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Theories of truth approximation in terms of truthlikeness (or verisimilitude) almost always deal with (non-probabilistically) approaching deterministic truths, either actual or nomic. This paper deals first with approaching a probabilistic nomic truth, viz. a true probability distribution. It assumes a multinomial probabilistic context, hence with a lawlike true, but usually unknown, probability distribution. We will first show that this true multinomial distribution can be approached by Carnapian inductive probabilities. Next we will deal with the corresponding deterministic nomic truth, that is, the set of conceptually possible outcomes with a positive true probability. We will introduce Hintikkian inductive probabilities, based on a prior distribution over the relevant deterministic nomic theories and on conditional Carnapian inductive probabilities, and first show that they enable again probabilistic approximation of the true distribution. Finally, we will show, in terms of a kind of success theorem, based on Niiniluoto’s estimated distance from the truth, in what sense Hintikkian inductive probabilities enable the probabilistic approximation of the relevant deterministic nomic truth. In sum, the (realist) truth approximation perspective on Carnapian and Hintikkian inductive probabilities leads to the unification of the inductive probability field and the field of truth approximation.

Assuming that the target of theory oriented empirical science in general and of nomic truth approximation in particular is to characterize the boundary or demarcation between nomic possibilities and nomic impossibilities, I have presented, in my article entitled “Models, postulates, and generalized nomic truth approximation” (Kuipers in Synthese 193(10):3057–3077, 2016. https://doi.org/10.1007/s11229-015-0916-9), the ‘basic’ version of generalized nomic truth approximation, starting from ‘two-sided’ theories. Its main claim is that nomic truth approximation can perfectly be achieved by combining two prima facie opposing views on theories: (1) the traditional (Popperian) view: theories are (models of) postulates that exclude certain possibilities from being realizable, enabling explanation and prediction and (2) the model view: theories are sets of models that claim to (approximately) represent certain realizable possibilities. Nomic truth approximation, i.e. increasing truth-content and decreasing falsity-content, becomes in this way revising theories by revising their models and/or their postulates in the face of increasing evidence. The basic version of generalized nomic truth approximation is in many respects as simple as possible. Among other things, it does not take into account that one conceptual possibility may be more similar (or closer) to another than a third one (is to that other). However, for example, one theory may include a possibility that is more similar to a wrongly not included possibility than another theory can offer. Similarly, for wrongly not excluded possibilities. In this article it will be shown that such ‘refined’ considerations can be taken into account by adapted clauses based on a ternary similarity relation between possibilities (structures). This allows again abductive conclusions about refined truth approximation if a theory is persistently more successful in the refined sense than another. It will also be indicated and illustrated that this refined approach enables a specification to the effect that refined truth approximation can be obtained by the method of idealization and subsequent concretization. Finally, the basic and the refined approach will be evaluated with regard to some general principles and objections that have been discussed in the literature.

This chapter primarily deals with the conceptual prospects for generalizing the aim of abduction from the standard one of explaining surprising or anomalous observations to that of empirical progress or even truth approximation. It turns out that the main abduction task then becomes the instrumentalist task of theory revision aiming at an empirically more successful theory, relative to the available data, but not necessarily compatible with them. The rest, that is, genuine empirical progress as well as observational, referential and theoretical truth approximation, is a matter of evaluation and selection, and possibly new generation tasks for further improvement. The chapter concludes with a survey of possible points of departure, in AI and logic, for computational treatment of the instrumentalist task guided by the ‘comparative evaluation matrix’.

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 chapter 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. The resulting generalized and thereby simplified approach to nomic truth approximation was strongly stimulated by the related work on conjunctive theories of Gustavo Cevolani, Vincenzo Crupi and Roberto Festa (Erkenntnis, 2011), in particular their systematic way of defining the truth- and falsity-content of a theory. It will be shown that explicating ‘(nomic) truth approximation’ and ‘empirical progress’ in these terms does justice to some basic instrumentalist/empiricist and realist conditions of adequacy.

Straightforward theory revision, taking into account as effectively as possible the established nomic possibilities and, on their basis induced empirical laws, is conducive for nomic truth approximation. The question this paper asks is: is it possible to reconstruct the relevant theory revision steps, on the basis of incoming evidence, in the terms of so-called belief revision and, more specifically, in the terms of belief set revision, also called the AGM-approach? Assumin g exclusion theories, a positive answer will be given in two rounds, first for the case in which the initial theory is compatible with the established empirical laws, then for the case in which it is incompatible with at least one such a law.

Nomic truth approximation and its relation to confirmation and empirical progress were the leading themes of my book From instrumentalism to constructive realism of 2000. The present book offers again a synthesis of articles. It concerns in the first place six chapters ( 2, 3, 4, 5, 6, and 7) dealing with a very broad generalization of the theory of nomic truth approximation, discovered in 2012. Moreover it presents adapted and otherwise revised versions of nine previously published articles (8–16) about extensions, related kinds of reasoning, including belief revision, and the resulting epistemological position. This introductory chapter starts with a global description of this background. Then it presents the main conclusions of From instrumentalism to constructive realism, which roughly remain valid. This is followed by an overview of all chapters and next a list with references to the origin of them. It concludes with some indications for new research in order to compare the presented approach with other approaches.

Assuming that the target of theory oriented empirical science in general and of nomic truth approximation in particular is to characterize the boundary or demarcation between nomic possibilities and nomic impossibilities, I have presented in Chap. 4 (“Models, postulates, and generalized nomic truth approximation”) the ‘basic’ version of generalized nomic truth approximation, starting from ‘two-sided’ theories. Its main claim is that nomic truth approximation can perfectly be achieved by combining two prima facie opposing views on theories: (1) the traditional (Popperian) view: theories are (sets of models of) postulates that exclude certain possibilities from being realizable, enabling explanation and prediction and (2) the model view: theories are sets of models that claim to (approximately) represent certain realizable possibilities. Nomic truth approximation, i.e., increasing truth-content and decreasing falsity-content, becomes in this way revising theories by revising their models and/or their postulates in the face of increasing evidence.

The theory of nomic truth approximation essentially deals primarily with truth approximation by theory revision for a fixed domain. However, variable domains and a fixed theory can also be taken into account, where the main changes concern domain extensions and restrictions. In this chapter I will present a coherent set of definitions of ‘more truthlikeness’, ‘empirical progress’ and ‘truth approximation’ due to a revision of the ‘domain of intended applications’. This set of definitions seems to be the natural counterpart of the basic definitions of similar notions as far as theory revision is concerned.

Assuming that the target of theory oriented empirical science in general and of nomic truth approximation in particular is to characterize the boundary or demarcation between nomic possibilities and nomic impossibilities, I have presented in Chap. 4 (“Models, postulates, and generalized nomic truth approximation”) the ‘basic’ version of generalized nomic truth approximation, starting from ‘two-sided’ theories. Its main claim is that nomic truth approximation can perfectly be achieved by combining two prima facie opposing views on theories: (1) the traditional (Popperian) view: theories are (sets of models of) postulates that exclude certain possibilities from being realizable, enabling explanation and prediction and (2) the model view: theories are sets of models that claim to (approximately) represent certain realizable possibilities. Nomic truth approximation, i.e., increasing truth-content and decreasing falsity-content, becomes in this way revising theories by revising their models and/or their postulates in the face of increasing evidence.

Arguments for and against scientific realism usually presuppose as the main epistemic claim about theories that we may have good reasons to conclude that they are true, or at least approximately true. The antirealist charges against this claim are not easy to counter. In this chapter it is argued that the defense of realism is much easier if we relativize its epistemic claim in the light of (theories of) truth approximation. From this comparative realist perspective the main epistemic claim becomes that we may have good reasons to conclude that successor theories are closer to the truth than their predecessors. For example, although Einstein’s theory of general relativity may still be false, and not even approximately true, we have good reasons to assume that it is closer to the truth than Newton’s theory of gravitation. A similar relativization of claims that theoretical terms refer to things in the world is argued for in terms of ‘being closer to the referential truth’. For both purposes it is also plausible to relativize ‘being empirically successful’ to the comparative notion of ‘being (persistently) empirically more successful’.

Assuming that the target of theory oriented empirical science in general and of nomic truth approximation in particular is to characterize the boundary or demarcation between nomic possibilities and nomic impossibilities, I have presented in Chap. 4 (“Models, postulates, and generalized nomic truth approximation”) the ‘basic’ version of generalized nomic truth approximation, starting from ‘two-sided’ theories. Its main claim is that nomic truth approximation can perfectly be achieved by combining two prima facie opposing views on theories: (1) the traditional (Popperian) view: theories are (sets of models of) postulates that exclude certain possibilities from being realizable, enabling explanation and prediction and (2) the model view: theories are sets of models that claim to (approximately) represent certain realizable possibilities. Nomic truth approximation, i.e., increasing truth-content and decreasing falsity-content, becomes in this way revising theories by revising their models and/or their postulates in the face of increasing evidence.

The account of nomic truth approximation by empirical progress presented in Chap. 2 is based on the exclusion claim of theories. It is a simplified and generalized version of the one presented in my book From Instrumentalism to Constructive Realism of 2000. However, its core itself, called the basic account, is based on several idealizations. In this chapter I will sketch a number of possible concretizations, to begin with a two-sided, otherwise also basic, version, based on an exclusion and inclusion claim, followed by quantification, refinement and stratification of this two-sided version. They will be elaborated in Chaps. 4, 5, 6, and 7.

In this chapter I will draw a very general picture of an overall pragmatic theory of justification from the viewpoint of truth approximation. Crucial is the comparative definition of success according to which a theory P* is more successful than a theory P if and only if (1) P* shares all general successes of P, that is all empirically confirmed general implications, and has some extra ones, and (2) P shares all empirical counter-examples of P*. The rule of success says that if so far P* has been more successful than P, then – for the time being – accept P* as the better one and eliminate P. It is further argued that according to the pragmatic-instrumentalist view of scientific theories, theory elimination should exclusively be guided by the rule of success rather than by Popper’s rule of falsification – for it is pragmatically reasonable to accept the most successful theory available, even if this theory is known to be false. In the last part of the chapter, it is argued that the connection between pragmatic theory-success and truth approximation is otherwise rather close to Popper: at least ‘almost’ always, a theory which is closer to the truth will also turn out to be the pragmatically more successful theory. However, in the upshot, ‘being false’ and hence ‘being falsified’ are not very important for truth approximation.

The result of a test of a general hypothesis can be positive, negative or neutral. The first, qualitative, task of confirmation theory is to explicate these types of test result. However, as soon as one also takes individual hypotheses into consideration, the interest shifts to the second, quantitative, task of confirmation theory: probabilistically evaluating individual and general hypotheses in the light of an increasing number of test results. This immediately suggests conceiving of the confirmation of an hypothesis as increasing its probability due to new evidence. Rudolf Carnap initiated a research programme in quantitative confirmation theory by designing a continuum of probability systems with plausible probabilistic properties for the hypothesis that the next test result will be of a certain kind. This continuum of inductive systems has guided the search for optimum systems and for systems that take analogy into account. Carnapian systems, however, assign zero probability to universal hypotheses. Jaakko Hintikka was the first to reconsider the confirmation of such hypotheses and using Carnap’s continuum for this purpose has set the stage for a whole spectrum of inductive systems of this type.

Could some evidence confirm a conjunction of two hypotheses more than it confirms either of the hypotheses separately? We show that it might, moreover under conditions that are the same for ten different measures of confirmation. Further, we demonstrate that it is even possible for the conjunction of two disconfirmed hypotheses to be confirmed by the same evidence.

Recently, Gustavo Cevolani, Vincenzo Crupi and Roberto Festa have shown that their account of verisimilitude of ‘conjunctive theories’ of a finite propositional language can be nicely linked to a variant of AGM belief set revision, viz. belief base revision, in the sense that the latter kind of revision is functional for truth approximation according to the conjunctive account (Cevolani et al., Erkenntnis 75(2):183–222, 2011). In the present paper I offer a generalization of these ideas to the case of approaching any divide of a (finite or infinite) universe, allowing several interpretations, besides true (false) atomic propositions, notably nomic states (not) in equilibrium, nomic (im)possibilities, (non-)instantiated ‘Q-predicates’ of a monadic language. It shows how and why approximation of ‘the true boundary’ takes place by belief base revision guided by evidence.

This paper primarily deals with theconceptual prospects for generalizing the aim ofabduction from the standard one of explainingsurprising or anomalous observations to that ofempirical progress or even truth approximation. Itturns out that the main abduction task then becomesthe instrumentalist task of theory revision aiming atan empirically more successful theory, relative to theavailable data, but not necessarily compatible withthem. The rest, that is, genuine empirical progress aswell as observational, referential and theoreticaltruth approximation, is a matter of evaluation andselection, and possibly new generation tasks forfurther improvement. The paper concludes with a surveyof possible points of departure, in AI and logic, forcomputational treatment of the instrumentalist taskguided by the `comparative evaluation matrix'.

En reponse a la critique de son article developpee par T. Mormann («The refined qualitative theory of thruth approximation does not deliver: remarks on Kuipers», in «ERkenntnis», 47, 2, 1997, pp. 145-79), l'A. montre que la version raffinee de la theorie de l'approximation qualitative de la verite qu'il defend ne depend ni du nombre de structures constituant les theories ni des distances reelles entre ces structures. Examinant les propositions de Niiniluoto sur la vraisemblance et les probabilites posterieures, l'A. defend son approche comparative de l'approximation qualitative avec l'approximation naive comprise comme paradigme de la relation de structuralite

Publisher Summary This chapter discusses laws, theories, and research programs. In the empirical sciences, the informal distinction between observational laws and proper theories plays a crucial role. Observational laws are supposed to describe observationally—usually experimentally—established regularities. Different names for roughly the same concept are: empirical, experimental or phenomenological laws, reproducible effects, inductive generalizations, and general facts. Proper theories or systems of theoretical laws (together with definitions and other conventions), on the other hand, are supposed to explain such laws and to predict new ones, by postulating underlying mechanisms. The law-distinction forms a crucial construction principle for the hierarchy of knowledge. Therefore, it is an important heuristic factor in the dynamics of knowledge development. However, it has occasioned philosophers of science much brain-racking to explicate the law-distinction in a defensible way. Without doubt, the distinction is strongly related to the distinction between observational (or empirical or experimental) and theoretical terms. Whereas proper theories introduce theoretical terms, observational laws do not.

The philosophy of science has lost its self-confidence. Structures in Science (2001) is an advanced textbook that explicates, updates and integrates the best insights of logical empiricism and its main critics. This “neo-classical approach” aims at providing heuristic patterns for research.The book introduces four ideal types of research programs (descriptive, explanatory, design and explicative) and reanimates the distinction between observational laws and proper theories without assuming a theory-free language. It explicates various patterns of explanation by subsumption and specification as well as structures in reductive and other types of interlevel research. Its threefold analysis of theory evaluation leads to new characterizations of confirmation, empirical progress, and truth approximation. What emerges are partial analogies between progress in nomological research, presented in detail in From Instrumentalism to Constructive Realism (2000) and progress in explicative and design research. Finally, special chapters are devoted to design research programs, computational philosophy of science, the structuralist approach to theories, and research ethics. The present synopsis of Structures in Science highlights the main topics, the final emphasis being on design research and research ethics.

Surprisingly enough, modified versions of the confirmation theory of Carnap and Hempel and the truth approximation theory of Popper turn out to be smoothly synthesizable. The glue between confirmation and truth approximation appears to be the instrumentalist methodology, rather than the falsificationist one. By evaluating theories separately and comparatively in terms of their successes and problems (hence even if they are already falsified), the instrumentalist methodology provides - both in theory and in practice - the straight route for short-term empirical progress in science in the spirit of Laudan. However, it is argued that such progress is also functional for all kinds of truth approximation: observational, referential, and theoretical. This sheds new light on the long-term dynamic of science and hence on the relation between the main epistemological positions, viz., instrumentalism (Toulmin, Laudan), constructive empiricism (van Fraassen), referential realism (Hacking and Cartwright), and theory realism of a nonessentialist nature (Popper), here called constructive realism. In From Instrumentalism to Constructive Realism (2000) the above story is presented in great detail. The present synopsis highlights the main ways of theory evaluation presented in that book, viz. evaluation in terms of confirmation (or falsification), empirical progress and truth approximation.

An interesting consequence of the theory of nomic truth approximation, as developed in my From Instrumentalism to Constructive Realism (Kuipers T, From instrumentalism to constructive realism. On some relations between confirmation, empirical progress, and truth approximation. Synthese Library 287. Kluwer Academic Publishers, Dordrecht, 2000), and enriched in this book, concerns so-called ‘inference to the best explanation’ (IBE). It can be argued that this popular rule among scientific realists can better be replaced by, various kinds of, ‘inference to the best theory’ (IBT). This chapter provides a survey of observational, theoretical, and referential kinds of IBT and discusses when and in what sense they may be seen as cases of ‘abduction’, that is, arguments for which an ‘abductive’ formulation and, hence, justification can be given in the minimal sense of Peirce.

In Chapter 1 we introduced four types of research programs: descriptive, explanatory, design and explicative research programs. So far the emphasis was on descriptive and explanatory programs. Moreover, several (groups of) chapters illustrated explicative programs in philosophy of science. Design research programs, however, only received a little attention in the previous chapter.

In the empirical sciences the informal distinction between observational laws and proper theories plays a crucial role. Observational laws are supposed to describe observationally, usually experimentally, established regularities. Different names for roughly the same concept are: empirical, experimental or phenomenological laws, reproducible effects, inductive generalizations, general facts. Proper theories or systems of theoretical laws, on the other hand, are supposed to explain such laws and to predict new ones, by postulating underlying mechanisms. For easy reference, we will call this distinction the law-distinction. The law-distinction forms a crucial construction principle for the hierarchy of knowledge and therefore an important heuristic factor in the dynamics of knowledge development. However, it has occasioned philosophers of science much brain-racking to explicate the law-distinction in a defensible way. Without doubt, the distinction is strongly related to the distinction between observational (or empirical or experimental) and theoretical terms. Whereas proper theories introduce theoretical terms, observational laws do not. But how can one make sense of this term-distinction?

This chapter gives a systematic introduction to the structuralist reconstruction of empirical theories. In a number of chapters in this book and in ICR we have used the structuralist representation, without using its refined conceptual framework in all detail. Notably, in ICR (Chapter 11), where we deal with truth approximation in the old quantum theory and in capital structure theory, both (sequences of) theories were represented in the structuralist way.

Although philosophers of science almost always use to identify scientific research with description and explanation, this distinction is not only vague (see Part II), but, as we have stressed in Chapter 1, there are also other important types of research, notably design research and concept explication. Description and explanation may concern the actual or historical world, or they may concern the nomic world, that is, the world of what is nomically (physically, chemically, biologically, etc.) possible. In such case, they constitute nomological research. Whereas the previous parts contain much that is also useful for the research focussing on the actual world, the possibility of progress and ‘actual truth’ approximation in such research will be neglected in this chapter. The reader is referred to ICR (Chapters 7, 8 and 10) for brief discussions of these topics. This chapter deals first of all with nomological research, that is, by definition, research directed, by laws and theories, at the truth about (some part or aspect of) the nomic world, called the ‘nomic truth’. The chapter deals, moreover, though only briefly, with design and explicative research. Recall that design research includes all kinds of constructive research intended to make or improve certain products or processes. Explicative research aims at the construction of a simple, precise and useful concept that resembles an intuitive concept as much as possible.

We will start this chapter with a brief exposition of HD(hypothetico-deductive)testing, that is, the HD-method of testing hypotheses, and indicate the related qualitative explication of confirmation as well as quantitative extensions. HD-testing attempts to give an answer to one of the questions that one may be interested in, the truth question, which may be qualified according to the relevant epistemological position.1 However, the (theory) realist, for instance, is not only interested in the truth question, but also in some other questions. To begin with, there is the more refined question of which (individual or general) facts2 the hypothesis explains (its explanatory successes) and with which facts it is in conflict (its failures); the success question for short. We will show in this chapter that the HD-method can also be used in such a way that it is functional for (partially) answering this question. This method is called HD-evaluation, and uses HD-testing. Since the realist ultimately aims to approach the strongest true hypothesis, if any, i.e. the (theoretical-cum-observational) truth about the subject matter, the plausible third aim of the HD-method is to help answer the question of how far a hypothesis is from the truth, the truth approximation question Here the truth will be taken in a relatively modest sense, viz. relative to a given domain and conceptual frame. In the next chapter we will make plausible that HD-evaluation is also functional for answering the truth approximation question.

Computational philosophy of science is a collaboration between the philosophy of science and cognitive science. Cognitive science is itself a joint venture involving cognitive psychology, artificial intelligence research, neuroscience, linguistics, anthropology, and philosophy (Stillings et al., 1987), suggesting many interesting problems for philosophers of science (see Bechtel, 1988a). As illustrated in Chapter 6, Bechtel pays quite a lot of attention to reductionistic and nonreductionistic possibilities for co-operation between the disciplines involved.

There is no doubt that explanation by subsumption in general and theoretical explanation of observational laws in particular are very important in the empirical sciences. However, it is not at all evident that all respectable kinds of explanation occurring in the scientific literature obey the basic pattern of explanation by subsumption, as dealt with in Chapter 3.

In the philosophy of science literature produced since 1960, emphasis has fallen on the reduction of laws, notwithstanding the fact that something like concept reduction frequently appears as a necessary intermediate link.

I discuss in a systematic order the most important epistemological positions in the instrumentalism-realism debate, viz., instrumentalism, constructive empiricism, referential realism, and theory realism. My conclusions are as follows. There are good reasons for the instrumentalist to become a constructive empiricist. In turn, the constructive empiricist is forced to become a referential realist in order to give deeper explanations of success differences. Consequently, there are further good reasons for the referential realist to become a theory realist.

Science and ethics have a complex relation. They cannot easily be separated. Although pleas for completely autonomous science as well as for extreme ethical dirigism still occur, an interaction model for science and ethics is more and more widely propagated. This model is not only presented as characteristic of the factual relation, but also as desirable, that is, in the interest of science and society, of course, under rationally defensible conditions. It concerns in particular moral dilemmas with respect to: Incorruptible professional behavior in scientific research. The present and future well-being of humans and (laboratory, domestic, and wild) animals and the state of their environment. The communication between science and society.

One of the most important insights of the philosophy of science since about 1960 was the awareness that the development of science should not be described in terms of the development of specific hypotheses and theories, but in more encompassing terms. The two main proponents of this insight are Kuhn and Lakatos. Kuhn first preferred the term ‘paradigm’ and later ‘disciplinary matrix’ (Kuhn, 1962/1969). Lakatos (1970, 1978) introduced the notion of a ‘research program’. Half a dozen other terms are used to denote roughly the same concept, although their details may differ.

The standard view on the explanation of individual facts, such as events and states of affairs, is that of (deductive-)nomological explanation or explanation by subsumption under a law or theory in combination with one or more so-called initial conditions. This standard of explanation is also called the covering-law model of explanation. Formally, the underlying argument uses the rule of inference called universal instantiation (UI), followed by the rule called Modus Ponens (MP). That is, the most elementary form of explanation by subsumption is given by the valid argument

In Chapter 1 we already indicated several goals and types of symmetric and asymmetric co-operation between research programs, where the programs may or may not deal with different levels of aggregation or even belong to different disciplines. Moreover, they may or may not use the same or different styles of description and explanation. In this chapter we will elaborate these themes, with emphasis on mind-body-research.

In Chapter 7 we presented the basic definition of nomic truthlikeness1 of theories in terms of the structuralist approach to theories. It surpassed first of all the main problem of Popper’s original definition, which was inadequate, for it did not leave room for false theories, i.e., theories having actual mistakes, i.e., real counter-examples. The basic definition was, moreover, attractive in other conceptual, logical and methodological respects, as we have shown in Part III. Moreover, for true theories, the basic definition captures ‘scientific common sense’ and practice. Improving a true theory, in a cautious way, amounts to strengthening it without making it false, i.e., without loosing correct models. Hence, in perfect agreement with the basic definition, improving a true theory is primarily a matter of dropping mistaken models, without introducing new ones.

In Subsection 10.2.2. we have elaborated our scepsis about the idea of quantitative truth approximation, mainly because it presupposes real-valued distances between structures. Since such distances are not presupposed by quantitative confirmation we are even more skeptical about quantitative approaches to truth approximation than to confirmation. However, we will nevertheless present some main lines of such approaches. We will first deal, in Section 12.1., with quantitative actual truthlikeness and truth approximation, which is relatively unproblematic when there is a plausible distance function on Mp. For the basic nomic case, to be dealt with in Subsection 12.2.1. and part of Subsection 12.3.1., the situation is also relatively clear, but the refined nomic case faces us with essentially two types of problems. First, for the idea of quantitative truthlikeness, discussed in Subsection 12.2.2., we have to find a plausible distance function between two theories, and hence between two subsets of Mp, based on a distance function on Mp. Even guided by some plausible principles, our best refined proposal is not completely satisfactory. Second, we have to choose between two essentially different ways of measuring the quantitative success of theories: a non-probabilistic and a probabilistic way. In the first case, presented in Subsection 12.3.1., the success, or better the failure, of a theory in meeting the available evidence is quantitatively expressed in a similar way as the chosen distance function between two theories. The second case, presented in Subsection 12.3.2., arises from an adaptation of Niiniluoto’s plausible idea of estimating the distance of a theory from the complete, hence actual, truth on the basis of the available evidence. It will be shown that there is, in the case that probabilities as well as distances are plausible, a very sophisticated way of doing so, where ideas of Carnap, Hintikka and Festa are combined, leading to ‘double’ truth approximation. However, it will also be argued that this primarily concerns improper cases of nomic truth approximation, that is, it amounts to refined actual truth approximation in two respects.

When Karl Popper published in 1963 in Chapter 10 of Conjectures and Refutations his definition of ‘closer to the truth’ this was an important intellectual event, but not a shocking one. Everybody could react by saying that the definition was as it should be, and even that it could have been expected. For plausible the definition was indeed: a theory is closer to the truth than another if the true consequences of the first include those of the second and the false consequences of the second include those of the first.

In the previous two chapters we elaborated the ideas of qualitative and quantitative theories of confirmation along the following lines. According to the hypothetico-deductive method a theory is tested by examining its implications. As will be further elaborated in the next chapter, the result of an individual test of a general hypothesis stated in observation terms, and hence of a general test implication of a theory, can be positive, negative or neutral. If it is neutral the test was not well devised, if it is negative, the hypothesis, and hence the theory, has been falsified. Qualitative confirmation theory primarily aims at further explicating the intuitive notions of neutral and positive test results. Some paradoxical features discovered by Hempel (1945/1965) and some queer predicates defined by Goodman (1955) show that this is not an easy task. In Chapter 2 we have presented a qualitative theory (with comparative component) which could deal with these problems. Quantitative,more specifically, probabilistic theories of confirmation aim at explicating the idea of confirmation as increasing probability, odds, or some other measure, due to new evidence. In Chapter 3 we have seen that one of the Bayesian theories of confirmation, the r-theory, is satisfactory in that it generates all main features of the qualitative theory.

In this chapter we present some topics concerning truthlikeness and truth approximation in terms of intuitions that play, at least in our opinion, an important role in scientific and philosophical theorizing. In Section 8.1. we will spell out in model-theoretical terms that the basic definition of ‘more truthlike’ in the nomic sense not only has an intuitively appealing ‘model foundation’, but also, at least partially, a conceptually plausible ‘consequence foundation’. Moreover, combining the relevant parts of both leads to a very appealing ‘dual foundation’, the more so since the relevant methodological notions, viz., ‘more successful’ and its ingredients provided by HD-evaluation, can be given a similar dual foundation. In Section 8.2. we will argue that the definition of basic truthlikeness can be reinterpreted as an intra-level explication, for the actual as well as the nomic perspective, of the idea of a correspondence theory of truth. In contrast to the usual inter-level reading, the intra-level reading makes straightforward sense of the intuitions that ‘true/false’ is a matter. of corresponding or not corresponding to the facts and that ‘closer to the truth’ is a matter of better corresponding to the facts. Finally, in Section 8.3. we will show that the basic definitions of ‘more truthlike’ and ‘more successful’ lead to straightforward explications of several dialectical notions, such as ‘dialectical negation’, ‘double negation’, the triad ‘thesis-antithesis-synthesis’, and ‘the absolute’. In fact, logical, methodological and ontological explications of such notions can be given for the actual as well as for the nomic perspective.

According to the leading expositions of the hypothetico-deductive (HD-)method by Hempel (1966), Popper (1934/1959) and De Groot (1961/1969), the aim of the HD-method is to answer the question whether a hypothesis is true or false, that is, it is a method of testing. On closer inspection, this formulation of the aim of the HD-method is not only laden with the epistemological assumption of theory realism according to which it generally makes sense to aim at true hypotheses, but it also mentions only one of the realist aims. One other aim of the HD-method for the realist, an essentially more refined aim, is to answer the question of which facts are explained by the hypothesis and which facts are in conflict with it. In the line of this second aim, we will show in Part II that the HD-method can also be used to evaluate theories, separately and in comparison with other theories, among others, in terms of their general successes and individual problems, where theories are conceived as hypotheses of a strong kind. In Part III and IV we will argue that the methodology of (comparative) HD-evaluation of theories is even functional for truth approximation, the ultimate aim of the realist.

In this chapter we will extend the analysis of the previous chapter to the comparison of theories, giving rise to a definition of empirical progress and a sophisticated distinction between scientific and pseudoscientific behavior.

In the previous chapter we showed that the refined definition has real life scientific examples, viz., by pointing out that the Law of van der Waals provides a perfect case of (refined) potential truth approximation, in particular by (double) concretization. Recall that a sequence of three theories is called a case of potential truth approximation when the second is closer to the third than the first is to the third. Two other scientific examples of such sequences have been studied as well. In (Hettema and Kuipers 1995) it is demonstrated that Sommerfeld’s reconstruction of the sequence consisting of the theories of Rutherford, of Bohr and his own theory, i.e., the successive stages of what now is called ‘the old quantum theory’, is also a case of potential truth approximation. In the previous chapter (Section 10.4.) we explained that the technical definition of basic and refined truth approximation can also be directed at so-called provable interesting truths. In (Cools, Hamminga and Kuipers 1994) it is shown that the theory of the capital structure of firms of Modigliani and Miller is closer to a provable interesting truth than the original theory of Kraus and Litzenberger.