The epistemic conception of scientific progress equates progress with accumulation of scientific knowledge. I argue that the epistemic conception fails to fully capture scientific progress: theoretical progress, in particular, can transcend scientific knowledge in important ways. Sometimes theoretical progress can be a matter of new theories ‘latching better onto unobservable reality’ in a way that need not be a matter of new knowledge. Recognising this further dimension of theoretical progress is particularly significant for understanding scientific realism, since realism is naturally (...) construed as the claim that science makes theoretical progress. Some prominent realist positions are best understood in terms of commitment to theoretical progress that cannot be equated with accumulation of scientific knowledge. (shrink)

The epistemic conception of scientific progress equates progress with accumulation of scientific knowledge. I argue that the epistemic conception fails to fully capture scientific progress: theoretical progress, in particular, can transcend scientific knowledge in important ways. Sometimes theoretical progress can be a matter of new theories ‘latching better onto unobservable reality’ in a way that need not be a matter of new knowledge. Recognising this further dimension of theoretical progress is particularly significant for understanding scientific realism, since realism is naturally (...) construed as the claim that science makes theoretical progress. Some prominent realist positions are best understood in terms of commitment to theoretical progress that cannot be equated with accumulation of scientific knowledge. (shrink)

This paper describes a theory of accuracy or approximate truth and applies it to problems in the realist interpretation of scientific theories. It argues not only that realism requires approximate truth, but that an adequate theory of approximation also presupposes some elements of a realist interpretation of theories. The paper distinguishes approximate truth from vagueness, probability and verisimilitude, and applies it to problems of confirmation and deduction from inaccurate premises. Basic results are cited, but details appear elsewhere. Objections are surveyed, (...) including arguments by Miller, Laymon, and Laudan. Comparison is made with Niiniluoto's theory of verisimilitude, and the utility of his theory for realism assessed. (shrink)

I present a new definition of verisimilitude, framed in terms of causes. Roughly speaking, according to it a scientific model is approximately true if it captures accurately the strengths of the causes present in any given situation. Against much of the literature, I argue that any satisfactory account of verisimilitude must inevitably restrict its judgments to context-specific models rather than general theories. We may still endorse—and only need—a relativized notion of scientific progress, understood now not as global advance but rather (...) as the mastering of particular problems. This also sheds new light on longstanding difficulties surrounding language-dependence and models committed to false ontologies. (shrink)

The 1994 US spectrum auction is now a paradigmatic case of the successful use of microeconomic theory for policy-making. We use a detailed analysis of it to review standard accounts in philosophy of science of how idealized models are connected to messy reality. We show that in order to understand what made the design of the spectrum auction successful, a new such account is required, and we present it here. Of especial interest is the light this sheds on the issue (...) of progress in economics. In particular, it enables us to get clear on exactly what has been progressing, and on exactly what theory has – and has not – contributed to that. This in turn has important implications for just what it is about economic theory that we should value. (shrink)

I provide and defend a hyper-intensional account of verisimilitude within the truthmaker framework.

Some think that issues to do with scientific method are last century's stale debate; Popper was an advocate of methodology, but Kuhn, Feyerabend, and others are alleged to have brought the debate about its status to an end. The papers in this volume show that issues in methodology are still very much alive. Some of the papers reinvestigate issues in the debate over methodology, while others set out new ways in which the debate has developed in the last decade. The (...) book will be of interest to philosophers and scientists alike in the reassessment it provides of earlier debates about method and current directions of research. (shrink)

The philosophical significance of the procedure of applying Akaike Information Criterion (AIC) to curve-fitting problems is evaluated. The theoretical justification for using AIC (the so-called Akaike's theorem) is presented in a rigorous way, and its range of validity is assessed by presenting both instances in which it is valid and counter-examples in which it is invalid. The philosophical relevance of the justification that this result gives for making one particular choice between simple and complicated hypotheses is emphasized. In addition, recent (...) claims that the methods based on Akaike's theorem are relevant to other philosophical problems associated with the notion of simplicity are presented and evaluated. (shrink)

Value is either additive or else it is subject to organic unity. In general we have organic unity where a complex whole is not simply the sum of its parts. Value exhibits organic unity if the value of a complex, whether a complex state or complex quality, is greater or less than the sum of the values of its components or parts. Whether or not value is additive might be thought to be of purely metaphysical interest, but it is also (...) connected with important aspects of evaluative reasoning. Additivity is closely connected with principles of bare difference and separability which are often tacitly assumed in value theory. The author spells out these principles and trace their connections with additivity and organic unity. The author then develops an unpleasant paradox of additivity. Additivity apparently entails nihilism: that nothing is more valuable than anything else. Additivity involves a kind of axiological atomism -- that complexes decompose into components or factors; that these factors possess value independently of their role in valuable complexes; and that the factors do not interact in their production of overall value. In order to avoid the paradox it seems as though the factors have to be akin to the metaphysically privileged states of logical atomism -- a doctrine that does not enjoy widespread support. The paradox poses a problem not only for the notions of organic unity and additivity, but also for the closely related bare-difference principles which lie at the heart of value theory and of its application. The author proposes a way of eliminating the paradox, and thereby saving additivity and separability, without presupposing an unpalatable variant of logical atomism. The author closes with the proposal to treat principles of additivity as regulative ideals in our search for intrinsic values. (shrink)

An explicit philosophy and meta-philosophy of positivism, empiricism and popperianism is provided. Early popperianism is argued to be essentially a form of empiricism, the deviations from empiricism are traced. In contrast, the meta-philosophy and philosophy of an evolutionary naturalistic realism is developed and it is shown how the maximal conflict of this doctrine with all forms of empiricism at the meta-philosophical level both accounts for the form of its development at the philosophical level and its defense against attack from nonrealist (...) quarters. Following an earlier article on realism of similar theme (Synthese 26 (1974), 409) this paper then further explores the ramifications of a thoroughgoing realist position. (shrink)

This paper proposes a solution to David Miller's Minnesotan-Arizonan demonstration of the language dependence of truthlikeness (Miller 1974), along with Miller's first-order demonstration of the same (Miller 1978). It is assumed, with Peter Urbach, that the implication of these demonstrations is that the very notion of truthlikeness is intrinsically language dependent and thus non-objective. As such, truthlikeness cannot supply a basis for an objective account of scientific progress. I argue that, while Miller is correct in arguing that the number of (...) true atomic sentences of a false theory is language dependent, the number of known sentences (under certain straightforward assumptions) is conserved by translation; degree of knowledge, unlike truthlikeness, is thus a linguistically invariant notion. It is concluded that the objectivity of scientific progress must be grounded on the fact (noted in Cohen 1980) that knowledge, not mere truth, is the aim of science. (shrink)

I. A. Kieseppä's criticism of the methodological use of the theory of verisimilitude, and D. B. Resnik's arguments against the explanation of scientific method by appeal to scientific aims are critically considered. Since the notion of verisimilitude was introduced as an attempt to show that science can be seen as a rational enterprise in the pursuit of truth, defenders of the verisimilitude programme need to show that scientific norms can be interpreted as rules that try to increase the degree of (...) truthlikeness of scientific theories. This possibility is explored for several approaches to the problem of verisimilitude. (shrink)

Judgment aggregation problems are language dependent in that they may be framed in different yet equivalent ways. We formalize this dependence via the notion of translation invariance, adopted from the philosophy of science, and we argue for the normative desirability of translation invariance. We characterize the class of translation invariant aggregation functions in the canonical judgment aggregation model, which requires collective judgments to be complete. Since there are reasonable translation invariant aggregation functions, our result can be viewed as a possibility (...) theorem. At the same time, we show that translation invariance does have certain normatively undesirable consequences (e.g. failure of anonymity). We present a way of circumventing them by moving to a more general model of judgment aggregation, one that allows for incomplete collective judgments. (shrink)

This paper began with the simple object of finding an account that allowed us to compare incompatible false theories. This we achieved with ρ. But that relation is language — or interest — dependent. ρ' is free from this limitation; though thus liberated it is perhaps rather unconcerned about what is true, and further fails to deliver certain intuitive comparisons. Whether ρ is to be preferred to ρ' or vice versa, seems to me a largely fruitless question: In fact it (...) might be that as we seek to free the idea of verisimilitude from any dependence on language, interests or problems, so we gradually collapse it to nothing more than logical power. (shrink)

The problem of distance from the truth, and more generally distance between hypotheses, is considered here with respect to the case of quantitative hypotheses concerning the value of a given scientific quantity.Our main goal consists in the explication of the concept of distance D(I, ) between an interval hypothesis I and a point hypothesis . In particular, we attempt to give an axiomatic foundation of this notion on the basis of a small number of adequacy conditions.

This paper is both a survey and a review of the current state of the debate concerning verisimilitude. As a survey it is intended for the interested outsider who wants both easy access to and some comparison between the respective approaches. As a review it covers the first three books on the topic: those of Oddie. Niiniluoto and Kuipers.

Recent solutions to the curve-fitting problem, described in Forster and Sober ([1995]), trade off the simplicity and fit of hypotheses by defining simplicity as the paucity of adjustable parameters. Scott De Vito ([1997]) charges that these solutions are 'conventional' because he thinks that the number of adjustable parameters may change when the hypotheses are described differently. This he believes is exactly what is illustrated in Goodman's new riddle of induction, otherwise known as the grue problem. However, the 'number of adjustable (...) parameters' is actually a loose way of referring to a quantity that is not language dependent. The quantity arises out of Akaike's theorem in a way that ensures its language invariance. (shrink)

J. P. Z. Bonilla's methodological approach to truthlikeness is evaluated critically. On a more general level, various senses in which the theory of truthlikeness could be seen as a theory concerned with methodology are distinguished, and it is argued that providing speical sciences with methodological tools is unrealistic as an aim of the theory of verisimilitude. Rather, when developing this theory, one should rest contnet with the more modest aim of conceptual analysis, or of providing explications for the relational concept (...) of being closer to the truth. In addition, some remarks will be made on the difficulties which the similarity approach to truthlikeness has in realizing this aim and which are caused by the important role that Hintikka's constituents have in it. (shrink)

When can one say that a new theory is truer than the old one it contradicts, even though neither is absolutely true? We are primarily concerned with the case in which the conflicting theories offer answers to the same questions, and so we do not introduce considerations of "logical width". We propose that part of the new theory is truer than part of the old one when the former part gets right whatever the latter-part got right while the former does (...) not make any new mistakes. Pragmatic considerations will determine the relative importance of whatever new mistakes the new theory does make as a whole. To avoid artificial counterexamples, we restrict the parts compared to those that are "convex". A "convex" theory is one that holds in all cases intermediate between any two cases in which it holds. (shrink)

In this article, I explain how a variant of David Miller's argument concerning the language dependence of the accuracy of predictions can be applied to Joyce's notion of the accuracy of “estimates of numerical truth-values”. This leads to a potential problem for Joyce's accuracy-dominance-based argument for the conclusion that credences should obey the probability calculus.

The importance of the comparative notion of versimilitude, or truthlikeness, for a realist conception of knowledge follows from two modest ‘realist’ assumptions, namely, that the aim of an enquiry, as an enquiry, is the truth of some matter; and that one false theory may realize this aim better than another. However, there seem to be two ways in which one (false) theory can realize this aim better than another. One (false) theory can be closer to the truth than another either (...) by being preponderantly more accurate in its predictions or by providing more comprehensive information about the system (or class of systems) at issue. This paper presents a model-theoretic approach to the analysis of the comprehensiveness-related component of the comparative notion of versimilitude. The machinery of the ‘semantic’ view of theories is applied to the problem of providing necessary and sufficient conditions for the truth of sentences of the form, ‘B is truth-increasing with respect to A’, where A and B are taken to be sets of structures. (shrink)

The paper provides a formal proof that efficient estimates of parameters, which vary as as little as possible when measurements are repeated, may be expected to provide more accurate predictions. The definition of predictive accuracy is motivated by the work of Akaike (1973). Surprisingly, the same explanation provides a novel solution for a well known problem for standard theories of scientific confirmation — the Ravens Paradox. This is significant in light of the fact that standard Bayesian analyses of the paradox (...) fail to account for the predictive utility of universal laws like All ravens are black. (shrink)

A qualitative theory of truthlikeness, based on a family of quantitative measures, is developed for hypotheses that are concerned with the values of a finite number of real-valued quantities. Representing hypotheses by subsets of $R^{n}$ , I first show that a straightforward application of the basic ideas of the similarity approach to truthlikeness does not work out for hypotheses with zero n-dimensional Lebesgue measure. However, it is easy to give a counterpart for the average measure preferred by Pavel Tichý and (...) Graham Oddie in terms of Hausdorff measures. The task of finding analogies of the min-sum-measure preferred by Ilkka Niiniluoto is more interesting. I present and discuss analogies of two different types for this measure. (shrink)

David Miller has demonstrated to the satisfaction of a variety of philosophers that the accuracy of false quantitative theories is language dependent (cf. Miller 1975). This demonstration renders the accuracy-based mode of comparison for such theories obsolete. The purpose of this essay is to supply an alternate basis for theory comparison which in this paper is deemed the knowledge-based mode of quantitative theory comparison. It is argued that the status of a quantitative theory as knowledge depends primarily on the soundness (...) of the measurement procedure which produced the theory; such soundness is invariant, on my view, under Milleresque translations. This point is the basis for the linguistic invariance of knowledgelikeness. When the aim of science is not construed simply in terms of the truthlikeness or accuracy of theories, but in terms of the knowledge such theories embody, Miller's language dependence problem is overcome. One result of this analysis is that the possibility of objective scientific progress is restored, a possibility that Miller's analysis has prima facie defeated. (shrink)

Classical logic is explosive in the face of contradiction, yet we find ourselves using inconsistent theories. Mark Colyvan, one of the prominent advocates of the indispensability argument for realism about mathematical objects, suggests that such use can be garnered to develop an argument for commitment to inconsistent objects and, because of that, a paraconsistent underlying logic. I argue to the contrary that it is open to a classical logician to make distinctions, also needed by the paraconsistent logician, which allow a (...) more nuanced ranking of theories in which inconsistent theories can have different degrees of usefulness and productivity. Facing inconsistency does not force us to adopt an underlying paraconsistent logic. Moreover we will see that the argument to best explanation deployed by Colyvan in this context is unsuccessful. I suggest that Quinean approach which Colyvan champions will not lead to the revolutionary doctrines Colyvan endorses. (shrink)

D. Miller's demonstrations of the language dependence of truthlikeness raise a profound problem for the claim that scientific progress is objective. In two recent papers (Barnes 1990, 1991) I argue that the objectivity of progress may be grounded on the claim that the aim of science is not merely truth but knowledge; progress thus construed is objective in an epistemic sense. In this paper I construct a new solution to Miller's problem grounded on the notion of "approximate causal explanation" which (...) allows for linguistically invariant progress outside an epistemic context. I suggest that the notion of "approximate causal explanation" provides the resources for a more robust theory of progress than that provided by the notion of "approximate truth.". (shrink)

The ArgumentIdealizations and approximations are an indispensable tool for the scientist. This paper argues that idealizations and approximations are equally indispensable for the philosopher of science. In particular, it is shown that the deductive model of scientific theories is an idealization in precisely the same sense that frictionless motion is an idealization in mechanics. By its very nature, an idealization cannot be criticized as not being absolutely true to the facts, for it need not be. Thus, the usual type of (...) criticism levelled against the deductive model is shown to be irrelevant. The main step in the argument consists in using a logic of approximation to clarify the notions of approximate implication and approximate entailment, which are usually rejected as hopelessly vague. More generally, the conception underlying the paper is that the desiderata of science should be formulated in such a way that complying with them could be a matter of degree, and that pragmatic considerations could be taken into account. This conception is motivated by the wish to introduce some quantitative concepts in the philosophy of science, thereby bringing the philosopher's perspective closer to that of the scientist. (shrink)

Examples of hypotheses about the number of planets are frequently used to introduce the topic of (actual) truthlikeness but never analyzed in detail. In this paper we first deal with the truthlikeness of singular quantity hypotheses, with reference to several ‘the number of planets’ examples, such as ‘The number of planets is 10 _versus_ 10 billion (instead of 8).’ For the relevant ratio scale of quantities we will propose two, strongly related, normalized metrics, the proportional metric and the (simplest and (...) hence favorite) fractional metric, to express e.g. the distance from a hypothetical number to the true number of planets, i.e. the distance between quantities. We argue that they are, in view of the examples and plausible conditions of adequacy, much more appropriate, than the standardly suggested, normalized absolute difference, metric. Next we deal with disjunctive hypotheses, such as ‘The number of planets is between 7 and 10 inclusive is much more truthlike than between 1 and 10 billion inclusive.’ We compare three (clusters of) general ways of dealing with such hypotheses, one from Ilkka Niiniluoto, one from Pavel Tichý and Graham Oddie, and a trio of ways from Theo Kuipers. Using primarily the fractional metric, we conclude that all five measures can be used for expressing the distance of disjunctive hypotheses from the actual truth, that all of them have their strong and weak points, but that (the combined) one of the trio is, in view of principle and practical considerations, the most plausible measure. (shrink)

In this paper, I examine the possibility of accounting for the rationality of belief-formation by utilising decision-theoretic considerations. I consider the utilities to be used by such an approach, propose to employ verisimilitude as a measure of cognitive utility, and suggest a natural way of generalising any measure of verisimilitude defined on propositions to partial belief-systems, a generalisation which may enable us to incorporate Popper's insightful notion of verisimilitude within a Bayesian framework. I examine a dilemma generated by the decision-theoretic (...) procedure and consider an adequacy condition (immodesty) designed to ameliorate one of its horns. Finally, I argue in a sceptical vein that no adequate verisimilitude measure can be used decision-theoretically. (shrink)

L'article propose une analyse comparative des positions de Popper et de Kuhn sur quatre points liés : la comparaison des « cadres théoriques » ou des « paradigmes scientifiques » ; le pouvoir contraignant que sont susceptibles d'avoir les arguments invoqués dans les discussions critiques ; les éléments qui, en fin de compte, pèsent ou doivent peser sur les praticiens lorsqu'il s'agit d'élire un cadre théorique au détriment d'un autre ; et enfin, la conception qu'ont chacun des deux auteurs de (...) ce que Kuhn a nommé la « science normale ». On commence par discuter en détails les comptes-rendus dans lesquels Popper explique pourquoi les choix scientifiques de Galilée et de Kepler sont conformes à la méthodologie falsificationniste. On montre de là, premièrement, qu'à suivre à la lettre ces comptes-rendus, il faut conclure que les données expérimentales non problématiques – les énoncés de base unanimement acceptés – ne suffisent pas à elles seules à imposer en droit la falsification empirique (puisque sur la base des mêmes mesures de Tycho Brahé, Kepler a eu raison d'admettre que l'hypothèse des orbites circulaires était empiriquement réfutée, tandis que Galilée a eu raison d'admettre le contraire). On montre alors, deuxièmement, qu'au-delà des différences importantes entre Popper et Kuhn au niveau des lexiques et de l'idéal de scientificité, la méthodologie poppérienne, bien qu'elle se revendique « logique », ne donne concrètement pas plus que la conception kuhnienne pourtant si souvent stigmatisée comme « sociologique », lorsqu'il s'agit d'indiquer ce qu'un bon scientifique devrait faire, que ce soit en période « révolutionnaire » ou en période de « science normale ». (shrink)

Popper's account of refutation is the linchpin of his famous view that the method of science is the method of conjecture and refutation. This thesis critically examines his account of refutation, and in particular the practice he deprecates as avoiding a refutation. I try to explain how he comes to hold the views that he does about these matters; how he seeks to make them plausible; how he has influenced others to accept his mistakes, and how some of the ideas (...) or responses to Popper of such people are thus similarly mistaken. I draw some distinctions necessary to the provision of an adequate account of the so-called practice of avoiding a refutation, and try to rid the debate about this practice of at least one red herring. I analyse one case of 'avoiding' a refutation in detail to show how the rationality of scientific practice eludes both Popper and many of his commentators. Popper's skepticism about contingent knowledge prevents him from providing an acceptable account of contingent refutation, and so his method is really the method of conjecture and conjecture. He cannot do without the concepts of knowledge and refutation, however, if his account of science is to be plausible or persuasive, and so he equivocates between, amongst other things, refutation as disproof and refutation as the weaker notion of discorroboration. I criticise David Stove's account of this matter, in particular to show how he misses this point. An additional advantage Popper would secure from this equivocation is that if refutations were mere discorroborations they would be easier to achieve, and hence more common in science, than is the case. On Popper's weak notion of refutation, it would be possible to refute true theories since corroboration does not entail truth. There are two other related doctrines Popper holds about refutation which, if accepted, make some refutations seem easier to obtain than is the case. I call these doctrines 'Strong Popperian Falsificationism' (SPF) and 'Weak Popperian Falsificationism' (WPF). SPF is the false doctrine that if a prediction from some theory is refuted then that theory is refuted. Popper does not always endorse SPF. In particular, when confronted with a counterexample to it, he retreats to WPF, which is the false doctrine that if a prediction from some theory is refuted then that theory is prima facie refuted. WPF , or even SPF, can seem plausible if one has in mind predictions derived from theories in strong or conclusive tests of those theories, which I suggest Popper characteristically does. v Popper is disposed to describe any such case of predictive failure which does not lead to the refutation of the theory concerned as one in which that refutation has been avoided. To reinforce his portrayal of the refutation, or the attempted refutation, of major scientific theories as the rational core of scientific practice, Popper treats the so-called practice of avoiding a refutation as untypical of science, and much so-called avoidance he dismisses as unscientific or pseudo-scientific. I argue that his notion of avoiding a refutation is incoherent. Popper is further driven to believe that such avoidance is possible, however, because he conflates sentences with propositions and propositions with propositional beliefs. Also, he wishes to avoid being saddled with the relativisim that is a consequence of his weak account of refutation as discorroboration. Popper believes that ad hoc hypotheses are the most important of the unscientific means of avoiding a refutation. I argue that his account of such hypotheses is also incoherent, and that several hypotheses thought to be ad hoc in his sense are not. Such hypotheses appear to be so largely because of Popper's use of rhetoric and partly because these hypotheses are unacceptable for other reasons. I conclude that to know that a hypothesis is ad hoc in Popper's sense does not illuminate scientific practice. Popper has also attempted to explicate ad hocness in terms of some undesirable, or allegedly undesirable, properties of hypotheses or the explanations they would provide. The first such property is circularity, which is undesirable; the second such property is reduction in empirical content, which is not. In the former case I argue that non-circularity is clearly preferable to non-ad hocness as a criterion for a satisfactory explanation or explanans, as the case may be, and in the latter case that Popper is barking up the wrong tree. Some cases of so-called avoidance are obviously not unscientific. The discovery of Neptune from a prediction based on the reasonable belief that there were residual perturbations in the motion of Uranus is an important case in point, and one that is much discussed in the literature. The manifest failure of astronomers to account for Uranus's motion did not lead to the refutation of Newton's law of gravitiation, yet significant scientific progress obviously did result. Retreating to WPF, Popper claims that Newton's law was prima facie refuted. In general, astronomers have never shared this view, and they are correct in not doing so. I argue that the law of gravitation would have been prima facie refuted only if there had been good reason at the time to believe as false what is true, namely, that an unknown trans-Uranian planet was the cause of those Uranian residuals. Knowledge of the trans-Uranian region was then so slight that it was merely a convenient assumption, one which there was little reason to believe was false, that the known influences on Uranus's motion were the only such influences. I conclude that in believing vi or supposing that it was this assumption that was false, rather than the law of gravitation, Leverrier and Adams, the co-predictors of Neptune, were acting rationally and intelligently. Popper's commentators offer a variety of accounts of the alleged practice of avoiding a refutation, and of this case in particular. I analyse a sample of their accounts to show how common is the acceptance of some of Popper's basic mistakes, even amongst those who claim to reject his falsificationism, and to display the effects on their accounts of this acceptance of his mistakes. Many commentators recognize that anomalies are typically dealt with by changes in the boundary conditions or in other of the auxiliary propositions employed. Where many still go wrong, however, is in retaining the presupposition of WPF which encouraged Popper to hold the contradictory view about anomalies in the first place. Thus Imre Lakatos and others, for example, have developed a 'siege mentality' about major scientific theories; they see them as under continual threat of refutation from anomalies, and so come to believe that dogmatism is essential in science if such theories are to survive as they do. I examine various such doomed attempts to reconcile Popper with the history of science. It is a common failure in this literature to conflate or to fail to see the need to distinguish a belief from a supposition, and an epistemic reason from a pragmatic reason. I argue that only if one does draw these distinctions can one give an adequate account of how anomalies are rationally dealt with in science. The other important strand in Popper's thinking about 'avoidance' of refutation which has seriously misled some of his commentators is his unfounded belief in the dangers of ad hoc hypotheses. I examine the accounts that a sample of such commentators provide of the trans-Uranian planet hypotheses of Leverrier and Adams. These commentators imply or assert what Popper only hints at, namely, that there is something fishy about this hypothesis. I provide a further defence of the rationality of entertaining this hypothesis at the time. I conclude with a few remarks about Popper's dilemma in respect of scientific practice and his long standing emphasis on refutations. (shrink)

In Part I of this paper, I assumed we begin with a signature $$P = \{P_i\}$$ P = { P i } and the corresponding language $$L_P$$ L P, and introduced the following notions: a definition system$$d_{\Phi }$$ d Φ for a set of new predicate symbols $$Q_i$$ Q i, given by a set $$\Phi = \{\phi _i\}$$ Φ = { ϕ i } of defining $$L_P$$ L P -formulas \leftrightarrow \phi _i)$$ ∀ x ¯ ↔ ϕ i ) ); (...) a corresponding translation function$$\tau _{\Phi }: L_Q \rightarrow L_P$$ τ Φ : L Q → L P ; the corresponding definitional image operator$$D_{\Phi }$$ D Φ, applicable to $$L_P$$ L P -structures and $$L_P$$ L P -theories; and the notion of definitional equivalence itself: for structures $$A + d_{\Phi } \equiv B + d_{\Theta }$$ A + d Φ ≡ B + d Θ ; for theories, $$T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$$ T 1 + d Φ ≡ T 2 + d Θ. Some results relating these notions were given, ending with two characterizations for definitional equivalence. In this second part, we explain the notion of a representation basis. Suppose a set $$\Phi = \{\phi _i\}$$ Φ = { ϕ i } of $$L_P$$ L P -formulas is given, and $$\Theta = \{\theta _i\}$$ Θ = { θ i } is a set of $$L_Q$$ L Q -formulas. Then the original set $$\Phi $$ Φ is called a representation basis for an $$L_P$$ L P -structure A with inverse $$\Theta $$ Θ iff an inverse explicit definition $$\forall \overline{x} \leftrightarrow \theta _i)$$ ∀ x ¯ ↔ θ i ) is true in $$A + d_{\Phi }$$ A + d Φ, for each $$P_i$$ P i. Similarly, the set $$\Phi $$ Φ is called a representation basis for a $$L_P$$ L P -theory T with inverse $$\Theta $$ Θ iff each explicit definition $$\forall \overline{x} \leftrightarrow \theta _i)$$ ∀ x ¯ ↔ θ i ) is provable in $$T + d_{\Phi }$$ T + d Φ. Some results about representation bases, the mappings they induce and their relationship with the notion of definitional equivalence are given. In particular, we show that $$T_1$$ T 1 is definitionally equivalent to $$T_2$$ T 2, with respect to $$\Phi $$ Φ and $$\Theta $$ Θ, if and only if $$\Phi $$ Φ is a representation basis for $$T_1$$ T 1 with inverse $$\Theta $$ Θ and $$T_2 \equiv D_{\Phi }T_1$$ T 2 ≡ D Φ T 1. (shrink)