The modern discussion on the concept of truthlikeness was started in 1960. In his influential Word and Object, W. V. O. Quine argued that Charles Peirce's definition of truth as the limit of inquiry is faulty for the reason that the notion 'nearer than' is only "defined for numbers and not for theories". In his contribution to the 1960 International Congress for Logic, Methodology, and Philosophy of Science at Stan­ ford, Karl Popper defended the opposite view by defining a compara­tive (...) notion of verisimilitude for theories. The concept of verisimilitude was originally introduced by the Ancient sceptics to moderate their radical thesis of the inaccessibility of truth. But soon verisimilitudo, indicating likeness to the truth, was confused with probabilitas, which expresses an opiniotative attitude weaker than full certainty. The idea of truthlikeness fell in disrepute also as a result of the careless, often confused and metaphysically loaded way in which many philosophers used - and still use - such concepts as 'degree of truth', 'approximate truth', 'partial truth', and 'approach to the truth'. Popper's great achievement was his insight that the criticism against truthlikeness - by those who urge that it is meaningless to speak about 'closeness to truth' - is more based on prejudice than argument. (shrink)

A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...) unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world. (shrink)

This paper sets out a new framework for discussing a long-standing problem in the philosophy of mathematics, namely the connection between the physical world and a mathematical domain when the mathematics is applied in science. I argue that considering counterfactual situations raises some interesting challenges for some approaches to applications, and consider an approach that avoids these challenges.

The numerical representations of measurement, geometry and kinematics are here subsumed under a general theory of representation. The standard theories of meaningfulness of representational propositions in these three areas are shown to be special cases of two theories of meaningfulness for arbitrary representational propositions: the theories based on unstructured and on structured representation respectively. The foundations of the standard theories of meaningfulness are critically analyzed and two basic assumptions are isolated which do not seem to have received adequate justification: the (...) assumption that a proposition invariant under the appropriate group is therefore meaningful, and the assumption that representations should be unique up to a transformation of the appropriate group. A general theory of representational meaningfulness is offered, based on a semantic and syntactic analysis of representational propositions. Two neglected features of representational propositions are formalized and made use of: (a) that such propositions are induced by more general propositions defined for other structures than the one being represented, and (b) that the true purpose of representation is the application of the theory of the representing system to the represented system. On the basis of these developments, justifications are offered for the two problematic assumptions made by the existing theories. (shrink)

A ubiquitous argument against mental-state accounts of well-being is based on the notion that mental states like happiness and satisfaction simply cannot be measured. The purpose of this paper is to articulate and to assess this “argument from measurability.” My main thesis is that the argument fails: on the most charitable interpretation, it relies on the false proposition that measurement requires the existence of an observable ordering satisfying conditions like transitivity. The failure of the argument from measurability, however, does not (...) translate into a defense of mental-state accounts as accounts of well-being or of measures of happiness and satisfaction as measures of well-being. Indeed, I argue, the ubiquity of the argument from measurability may have obscured other, very real problems associated with mental-state accounts of well-being – above all, that happiness and satisfaction fail to track well-being – and with measures of happiness and satisfaction – above all, the tendency toward reification. I conclude that the central problem associated with the measurement of, e.g., happiness as a subjectively experienced mental state is not that it is too hard to measure, but rather that it is too easy to measure. (shrink)

In this paper, we explore semantics for comparative epistemic modals that avoid the entailment problems shown to result from Kratzer’s (1991) semantics by Yalcin (2006, 2009, 2010). In contrast to the alternative semantics presented by Yalcin and Lassiter (2010, 2011), based on finitely additive probability measures, we introduce semantics based on qualitatively additive measures, as well as semantics based on purely qualitative orderings, including orderings on propositions derived from orderings on worlds in the tradition of Kratzer (1991). All of these (...) semantics avoid the entailment problems that result from Kratzer’s semantics. Our discussion focuses on methodological issues concerning the choice between different semantics. (shrink)

Measurement is a process aimed at acquiring and codifying information about properties of empirical entities. In this paper we provide an interpretation of such a process comparing it with what is nowadays considered the standard measurement theory, i.e., representational theory of measurement. It is maintained here that this theory has its own merits but it is incomplete and too abstract, its main weakness being the scant attention reserved to the empirical side of measurement, i.e., to measurement systems and to the (...) ways in which the interactions of such systems with the entities under measurement provide a structure to an empirical domain. In particular it is claimed that (1) it is on the ground of the interaction with a measurement system that a partition can be induced on the domain of entities under measurement and that relations among such entities can be established, and that (2) it is the usage of measurement systems that guarantees a degree of objectivity and intersubjectivity to measurement results. As modeled in this paper, measurement systems link the abstract theory of measuring, as developed in representational terms, and the practice of measuring, as coded in standard documents such as the International Vocabulary of Metrology. (shrink)

Axiomatic measurement theories are commonly interpreted as claiming that, in order to quantify an empirical domain, the qualitative structure of data about that domain must be mapped to a numerical structure. Such mapping is supposed to be established independently, i.e., without presupposing that the domain can be quantified. This interpretation is based on two myths: that it is possible to independently infer the qualitative structure of objects from empirical data, and that the adequacy of numerical representations can only be justified (...) by mapping such qualitative structures to numerical ones. I dispel the myths and show that axiomatic measurement theories provide an inadequate characterization of the kind of evidence required to detect quantities. (shrink)

This article bears on four topics: observational predicates and phenomenal properties, vagueness, strict finitism as a philosophy of mathematics, and the analysis of feasible computability. It is argued that reactions to strict finitism point towards a semantics for vague predicates in the form of nonstandard models of weak arithmetical theories of the sort originally introduced to characterize the notion of feasibility as understood in computational complexity theory. The approach described eschews the use of nonclassical logic and related devices like degrees (...) of truth or supervaluation. Like epistemic approaches to vagueness, it may thus be smoothly integrated with the use of classical model theory as widely employed in natural language semantics. But unlike epistemicism, the described approach fails to imply either the existence of sharp boundaries or the failure of tolerance for soritical predicates. Applications ofmeasurement theory(in the sense of Krantz, Luce, Suppes, & Tversky (1971)) to vagueness in the nonstandard setting are also explored. (shrink)

Part II: Suppes and the mature theory. Representation and uniqueness.

Developing some suggestions of Ramsey (1925), elementary logic is formulated with respect to an arbitrary categorial system rather than the categorial system of Logical Atomism which is retained in standard elementary logic. Among the many types of non-standard categorial systems allowed by this formalism, it is argued that elementary logic with predicates of variable degree occupies a distinguished position, both for formal reasons and because of its potential value for application of formal logic to natural language and natural science. This (...) is illustrated by use of such a logic to construct a theory of quantity which is argued to be scientifically superior to existing theories of quantity based on standard categorial systems, since it yields real-valued scales without the need for unrealistic existence assumptions. This provides empirical evidence for the hypothesis that the categorial structure of the physical world itself is non-standard in this sense. (shrink)

Modifications of current theories of ordinal, interval and extensive measurement are presented, which aim to accomodate the empirical fact that perfectly exact measurement is not possible (which is inconsistent with current theories). The modification consists in dropping the assumption that equality (in measure) is observable, but continuing to assume that inequality (greater or lesser) can be observed. The modifications are formulated mathematically, and the central problems of formal measurement theory--the existence and uniqueness of numerical measures consistent with data--are re-examined. Some (...) results also are given on a problem which does not arise in current theories: namely that of determining limits of accuracy attainable on the basis of observations. (shrink)

In this paper I argue that it is finally time to move beyond the Nagelian framework and to break new ground in thinking about epistemic reduction in biology. I will do so, not by simply repeating all the old objections that have been raised against Ernest Nagel’s classical model of theory reduction. Rather, I grant that a proponent of Nagel’s approach can handle several of these problems but that, nevertheless, Nagel’s general way of thinking about epistemic reduction in terms of (...) theories and their logical relations is entirely inadequate with respect to what is going on in actual biological research practice. (shrink)

Part II: Suppes and the mature theory. Representation and uniqueness.

Generalizations of partial meet contraction are introduced that start out from the observation that only some of the logically closed subsets of the original belief set are at all viable as contraction outcomes. Belief contraction should proceed by selection among these viable options. Several contraction operators that are based on such selection mechanisms are introduced and then axiomatically characterized. These constructions are more general than the belief base approach. It is shown that partial meet contraction is exactly characterized by adding (...) to one of these constructions the condition that all logically closed subsets of the belief set can be obtained as the outcome of a single (multiple) contraction. Examples are provided showing the counter-intuitive consequences of that condition, thus confirming the credibility of the proposed generalization of the AGM framework. (shrink)

This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving (...) operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. The second part is concerned with physical space and space-time and deals mainly with topics concerning the axiomatic theory of bodies, the operational foundations of special relativity and the conceptual foundations of elementary physics. (shrink)

The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper topology. However, this condition fails to be sufficient. Instead of search (...) for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements. (shrink)

The formal methods of the representational theory of measurement (RTM) are applied to the extensive scales of physical science, with some modifications of interpretation and of formalism. The interpretative modification is in the direction of theoretical realism rather than the narrow empiricism which is characteristic of RTM. The formal issues concern the formal representational conditions which extensive scales should be assumed to satisfy; I argue in the physical case for conditions related to weak rather than strong extensive measurement, in the (...) sense of Holman 1969 and Colonius 1978. The problem of justifying representational conditions is addressed in more detail than is customary in the RTM literature; this continues the study of the foundations of RTM begun in an earlier paper. The most important formal consequence of the present interpretation of physical extensive scales is that the basic existence and uniqueness properties of scales (representation theorem) may be derived without appeal to an Archimedean axiom; this parallels a conclusion drawn by Narens for representations of qualitative probability. It is concluded that there is no physical basis for postulation of an Archimedean axiom. (shrink)

This volume analyzes and studies how vagueness occurs and matters as a specific problem in the context of theories that are primarily about something else.

Shepard has supposed that the mind is stocked with innate knowledge of the world and that this knowledge figures prominently in the way we see the world. According to him, this internal knowledge is the legacy of a process of internalization; a process of natural selection over the evolutionary history of the species. Shepard has developed his proposal most fully in his analysis of the relation between kinematic geometry and the shape of the motion path in apparent motion displays. We (...) argue that Shepard has made a case for applying the principles of kinematic geometry to the perception of motion, but that he has not made the case for injecting these principles into the mind of the percipient. We offer a more modest interpretation of his important findings: that kinematic geometry may be a model of apparent motion. Inasmuch as our recommended interpretation does not lodge geometry in the mind of the percipient, the motivation of positing internalization, a process that moves kinematic geometry into the mind, is obviated. In our conclusion, we suggest that cognitive psychologists, in their embrace of internal mental universals and internalization may have been seduced by the siren call of metaphor. Key Words: apparent motion; imagery; internalization; inverse projection problem; kinematic geometry; measurement; metaphors of mind. (shrink)

The thesis of the empirical underdetermination of theories (U-thesis) maintains that there are incompatible theories which are empirically equivalent. Whether this is an interesting thesis depends on how the term incompatible is understood. In this paper a structural explication is proposed. More precisely, the U-thesis is studied in the framework of the model theoretic or emantic approach according to which theories are not to be taken as linguistic entities, but rather as families of mathematical structures. Theories of similarity structures are (...) studied as a paradigmatic case. The structural approach further reveals that the U-thesis is related to problems of uniqueness in the representational theory of measurement, questions of geometric conventionalism, and problems of structural underdetermination in mathematics. (shrink)

The author argues that vagueness in law is typically extravagant, in the sense that it is possible for two competent users of the language, who understand the facts of each case, to take such different views that there is not even any overlap between the cases that each disputant would identify as borderline. Extravagant vagueness is a necessary feature of legal systems. Some philosophers of law and philosophers of language claim that bivalence is a property of statements in the domains (...) that concern them (the domain of law in the former case, the whole domain of meaningful discourse in the latter). The author argues that the bivalence claim should be rejected. In philosophy of law, the motivation underlying the bivalence claim is an urge to assert the principle that the law must be capable of standing against arbitrary use of political power. The challenge – once the bivalence claim is rejected – is to articulate the law's opposition to arbitrariness in a way that is compatible with the possibility of indeterminacy in the application of vague laws. (shrink)

Of the many families of words that are thought to be vague, so-called observational predicates may be both the most fascinating and the most confounding. Roughly, observational predicates are terms that apply to objects on the basis of how those objects appear to us perceptually speaking. ‘Red’, ‘loud’, ‘sweet’, ‘acrid’, and ‘smooth’ are good examples. Delia Graff explains that a “predicate is observational just in case its applicability to an object (given a fixed context of evaluation) depends only on the (...) way that object appears” (2001, 3). By the same token observational predicates are, as Crispin Wright observes, terms “whose senses are taught entirely by ostension” (1976). Like other vague predicates, observational words appear to generate sorites paradoxes. Consider for example a series of 20 colored patches progressing from a clearly red one to a clearly orange one, so ordered that each patch is just noticeably different in hue from the one before. The following argument then seems forced upon us: (1) Patch #1 is red. (2) Any patch that differs only slightly in hue from a red patch is itself red. (3) Therefore patch #20 is red. Premise (2) expresses what Wright has called the tolerance of ‘red’: the application of the predicate tolerates small changes in a decisive parameter (here, hue). Of course, most vague predicates, hence most versions of the sorites, are not observational. For instance, given a series of. (shrink)

What are the pillars on which the success of modern science rest? Although philosophers have much discussed what is behind science’s success, this paper argues that much of the discussion is misdirected. The extant literature rightly regards the semantic and inferential tools of formal logic and probability theory as pillars of scientific rationality, in the sense that they reveal the justificatory structure of important aspects of scientific practice. As key elements of our rational reconstruction toolbox, they make a fundamental contribution (...) to our understanding of the success of science.At the same time, any science, however exact, is dominated by approximation, error, and uncertainty, a fact that makes one wonder how science can be so successful. This paper articulates and illustrates general themes—e.g., that truth-preserving arguments often fail to preserve approximate truth—that highlight the need for additional semantic resources. Thus, our proposal is that persistent failures to unravel the reasons behind the success of science in the face of pervasive error and uncertainty should be attributed to an insufficiently rich way of rationally reconstructing scientific and mathematical knowledge. What is missing? This paper claims that there is a third formal method of reasoning that constitutes a distinct pillar on which rests the success of science, namely, perturbation theory. The paper outlines how the representational and inferential tools of perturbation theory differ from those of logic and probability theory, and how they enable us to understand the apparently elusive aspects of the success of science.However, compared to its peers, perturbative reasoning has not received the attention it deserves. As the paper explains, this partly results from the circumstances in which perturbation theory is taught, and partly from the fact that perturbation theory first appears to be a vaguely related collection of methods offering no systematic semantic insight. In an attempt to show that this first impression is wrong, this paper presents its contribution to the semantic dimension of scientific representation and inference in terms of what I call “semantic layering.”. (shrink)

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable (...) us to treat formally errors arising in nominal and ordinal measurements. (shrink)

In the literature, there are many axiomatizations of qualitative probability. They all suffer certain defects: either they are too nonspecific and allow nonunique quantitative interpretations or are overspecific and rule out cases with unique quantitative interpretations. In this paper, it is shown that the class of qualitative probability structures with nonunique quantitative interpretations is not first order axiomatizable and that the class of qualitative probability structures with a unique quantitative interpretation is not a finite, first order extension of the theory (...) of qualitative probability. The idea behind the method of proof is quite general and can be used in other measurement situations. (shrink)

A simple interpretation of quantity calculus is given. Quantities are described as two-place functions from objects, states or processes (or some combination of them) into numbers that satisfy the mutual measurability property. Quantity calculus is based on a notational simplification of the concept of quantity. A key element of the simplification is that we consider units to be intentionally unspecified numbers that are measures of exactly specified objects, states or processes. This interpretation of quantity calculus combines all the advantages of (...) calculating with numerical values (since the values of quantities are numbers, we can do with them everything we do with numbers) and all the advantages of calculating with standardly conceived quantities (calculus is invariant to the choice of units and has built-in dimensional analysis). This also shows that the standard metaphysics and mathematics of quantities and their magnitudes is not needed for quantity calculus. At the end of the article, arguments are given that the concept of quantity as defined here is a pivotal concept in understanding the quantitative approach to nature. As an application of this interpretation of quantity calculus, an easy proof of dimensional homogeneity of physical laws is given. (shrink)

In this article I give an overview of some recent work in philosophy of science dedicated to analysing the scientific process in terms of (conceptual) mathematical models of theories and the various semantic relations between such models, scientific theories, and aspects of reality. In current philosophy of science, the most interesting questions centre around the ways in which writers distinguish between theories and the mathematical structures that interpret them and in which they are true, i.e. between scientific theories as linguistic (...) systems and their non-linguistic models. In philosophy of science literature there are two main approaches to the structure of scientific theories, the statement or syntactic approach -- advocated by Carnap, Hempel and Nagel -- and the non- statement or semantic approach --advocated, among others, by Suppes, the structuralists, Beth, Van Fraassen, Giere, Wojcicki. In conclusion, I briefly review some of the usual realist inspired questions about the possibility and character of relations between scientific theories and reality as implied by the various approaches I discuss in the course of the article. The models of a scientific theory should indeed be adequate to the phenomena, but if the theory is 'adequate' to (true in) its conceptual (mathematical) models as well, we have a model-theoretic realism that addresses the possible meaning and reference of 'theoretical entities' without relapsing into the metaphysics typical of the usual scientific realist approaches. (shrink)

Suppose it is possible to make approximate interpersonal comparisons of welfare gains and losses. Thus, if w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w$$\end{document}, x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document}, y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y$$\end{document} and z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z$$\end{document} are personal states, then it is sometimes possible to say “The welfare gain of the state change w⇝x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\leadsto x$$\end{document} is greater than the welfare gain of the state change y⇝z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\leadsto z$$\end{document}”. We can represent this by the formula “⪰\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\succeq $$\end{document}”, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} is a difference preorder: an incomplete preorder on the space of all possible personal state changes. A social state change is a bundle of personal state changes. A social difference preorder is an incomplete preorder on the space of social state changes, which satisfies Pareto and Anonymity axioms. We characterize a family of SDPs which are roughly utilitarian in nature. We also apply the model to redistributive taxation. (shrink)

a–b* c–d is taken to mean that your degree of preference for a over b is less than your degree of preference for c over d. Various properties of the strength-of-preference comparison relation * are examined along with properties of simple preferences defined from *. The investigation recognizes an individual's limited ability to make precise judgments. Several utility theorems relating a–b * c–d to u(a)–u(b) are included.

We study the reverse mathematics of interval orders. We establish the logical strength of the implications among various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2 \oplus 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it (...) contains neither 2 \oplus 2 nor 3 \oplus 1. (shrink)

Axiomatizations of ordered preference differences typically assume that preferences are ordered. However, the mere assumption that preference differences can be ordered says nothing about whether preferences themselves are ordered. Utility representations for ordered differences without ordered preferences are investigated.

Though he maintained a significant interest in theoretical aspects of measurement, Henry E. Kyburg, Jr. was critical of the representational theory that in many ways has come to dominate discussions concerning the foundations of measurement. In particular, Kyburg (in Savage and Ehrlich (eds) Philosophical and foundational issues in measurement theory, 1992 ) asserts that the representational theory of measurement, as introduced in (Scott and Suppes, Journal of Symbolic Logic, 23:113–128, 1958 ) and developed in (Krantz et al., Foundations of measurment: (...) additive and polynomial representations. Academic Press, 1971 ), cannot account for the measurement of error. The present work examines and responds to this charge. (shrink)