-belief revision, notably partial meet revision, using Adam Grove’s spheres approach (Grove, 1988) and Wlodek Rabinowizc’s similarity foundation of it (Rabinowicz, 1995). It seems plausible to try to use the same means to dovetail refined truth approximation (nomic and more generally) in their conjunctive form with belief base revision. Acknowledgements I like to thank Gustavo Cevolani, Roberto Festa, Jan Sprenger and two anonymous referees for their constructive remarks on the draft. With their consent, I start the introduction of the present paper with a few paragraphs derived from the joint paper with Cevolani and Festa (Cevolani et al., 2013), viz. of the beginning of the paper and of Section 2. References Cevolani, G., Crupi, V., & Festa, R. (2011). Verisimilitude and belief change for conjunctive theories. Erkenntnis, 75(2), 183–222. Cevolani, G., Festa, R., & Kuipers, T. (2013). Verisimilitude and belief change for nomic conjunctive theories. Synthese, 190, 3307–3324. Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17, 157–170. Hansson, S. O. (1999). A textbook of belief dynamics. Dordrecht: Kluwer. Kuipers, T. (1982). Approaching descriptive and theoretical truth. Erkenntnis, 18(3), 343–378.
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T.A.F. Kuipers / Studies in History and Philosophy of Science xxx (2014) xxx–xxx Kuipers, T. (1984). Approaching the truth with the rule of success. In P. Weingartner, & Chr. Pühringer (Eds.). Philosophy of Science—History of Science, Selection 7th LMPS Salzburg 1983, Philosophia Naturalis, 21, 2–4, 244– 253. Kuipers, T. (2000). From instrumentalism to constructive realism. Dordrecht: Kluwer AP. Kuipers, T. (2002). Beauty, a road to the truth. Synthese, 131(3), 291–328. Kuipers, T. (2004). Inference to the best theory, rather than inference to the best explanation. Kinds of abduction and induction. In F. Stadler (Ed.). Induction and Deduction in the Sciences (pp. 25–51), Proceedings of the ESF-workshop Induction and Deduction in the Sciences, Vienna, July, 2002. Dordrecht: Kluwer Academic Publishers. Kuipers, T. (2006). Theories looking for domains. Fact or fiction? Reversing structuralist truth approximation. In L. Magnani (Ed.). Model-based reasoning in science and engineering (pp. 33–50), Studies in logic, Vol. 2, College Publications. London: King’s College. Kuipers, T. (2011). Basic and refined nomic truth approximation by evidence-guided belief revision in AGM-terms. Erkenntnis, 75(2), 223–236. Kuipers, T. (2013). ‘‘Dovetailing belief base revision with (basic) truth approximation’’. In E. Weber, J. Meheus & D. Wouters (Eds.), Proceedings of the Logic, Reasoning and Rationality-Conference in Gent. Miller, D. (1974). Popper’s qualitative theory of verisimilitude. The British Journal for the Philosophy of Science, 25(2), 166–177.
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Miller, D. (1978). On distance from the truth as a true distance. In J. Hintikka et al. (Eds.), Essays on mathematical and philosophical logic (pp. 415–435). Dordrecht: Reidel. Niiniluoto, I. (1987). Truthlikeness. Dordrecht: Reidel. Niiniluoto, I. (1998). Verisimilitude: The third period. The British Journal for the Philosophy of Science, 49(1), 1–29. Niiniluoto, I. (2011). Scientific progress. In E. N. Zalta (Ed.). The Stanford encyclopedia of philosophy (Summer 2011 ed.).
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Empirical progress and nomic truth approximation revisited Theo A.F. Kuipers University of Groningen, c/o Platanenlaan 15, 2061 TP Bloemendaal, Netherlands
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Article history: Available online xxxx Keywords: Empirical progress Nomic truth approximation Truth content Falsity content Instrumentalism Realism Conceptual possibilities Nomic possibilities
a b s t r a c t In my From Instrumentalism to Constructive Realism (2000) I have shown how an instrumentalist account of empirical progress can be related to nomic truth approximation. However, it was assumed that a strong notion of nomic theories was needed for that analysis. In this paper it is shown, in terms of truth and falsity content, that the analysis already applies when, in line with scientific common sense, nomic theories are merely assumed to exclude certain conceptual possibilities as nomic possibilities. Ó 2014 Elsevier Ltd. All rights reserved.
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1. Introduction The intuitive idea underlying the notion of truth approximation can be expressed as follows: one theory is closer to the truth than another when the first says more things about the domain under investigation and more of all things said are true. The first formal definition of ‘closer to the truth’, or ‘more truthlike’ or ‘more verisimilar’, was proposed by Popper (1963). According to Popper, a theory is more truthlike than another if the former implies more true sentences and fewer false sentences than the latter. Notwithstanding its intuitive appeal, Popper’s definition was shown to be untenable by Tichy´ (1974) and Miller (1974), who independently proved that, according to this definition, a false theory can never be closer to the truth than another (true or false) theory. The Tichy´-Miller theorem opened the way to the post-Popperian approaches to truthlikeness, which have emerged since 1975. Such approaches escape the strictures pointed out by Tichy´ and Miller, allowing for a comparison of at least some false theories with regard to their closeness to the truth. Excellent surveys of most post-Popperian accounts of verisimilitude can be found in Niiniluoto (1998), Zwart (2001) and Oddie (2008).
In Kuipers (1982, 1984, 2000) I have developed the so-called nomic account. The intuitive idea underlying the notion of nomic truthlikeness can be expressed as follows. Given a domain of inquiry, let U be the set of all relevant conceptual possibilities which might occur within this domain. U may be construed as the conceptual frame of a given scientific inquiry, specifying the relevant kinds of objects, events or states of natural systems or artifacts under investigation. As an example, U may contain four kinds of object: ‘black raven’, ‘black non-raven’, ‘non-black raven’ and ‘non-black non-raven’. One may assume that there is a unique subset T of U, including precisely all nomic possibilities, i.e., all conceptual possibilities which are ‘really’ possible in the domain of inquiry. Here, ‘nomic possibilities’ may assume different meanings, depending on the particular context: e.g., T may concern the physical, chemical, biological, psychological or socio-economical possibilities of the domain. Of course, the set UT is then the set of the nomic impossibilities, i.e., the set of those conceptual possibilities which are impossible as a matter of fact. For example, T might contain the conceptual possibilities ‘black raven’, ‘black non-raven’, and ‘non-black non-raven’, whereas UT might contain the conceptual possibility ‘non-black raven’. Since theory-oriented
E-mail address: [email protected] URL: http://www.rug.nl/staff/t.a.f.kuipers http://dx.doi.org/10.1016/j.shpsa.2014.02.003 0039-3681/Ó 2014 Elsevier Ltd. All rights reserved.
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scientists usually aim at understanding and discovering the nomic features of the world, T can be construed as the target, or ‘the great unknown’, of scientific inquiry. Accordingly, T may be construed as the (whole) nomic truth about the domain and the nomic truthlikeness of a nomic theory may be defined as its similarity or nearness to the nomic truth T. The logical problem of nomic truthlikeness thus amounts to the explication of the idea that a given theory is closer to T than another. As long as T is unknown, solving the logical problem of nomic truthlikeness by an explication does not enable direct applications. The empirical problem of nomic truthlikeness is to relate the explication to empirical evidence in a convincing way, i.e., in such a way that we may have good empirical reasons to conclude, for the time being, that a new theory is closer to the nomic truth than an old one. That is, that we approximate the nomic truth better or more by the new one than by the old. It is plausible to try to link truth approximation in this tentative sense to (an explication of the idea of) empirical progress, a notion that is used by realists and instrumentalists. More specifically, explicating ‘empirical progress’ and ‘(nomic) truth approximation’ should do justice to some basic instrumentalist/empiricist and realist Conditions of Adequacy. CA-instrumentalist: the explication of ‘empirical progress’ (1) should at most make use of inductive steps and (2) should not be laden by realist notions, notably, ‘the truth’ and ‘closer to the truth’. CA-realist: the explication of ‘truth approximation’ and ‘empirical progress’ should be such that (1) ‘truth approximation’ explains ‘empirical progress’ and (2) ‘empirical progress’ supports the ‘truth approximation’-hypothesis. The first condition is important in order to convince instrumentalists that the realist intentions in the second condition pertain to their crucial notion of empirical progress. The notion of ‘estimated progress’ of Niiniluoto (1987, 2011) cannot work in this respect. It evidently does not satisfy the first condition, for it is defined as (the degree of) increase of estimated verisimilitude, a highly plausible, but typical realist notion for it is based on Niiniluoto’s wellknown quantitative measure of verisimilitude. After 30 years, I discovered in 2012 that my qualitative approach to nomic truth approximation and empirical progress (Kuipers, 1982, 1984, 2000) can be presented in a much more general way than I always thought. The definition of ‘closer to the truth’ can already be conceptually motivated by assuming that the claim of a theory only excludes certain conceptual possibilities as nomic possibilities, i.e. the ‘exclusion claim’. I always thought that the ‘inclusion claim’ had to be added that the not excluded possibilities were nomically possible. The new, simplified approach to nomic truth approximation was strongly stimulated by the related work of Gustavo Cevolani, Vincenzo Crupi and Roberto Festa (2011). In Section 2 the simplified story based on the exclusion claim will be presented in its ‘basic’ form, that is, the explication of nomic truth approximation and its relation to empirical progress assuming that there is just one language that generates the conceptual possibilities. In Section 3 the story will be concretized by taking the crucial (theory-relative) distinction between an observational language and a theoretical language into account. In this case, nomic truth approximation is of course explicated in terms of the theoretical language and empirical progress in terms of the observational language. Finally, in Section 4 several perspectives of the simplified account will be presented.
bulbs the language will have elementary propositions that enable to indicate which switches are on and which are off and also to indicate which bulbs give light and which do not. Among the conceptually possible states there will at any moment be one actual state, but several other states are also physically possible. To represent these states by one proposition or theory one will have to design a complex proposition, the nomic truth. All states in which this proposition is true are physically possible, all others are not. All propositions that can be formulated in the indicated language may be considered as candidates for being this nomic truth and, at least intuitively, one proposition may be closer to the truth than another. This may be a toy example for representing theory oriented science, but in present day epigenetics there is even a close analogy: genes are considered as switches that may be on or off. However this may be, only the general tenet of the example is relevant: theory oriented science is ultimately aiming at characterizing what is physically or biologically possible and theories are tested by experiments which are realizations of possibilities. Let U, as above, indicate the set of conceptual possibilities in a given context (e.g. the possible states of a system), generated by a descriptive vocabulary V in which U is, and subsets of U, e.g. X, Y, R, S, can be characterized. Let (bold) T indicate the subset of nomic, e.g. physical, possibilities, and hence cT (the complement of T, U T) the subset of nomic impossibilities. By the bold ‘T’ we indicate that we do not (yet) dispose of a characterization of it in terms of V. See Fig. 1. The target of research is identifying, if possible, T’s boundary in V-terms, indicated by (non-bold) T, hence T = T, assuming such a characterization exists, which I will do throughout in this paper. T will be called ‘the (explicit) truth’, for reasons that will become clear. In a nomic context attempts to characterize T in V-terms are primarily done by theories that exclude certain conceptual possibilities as nomic possibilities. Let theory X, or simply X, indicate a subset X of U, defined in V-terms, with the (exclusion) claim ‘‘T # X’’, or equivalently ‘‘cX # cT’’, i.e., all non-members of X are excluded as nomic possibilities. It is now plausible to define that X is true iff its claim is true, i.e. iff cX # cT, that is, iff cX cT = £, false otherwise. It is easy to see that there is at most one strongest true theory, called the true theory or the (explicit) truth, viz. the characterization of T in V-terms, if it exists, as before indicated by T, with non-bold ‘T’, i.e. the target of research! It is also plausible to define: the truth content of X, TC(X): the largest subset of cX which is also a subset of cT: cX \ cT, that is, the subset of those members of cX for which the claim cX # cT is true, the falsity content of X, FC(X): the largest subset of cX which is also a subset of T: cX \ T = cXcT, that is, the subset of those members of cX for which the claim cX # cT is false.
2. The basic story Let me start with my favorite toy example of theory oriented science. To represent an electric circuit with several switches and
Fig. 1. The set of conceptual possibilities U and the (unknown) subset of nomic possibilities T.
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Fig. 2. The truth content and the falsity content of theory X.
Note that the truth content of X as well as its falsity content are ‘outside’ X due to the fact that they derive from an exclusion claim. Their union equals the complement of X, cX, which I like to call the Popperian content of theory X, for it contains all conceptual possibilities that are excluded for the status of nomic possibility. Fig. 2 depicts the defined notions. Finally it is plausible to define: Y is at least as close to T, or to the truth, as X, iff: TC-clause:
the truth content of X is a subset of that of Y: cX \ cT # cY \ cT
TC(X) # TC(Y), that is, & FC-clause:
the falsity content of Y is a subset of that of X: cY cT # cX cT FC(Y) # FC(X), that is,
Y is closer to T, or to the truth, than X, iff: Y is at least as close to T as X & one of the clauses may be strengthened to a proper subset claim.
Of course, we speak of nomic truth approximation when a new theory is closer to the truth than an old one. It is easy to check that the TC- and the FC-clause of ‘at least as close’ are equivalent to: c (X [ T) # c (Y [ T) Y[T # X[T ? YT # XT
resp. resp. resp.
cY \ T # cX \ T TY # T–X T\X # T\Y
Hence, the single claim ‘‘cX # cT’’ already generates the two difference clauses (indicated by arrows), and hence the symmetric difference definition, viz. D(Y,T) # D(X,T) (Kuipers, 1982, 2000).1 To motivate the symmetric difference definition of ‘closer to the truth’ I always thought to also have to assume the ‘inclusion’ claim, X # T, hence together the ‘strong claim’: ‘‘X = T’’, but this turns out to be unnecessary. Consequently, the so far added ‘inclusion’ claim ‘‘X # T’’, leading to the claim ‘‘X = T’’, is totally redundant.2 Fig. 3 represents the situation that theory Y is closer to the truth than theory X as follows: shaded areas indicate empty sets and at least one of the two starred areas has to be non-empty. As is easy to check, the ///-shaded area is empty due to the TC-clause, which is equivalent to TC(X) TC(Y) = £, while the n n n-shaded area is empty due to the FC-clause, which is equivalent to FC(Y) FC(X) = £.
Fig. 3. Theory Y is closer to the truth than theory X.
So far, we have been engaged with the logical problem of verisimilitude, that is, explicating ‘being closer to the truth’, assuming that we dispose of the truth in one way or another. Now we turn to the epistemic problem of specifying empirical conditions that support the conclusion, at least for the time being, that one theory is closer to the unknown truth than another. As we will see, empirical progress is such a condition. In the nomic context the empirical data are asymmetric in the following sense. We can establish by experiments nomic possibilities. In fact, every experiment realizes by definition a nomic possibility. However, it is evident that we cannot establish nomic impossibilities in such a direct way. But in the empirical sciences we use to ‘induce’ nomic impossibilities indirectly by inductive (empirical) generalizations that we accept, for the time being, on the basis of ‘sufficient’ experimentation, notably by trying to realize counterexamples. For example, the observation of a black raven is the realization of a nomic possibility, whereas concluding that non-black ravens do not exist is an inductive generalization, viz. all ravens are black. We indicate the (asymmetric) data at a certain moment by R/S, where R indicates the set of realized nomic possibilities (e.g. realized physical possibilities) and S indicates the strongest law induced on the basis of R, that is, the set of conceptual possibilities that are not excluded by any of the accepted inductive generalizations. In this setup cS indicates the set of induced nomic impossibilities. Of course, we may always assume that R # S, for if not any element in R S would represent a realized counterexample to S and hence to at least one of its constituting inductive generalizations. The following definitions of accepted and rejected content are now plausible:
1 The symmetric difference of sets A and B, ADB, is standardly defined as (A B) [ (B A). It is easy to check that the conditions Y T # X T and T Y # T X together can be abbreviated to D(Y,T) # D(X,T). 2 Soon after 1982 I came to know of David Miller’s symmetric difference definition (Miller, 1978), which he however did not put in the nomic context, but in the context of ‘the actual or descriptive truth’.
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Fig. 4. The accepted content and the rejected content of theory X relative to evidence R/S.
the accepted content of X, AC(X): the set of ‘induced examples’ of X: cX \ cS = c(X [ S), that is, the subset of those members of cX which are also excluded by the accepted evidence, the rejected content of X, RC(X): the set of realized counterexamples of U: cX \ R = R X, that is, the subset of those members of cX for which the exclusion claim is rejected by the accepted evidence. Fig. 4 illustrates these notions, lying of course again outside X. Analogous to the previous truth-oriented comparative definitions, we can now define success-oriented comparative notions: Y is (empirically) at least as successful as X’, relative to R/S: AC-clause: the accepted content of X is a subset of that of Y: AC(X) # AC(Y), that is, cX \ cS # cY \ cS, or equivalently S[Y # S[X all induced examples of X are (induced) examples of Y, or equivalently all induced laws entailed by X are entailed by Y3 & RC-clause: the rejected content of Y is a subset of that of X: RC(Y) # RC(X), that is, R Y # R – X all realized counterexamples of Y are (realized) counterexamples of X
Fig. 5. Theory Y is more successful than theory X relative to evidence R/S.
Success Theorem: Assuming the Correct Data (CD-)hypothesis): (a) if Y is at least as close to the truth as X, Y is at least as successful, (b) if Y is closer to the truth than X, Y is or will become more successful. Proof (a): as suggested it is formally almost trivial to prove this; however, it is conceptually interesting to write out a proof in some detail. We start with a lemma, following from the CD-hypothesis. Lemma: R # T # S i.e., (CD-i) R # T & (CD-j) T # S, entails
Y is more successful than X, relative to R/S: Y is at least as successful as X & one of the clauses may be strengthened to a proper subset claim. Fig. 5 represents ‘theory Y is more successful than theory X’ as follows: shaded areas indicate empty sets, the -shaded area due to the AC-clause, which is equivalent to AC(X) AC(Y) = £, and the -shaded area due to the RC-clause, which is equivalent to RC(Y) RC(X) = £. Moreover, at least one of the two starred areas has to be non-empty. Now we can turn to the connection between ‘closer to the truth’ and ‘more successful’.4 To begin with, it is conceptually trivial that if R is correctly described and if S is correctly induced, R # T # S. This will be called the correct data (CD-) hypothesis, surely a very strong assumption, in particular the inductive component. However, now it is easy to prove the first clause of the following:
(i1)
RC(X) # FC(X):
cX cR # cX cT
(i2)
RC(Y) \ FC(X) # RC(X)
(cY cR) \ (cX cT) # cX cR
(j1)
AC(X) # TC(X):
cX \ cS # cX \ cT
(j2)
AC(X) \ TC(Y) # AC(Y):
(cX \ cS) \ (cY \ cT) # cY \ cS
In words: the accepted/rejected content of a theory is a subset of its truth/falsity content and the accepted/rejected content of one theory belonging to the truth/falsity content of another also belongs to the accepted/rejected content of that other. To prove (a), hence, assuming the CD-hypothesis and the TCand FC-clause, we get: AC(X) AC(X) & RC(Y) RC(Y)
# (j1) TC(X) # # AC(Y) # (i1) FC(Y) # # RC(X)
TC-clause
TC(Y)
and
hence,
by
(j2),
FC-clause
FC(X)
and
hence,
by
(i2),
3 Note first that cX \ cS = c(X [ S) and hence cX \ cS # cY \ cS is equivalent to c(X [ S) # c(Y [ S), hence to S [ Y # S [ X. Now if L is a superset of S [ X (S [ X # L) it is a superset of both S (S # L) and X (X # L). The former implies that L is at least implicitly an induced law, the latter that it is entailed by X. If S [ Y # S [ X, as assumed by the ACclause, any superset L of S [ X will be a superset of S [ Y, which explains the ‘induced law’ paraphrase of the clause. 4 From here on the section presents an adapted and simplified version of Sections 7.3.3 and 7.3.4 of Kuipers (2000).
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Fig. 6. A visualization of the proof of part (a) of the Success Theorem.
Proof (b): here we only give an informal indication. In view of the fact that Y is closer to the truth than X, there must exist a nomic possibility in (Y \ T) X or a nomic impossibility in X (Y [ T). Hence we may assume that by sufficient experimentation sooner or later such a nomic possibility in (Y \ T) X will be realized or such a nomic impossibility in X (Y [ T) will be induced (by accepting a new inductive generalization). To get a straightforward proof more specific conditions about the experimentation have to be assumed. Fig. 6 makes the proof of (a) visible. Assuming R # T # S, the emptiness of the simple shaded areas, representing ‘‘theory Y is at least as close to the truth as theory X’’, guarantees the emptiness of the areas shaded with double arrows, representing ‘‘theory Y is at least as successful as theory X’’. Now we are in the position to explicate the notion of empirical progress. If a new theory, Y, is more successful than the present one, X, relative to the available data R/S, this suggests the Empirical Progress (EP-)hypothesis, Y (is and) remains more successful than X, i.e., all (past) and future induced examples of X are examples of Y, all (past) and future realized counterexamples of Y are counterexamples of X, Y will have at least some extra induced examples, at least in the past, or X will have at least some extra realized counterexamples, at least in the past. Note that this is a comparative hypothesis, which can be tested by designing tests that may confirm or falsify the EP-hypothesis. In the latter case, it is a first illustration of ‘divided success’. The core idea of Empirical Progress now is the following: acceptance of the EP-hypothesis when it has been ‘sufficiently confirmed’ to be accepted as true, at least for the time being. Of course, being ‘sufficiently confirmed’ will always be a matter of debate. Recall that the explication of ‘empirical data’ and of ‘more successful’ only made use of descriptive and inductive steps. Moreover, the definition of Empirical Progress is based on two (comparative) inductive steps, viz. the two inductive generalizations of the EP-hypothesis. Finally, the definition does not depend on that of T, let alone on that of ‘closer to T’. Hence, the instrumentalist condition of adequacy (CA-instrumentalist) is satisfied. This brings us to the crucial point of my analysis: the connection between Truth Approximation (TA) and Empirical Progress (EP), i.e. the satisfaction of the realist condition of adequacy consisting of two requirements, TA should explain EP and EP should justify TA. 5
5
The first is simple, according to the Success Theorem, TA entails EP, hence it explains EP, at least in a basic sense. The immediate consequence of this is that EP suggests the Truth Approximation (TA-) hypothesis: Y is closer to the truth than X. The way of suggesting is straightforwardly abductive in the Peircean sense: given EP and given that TA explains EP, ‘‘hence, there is reason to suspect that [the TA-hypothesis] is true’’, to quote Peirce’s famous conclusion. The TA-hypothesis is to be tested by testing the EP-hypothesis, for it entails it. The EP-hypothesis even specifies all its empirical consequences. Moreover, the reverse consequences of the Success Theorem are such that EP not only suggests the TA-hypothesis, they also justify it to a substantial extent: 1. the Success Theorem not only makes it perfectly possible that
This section is heavily based on Kuipers (2000, Section 9.1.1, 2004, Section 3.1).
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Fig. 7. See text.
versa. Hence, let p indicate the projection function of members and subsets of Ut to Uo. For the projection of subsets X of Ut we will replace the standard notation p(X) by pX. From the realist point of view we assume that the primary target of research is some subset Tt, e.g. of nomic possibilities, with the aim to find a characterization in terms of Vt, i.e. Tt. Whether or not there is such a characterization, the realist will also assume that there is a subset To of Uo, if possible to be characterized in Vo, i.e. by some To. The instrumentalist has at most to assume that the target of research is To. The realist will claim that this must be pTt, i.e. To = pTt, and hence To = pTt. If To has no characterization in Vo, the instrumentalist will also be happy with an indirect characterization pTt, based on (non-bold) Tt, but of course merely as an instrument. Assuming theories are aiming at Tt, hence on the t-level, with the corresponding (weak) exclusion claim, there is again at most one strongest true theory, ‘the truth’, viz. Tt (non-bold). Hence, we can also take over the definitions of truth and falsity content, of ‘closer to the truth’ and of truth approximation; all with the adjective ‘theoretical’, when confusion might arise otherwise. On the o-level we can do all the same, now with the adjective ‘observational’. Moreover, the notions of ‘more successful’ and ‘empirical progress’ have now to be localized on this level. Hence, on the o-level the full basic story of Section 2 applies, including the connection between empirical progress and (observational) truth approximation. The crucial question is: what is the relation between theoretical and observational truth approximation and, hence, indirectly, what is the relation between theoretical truth approximation and empirical progress? To study this we need to focus on the relation between ‘closer to the truth’ on the two levels. It is not difficult to check that the TC-clause on the t-level is projected or preserved on the o-level. That is, if the TC of Y includes the TC of X, the TC of pY includes that of pX. This can be transformed into: ‘‘Y (X [ Tt) = £’’ implies ‘‘pY (pX [ pTt) = £’’, which is easy to check. However, the FC-clause is not simply preserved by projection. That is, if the FC of Y is a subset of the FC of X, the FC of
pY need not be a subset of the FC of pX. This can be transformed into: ‘‘(Tt \ X) Y = £’’ does not imply ‘‘(pTt \ pX) pY = £’’. It means that there exists the possibility of a ‘lucky hit’ of X relative to a counterexample of Y, that is, a counterexample to the claim that Y is at least as close to the truth as X on the observational level, despite the fact that Y is closer to the truth than X on the theoretical level. That is, there may be a theoretical (nomic) impossibility x, that is rightly excluded by Y, but not by X, however, such that there is also a theoretical possibility z, wrongly excluded by both, that has the same observational projection as x. A picture will be illuminating. Fig. 7 represents on the t-level ‘‘theory Y is at least as close to the truth as theory X’’ by two empty shaded areas. Regarding the TC-clause, it is easy to see that if the ///-shaded area on the t-level is empty, the projections are such that the similarly shaded area on the o-level must also be empty. Regarding the FC-clause, however, the emptiness of the n n n-shaded area on the t-level, leaves room for the non-emptiness of the formally corresponding area, witness the indicated possibility that the projections of x and z are the same, viz. a. Such potential lucky hits can be excluded in a useful general way by assuming that theory X is relatively correct, that is, p(X \ Tt) is not only, by the nature of projection, a subset of pX \ pTt, but also vice versa, and hence:
p(X \ Tt) = pX \ pTt
theory X is relatively correct
The name of the assumption is based on the fact that it precisely guarantees that (relative to pTt) correct items of theory pX (i.e., in the sense of correctly not excluded by theory pX) can be extended (relative to Tt, in the corresponding sense) to correct items of theory X. In other words, it guarantees that X has theoretically correct reproductions of all its observationally correct items. How strong is this assumption? It means something like: theory X is on the right track, as far as Vt relative to Vo is concerned. Apart from this, it is not a very strong condition, as argued in some detail on p. 213 of Kuipers (2000).
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Now it is easy to prove the following Projection Theorem: If theory Y is at least as close to the theoretical truth as theory X, and if X is relatively correct, then theory pY is at least as close to the observational truth as theory pY. Together with the Success Theorem we get a somewhat relativized claim about the consequence of theoretical truth approximation for more successfulness. Success Corollary: If theory Y is at least as close to the theoretical truth as theory X, and if X is relatively correct, then theory pY is always at least as successful as theory pY. For the opposite direction, we assume of course that empirical progress has been established in the sense described in Section 2, now on the o-level, hence it remains to satisfy the instrumentalist condition of adequacy (CA-instrumentalist). In the stratified situation empirical progress not only suggests the observational truth approximation (oTA-) hypothesis, but also the theoretical truth approximation (tTA-) hypothesis, having the same overall test implication, viz. the empirical progress (EP-) hypothesis, however now with the proviso that the discarded theory is relatively correct. Hence, the empirical progress conclusion now not only justifies ‘inference to the best theory (among the available theories) as the closest to the observational truth ‘, i.e. acceptance of the oTA-hypothesis, at least for the time being. It also justifies, though evidently in a weaker sense, ‘inference to the best theory as the closest to the theoretical truth’, i.e. acceptance of the tTA-hypothesis, at least for the time being. In which sense ‘weaker’ is difficult to spell out. However, it is at least possible to construe a case of theoretical truth approximation by some revision of the two theories.6 Note that potential lucky hits may wrongly block the theoretical truth approximation conclusion. It may occur that the AC (accepted content) of theory Y exceeds the AC of theory X, but that the former’s RC (rejected content) is not a subset of the latter’s RC, that is, that Y has extra counterexamples due to the fact that these items are lucky hits of X. This possibility gives reason for the realist to introduce the notion of one-sided or explanatory empirical progress: the AC of theory X is a proper subset of the AC of theory Y, that is, all induced laws entailed by X form a proper subset of the induced laws entailed by Y, even if the latter has extra counterexamples. The explanatory progress might be due to theoretical truth approximation, and the extra counterexamples might be cases of lucky hits. However this may be, since the Tt-hypothesis explains EP, at least when assuming relative correctness, and since EP supports the Tt-hypothesis, the realist condition (CA-realist) is also satisfied. So far for the stratified story. There is no reason to assume that the analysis of referential truth approximation given in Kuipers (2000) is influenced by replacing the strong claim of a theory by the exclusion claim. For the analysis of various forms of inference to the best theory and its relation to induction and abduction, see Kuipers (2004).
4. Perspectives In this section I will sketch a number of possible ramifications, extensions and connections with other fields.
6
7
To begin with, it is important to note that ‘closer to the truth’ and ‘more successful’ are partial order relations. Hence, theories will frequently not be comparable in either way. There are at least two plausible way-outs. The standard way is to design a quantitative concretization. In a finite context it is even plausible to just count number of elements, leading to the quantitative symmetric difference definition of ‘closer to the truth’, viz. |D(Y,T)| 6 |D(X,T)|. However, as soon as U is infinite, ad hoc elements, e.g. weighing factors and other parameters, are unavoidable, witness Niiniluoto’s otherwise impressive approach (Niiniluoto, 1987). However, from a methodological point of view it seems more important to have a strategy to deal with cases of ‘divided success’, that is, when the one theory is more successful in some respects and the other in other respects. The qualitative ideal suggests to try to apply the ‘principle of dialectics’ in this situation, that is: try to improve both in one stroke. In other words, try to design a new theory that is and remains more successful than both, that is, try to achieve genuine empirical progress, and hence truth approximation. There are two plausible qualitative concretizations of the basic version presented in Section 2 (cf. Kuipers, 2000). In Section 3 we have already presented stratification with an observational and a theoretical level, which is more or less crucial for the realism/ instrumentalism debate. As we saw, it leads to some substantial weakening of the connection between empirical progress and truth approximation, but the connection remains remarkable. Another concretization is the refinement by an underlying ternary similarity relation, making one counterexample less severe than another. Below we will see that it is interesting to see the refined analysis of 2000 in the light of the present simplified background of the symmetric difference definition. The possibility of the simplified approach opens the way for an alternative. In fact, it is not difficult to check that if we replace in Section 2 the exclusion claim (T # X) by the inclusion claim (X # T) and proceed conceptually otherwise in the same way, we get formally the same definitions of ‘closer to the truth’ and ‘more successful’. However, the formal versions of the TC/ACclauses and the FC/RC-clauses have been interchanged. Hence, in the stratified version, the FC-clause is now projected, and the projection of TC-clause needs qualification. The consequence of these two roads to the symmetric difference definition is that we can distinguish three types of theories and methods of (nomic) truth approximation, leaving room for concretizations, viz. truth approximation by theories based on an exclusion claim, based on an inclusion claim, and combined or ‘two-sided’ theories. The latter are theories consisting of two subsets of U,
X⁄ = df X-{x 2 X Y/p(x) 2 pY \ To} and Y⁄ = df X \ Y.
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monadic, the propositional, and the dichotomic interpretation. In the monadic interpretation the elements of U are interpreted as ‘Q-predicates’, i.e. predicates indicating (mutually exclusive and together exhaustive) kinds of objects. An exclusion claim now states that certain kinds are not instantiated in the relevant universe, whereas an inclusion claim states that certain (other) kinds are instantiated. As in the nomic interpretation, the evidence in the monadic case is by nature asymmetric: it can be shown directly that certain kinds are instantiated, but it can only indirectly, by inductive generalization, be concluded that certain other kinds are not instantiated. In the propositional interpretation the elements of U represent logically independent elementary propositions and an inclusion claim states that certain propositions are true and an exclusion claim that certain others are false. In this interpretation the evidence is most plausibly of symmetric nature: of certain elementary propositions it has been established that they are true and of others that they are false. Finally, in the dichotomic interpretation, there is a monadic predicate that divides U into two parts, e.g. equilibrium states and non-equilibrium states. Again the evidence is symmetric: by experiment it may be found out that such and such state is an equilibrium state whereas some other state is not. The exclusion method seems the most suitable for the nomic and monadic interpretation. In both cases, the method reflects what might be called ‘Popperian thinking’ and seems in accordance with scientific common sense. The inclusion method is also suitable for both; moreover, it seems suitable for domain revision. To begin with the former, inclusion theories seem to primarily concern theories that claim the physically possible existence of certain kinds of states or entities (Higgs particles) in the nomic context, and the actual existence of them in the monadic context. How and when empirical progress and truth approximation are achieved with these kinds of theories has still to be elaborated. In Kuipers (2006) I have shown that empirical progress and truth approximation is not only possible by theory revision, but also by domain revision. Of course, this was still based on the strong claim, but it is evident that, given a theory, represented by a set X, characterized in terms of V, the crucial claim about its intended domain of application I is of the form, ‘‘I # X’’, where I can be characterized in some sub-vocabulary of V. Hence, the claim that one domain of application (J) is ‘closer to the truth’ than another (I), now amounts, in its basic form, to the claim that the symmetric difference between J and X is a proper subset of that between I and X. As explained in the paper referred to, this is also relevant in the context of proving ‘interesting theorems’, where the ‘validity domain’ is initially unknown. One will start with claims about certain cases or situations for which the theorem can be proved. If successful one will try to prove it also for other, perhaps more realistic, cases. If unsuccessful, one will posit some other set of cases for which it can be proved. The two-sided method is suitable for propositional and dichotomic interpretation, assuming ‘symmetric data’. Again, how and when empirical progress and truth approximation are achieved with these kinds of theories has still to be elaborated. As already alluded to, in Kuipers (2000, chap. 10) I have presented a refined approach to empirical progress and (nomic) truth approximation based on an underlying ternary similarity relation, making one counterexample less severe than another. Of course, the point of departure was then the strong claim, combining the inclusion and the exclusion method in an extreme complementary way. From the present perspective, the refined definitions of ‘closer to the truth’ and ‘more successful’ seem to be primarily concretizations of the TC-/AC- and FC-/RC-clauses corresponding to inclusion theories, although it is possible to translate them, by taking suitable complements, as a concretization of the exclusion method. In the refined inclusion method new theories revise old theories
by changing the models of the theories to some extent, e.g. by taking new factors into account that have been neglected before. The new models are claimed to be more similar to ‘the true ones’ than those before. Hence, although the exclusion method may appeal to Popperian intuitions, revising theories in the way suggested also seems to reflect part of scientific common sense. Therefore, it is plausible to think that an adequate modeling of scientific common sense has to take into account both intuitions. This is perfectly possible from the present perspective: two-sided theories for which the basic TC-/AC- and FC-/RC-clauses are used for the exclusion subtheories, whereas the refined clauses are used for the inclusion subtheories. In terms of Zwart (2001), this is a way of combining the (Popperian) ‘content-approach’, with the ‘similarity approach’ to truth approximation, à la Niiniluoto (1987) and Oddie (1986). One may even speculate that Lakatosian research program thinking can be represented by such two-sided theories: progress is achieved by revising the corresponding inclusion theory by revising auxiliary hypotheses, however within the boundaries of the corresponding exclusion theory, forming the hard core. One of the relieving consequences of the possibility to base nomic truth approximation merely on the exclusion claim, is that my paper (Kuipers, 2011) in the special issue of Erkenntnis (75.2, 2011) on Belief Revision Aiming at Truth Approximation (ed. T. Kuipers and G. Schurz), entitled ‘‘Basic and refined nomic truth approximation by evidence-guided belief revision in AGM-terms’’ is not at all as ad hoc as I remarked at the end of that paper: after the revision of a theory by evidence according to the AGM-rules of belief revision, I wrongly thought to have to add the inclusion claim in order to prove nomic truth approximation. As a matter of fact, by withdrawing ‘the final step’, Kuipers (2011) presents ‘nomic truth approximation by belief revision’ for nomic exclusion theories, in a basic as well as a refined form. However, as has been demonstrated in Cevolani et al. (2011), starting from the conjunctive approach to truth approximation it is more plausible to dovetail it with belief base revision, i.e., belief revision as developed by Hansson (1999). This is presented in a very general way in Kuipers (2013) for two-sided theories and such that it is easy to derive how it can be restricted to one-sided theories. One remaining challenge is to show how these ‘basic’ versions can best be refined. The above indicated proposal to dovetail refined nomic truth approximation and belief revision was based on a refined form of AGM-belief revision, notably partial meet revision, using Adam Grove’s spheres approach (Grove, 1988) and Wlodek Rabinowizc’s similarity foundation of it (Rabinowicz, 1995). It seems plausible to try to use the same means to dovetail refined truth approximation (nomic and more generally) in their conjunctive form with belief base revision. Acknowledgements I like to thank Gustavo Cevolani, Roberto Festa, Jan Sprenger and two anonymous referees for their constructive remarks on the draft. With their consent, I start the introduction of the present paper with a few paragraphs derived from the joint paper with Cevolani and Festa (Cevolani et al., 2013), viz. of the beginning of the paper and of Section 2. References Cevolani, G., Crupi, V., & Festa, R. (2011). Verisimilitude and belief change for conjunctive theories. Erkenntnis, 75(2), 183–222. Cevolani, G., Festa, R., & Kuipers, T. (2013). Verisimilitude and belief change for nomic conjunctive theories. Synthese, 190, 3307–3324. Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17, 157–170. Hansson, S. O. (1999). A textbook of belief dynamics. Dordrecht: Kluwer. Kuipers, T. (1982). Approaching descriptive and theoretical truth. Erkenntnis, 18(3), 343–378.
Please cite this article in press as: Kuipers, T. A. F. Empirical progress and nomic truth approximation revisited. Studies in History and Philosophy of Science (2014), http://dx.doi.org/10.1016/j.shpsa.2014.02.003
T.A.F. Kuipers / Studies in History and Philosophy of Science xxx (2014) xxx–xxx Kuipers, T. (1984). Approaching the truth with the rule of success. In P. Weingartner, & Chr. Pühringer (Eds.). Philosophy of Science—History of Science, Selection 7th LMPS Salzburg 1983, Philosophia Naturalis, 21, 2–4, 244– 253. Kuipers, T. (2000). From instrumentalism to constructive realism. Dordrecht: Kluwer AP. Kuipers, T. (2002). Beauty, a road to the truth. Synthese, 131(3), 291–328. Kuipers, T. (2004). Inference to the best theory, rather than inference to the best explanation. Kinds of abduction and induction. In F. Stadler (Ed.). Induction and Deduction in the Sciences (pp. 25–51), Proceedings of the ESF-workshop Induction and Deduction in the Sciences, Vienna, July, 2002. Dordrecht: Kluwer Academic Publishers. Kuipers, T. (2006). Theories looking for domains. Fact or fiction? Reversing structuralist truth approximation. In L. Magnani (Ed.). Model-based reasoning in science and engineering (pp. 33–50), Studies in logic, Vol. 2, College Publications. London: King’s College. Kuipers, T. (2011). Basic and refined nomic truth approximation by evidence-guided belief revision in AGM-terms. Erkenntnis, 75(2), 223–236. Kuipers, T. (2013). ‘‘Dovetailing belief base revision with (basic) truth approximation’’. In E. Weber, J. Meheus & D. Wouters (Eds.), Proceedings of the Logic, Reasoning and Rationality-Conference in Gent. Miller, D. (1974). Popper’s qualitative theory of verisimilitude. The British Journal for the Philosophy of Science, 25(2), 166–177.
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Miller, D. (1978). On distance from the truth as a true distance. In J. Hintikka et al. (Eds.), Essays on mathematical and philosophical logic (pp. 415–435). Dordrecht: Reidel. Niiniluoto, I. (1987). Truthlikeness. Dordrecht: Reidel. Niiniluoto, I. (1998). Verisimilitude: The third period. The British Journal for the Philosophy of Science, 49(1), 1–29. Niiniluoto, I. (2011). Scientific progress. In E. N. Zalta (Ed.). The Stanford encyclopedia of philosophy (Summer 2011 ed.).
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