Many of our sources of knowledge only afford us knowledge that is inexact. When trying to see how tall something is, or to hear how far away something is, or to remember how long something lasted, we may come to know some facts about the approximate size, distance or duration of the thing in question but we don’t come to know exactly what its size, distance or duration is. In some such situations we also have some pointed knowledge of how (...) inexact our knowledge is. That is, we can knowledgeably pinpoint some exact claims that we do not know. We show that standard models of inexact knowledge leave little or no room for such pointed knowledge. We devise alternative models that are not afflicted by this shortcoming. (shrink)

This book offers a comprehensive overview of the structure, strategy and methods of assessment of orthodox theoretical economics. In Part I Professor Hausman explains how economists theorise, emphasising the essential underlying commitment of economists to a vision of economics as a separate science. In Part II he defends the view that the basic axioms of economics are 'inexact' since they deal only with the 'major' causes; unlike most writers on economic methodology, the author argues that it is the rules that (...) economists espouse rather than their practice that is at fault. Part III links the conception of economics as a separate science to the fact that economic theories offer reasons and justifications for human actions, not just their causes. With its lengthy appendix introducing relevant issues in philosophy of science, this book is a major addition to philosophy of economics and of social science. (shrink)

This paper supersedes an ealier version, entitled "A Non-Standard Semantics for Inexact Knowledge with Introspection", which appeared in the Proceedings of "Rationality and Knowledge". The definition of token semantics, in particular, has been modified, both for the single- and the multi-agent case.

I defend, to a certain extent, the traditional view that perceptual indiscriminability is non-transitive. The argument proceeds by considering important recent work by Benj Hellie: Hellie argues that colour perception represents ‘inexactly’, and that this results in violations of the transitivity of colour indiscriminability. I show that Hellie’s argument remains inconclusive, since he does not demonstrate conclusively that colour perception really does represent inexactly. My own argument for the non-transitivity of perceptual indiscriminability uses inexactness instead as one horn of (...) a dilemma: the key idea is that there is a class of perceptual experiences which might plausibly be supposed either to represent inexactly or to represent exactly—but which demonstrate the non-transitivity of perceptual indiscriminability either way. (shrink)

Most of our knowledge is inexact, and known by us to be so. An example of such known inexactness will be described in some detail. The description seems to entail a contradiction. However, the paradoxical reasoning rests on an assumption. It will be suggested that the description is correct and this assumption false. Its failure will be explained by means of a picture of inexact knowledge in which the notion of a margin for error is central. This picture suggests (...) diagnoses of other paradoxical arguments: Surprise Examinations, backwards inductions about Iterated Prisoner's Dilemma, and the Heap. (shrink)

Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as an inexact non-demonstrative science (...) is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. (shrink)

Cases of inexact observations have been used extensively in the recent literature on higher-order evidence and higher-order knowledge. I argue that the received understanding of inexact observations is mistaken. Although it is convenient to assume that such cases can be modeled statically, they should be analyzed as dynamic cases that involve change of knowledge. Consequently, the underlying logic should be dynamic epistemic logic, not its static counterpart. When reasoning about inexact knowledge, it is easy to confuse the initial situation, the (...) observation process, and the result of the observation; I analyze the three separately. This dynamic approach has far reaching implications: Williamson’s influential argument against the KK principle loses its force, and new insights can be gained regarding synchronic and diachronic introspection principles. (shrink)

The Inexact and Separate Science of Economics represents the most ambitious attempt to provide a systematic account of economic methodology since the first edition of Blaug's The Methodology of Economics. As such, it has been the subject of extensive critical commentary. For all the attention it has received, however, some important aspects of the book's thesis have not been developed properly. Two important ones are what might be called, following the terminology used in the experimental economics literature, the ‘framing effect’ (...) of Hausman's definition of economics, and the significance of Hausman's claim that economists are committed to developing economics as a ‘separate’ science. To understand these points it is important to make explicit the position from which Hausman approaches the philosophy of science. (shrink)

Williamson (2000a) has argued that posi- tive introspection is incompatible with in- exact knowledge. His argument relies on a margin-for-error requirement for inexact knowledge based on a intuitive safety prin- ciple for knowledge, but leads to the counter- intuitive conclusion that no possible creature could have both inexact knowledge and posi- tive introspection. Following Halpern (2004) I put forward an alternative margin-for-error requirement that preserves the safety require- ment while blocking Williamson’s argument. I argue that the infallibilist conception of knowledge (...) that underlies the new require- ment provides a better account of inexact knowledge and higher-order knowledge than both Williamson’s and Halpern’s. (shrink)

An inexact generalization like ‘ravens are black’ will be symbolized as a prepositional function with free variables thus: ‘Rx ⇒ Bx.’ The antecedent ‘Rx’ and consequent ‘Bx’ will themselves be called absolute formulas, while the result of writing the non-boolean connective ‘⇒’ between them is conditional. Absolute formulas are arbitrary first-order formulas and include the exact generalization ‘(x)(Rx → Bx)’ and sentences with individual constants like ‘Rc & Bc.’ On the other hand the non-boolean conditional ‘⇒’ can only occur as (...) the main connective in a formula. We shall also need to consider formulas with more than one free variable such as ‘xHy ⇒ xTy,’ which might express ‘if x is the husband of y then x is taller than y.’ Though it is inessential, it will simplify things to work in ‘n-languages’ with a finite number of individual constants c1,…, cn, which are interpreted as denoting the elements of the domains of the ‘n-models’ to be described below. (shrink)

The paper discusses the problems raised by the inexactness of experiential concepts for a deductivist account of theoretical explanation. The process of theoretical explanation is explicated in terms of the devising of exact forms of inexact concepts. Analysis of the adjustments of concepts and their exact forms to each other reveals an implicit criterion of adequacy for theories which is related to the principle of connectivity.

In a series of recent papers, Timothy Williamson has argued for the surprising conclusion that there are cases in which you know a proposition in spite of its being overwhelmingly improbable given what you know that you know it. His argument relies on certain formal models of our imprecise knowledge of the values of perceptible and measurable magnitudes. This paper suggests an alternative class of models that do not predict this sort of improbable knowing. I show that such models are (...) motivated by independently plausible principles in the epistemology of perception, the epistemology of estimation, and concerning the connection between knowledge and justified belief. (shrink)

I suppose that 'ravens are black' is an inexact generalization having a degree of truth measured by the proportion of ravens that are black, and a probability measured by its expected degree of truth in different 'possible worlds.' Given this, 'ravens are black' differs in truth, probability, and confirmation from 'non-black things are not ravens', and this suggests a new approach to Hempel's Paradox as well as to other aspects of confirmation. Basic concepts of a formal theory developing this approach (...) are described, and some of its fundamental laws are given, together with sketches of applications. (shrink)

THERE IS OFTEN UNCERTAINTY ABOUT WHETHER SOME PREDICATE APPLIES TO SOME PHYSICAL OBJECT OR STATE. THIS UNCERTAINTY MAY HAVE ANY OF THREE SOURCES - VAGUENESS OF A TERM, INEXACTNESS OF A CONCEPT, OR PRACTICAL DIFFICULTY IN DETERMINING ITS APPLICABILITY. VARIOUS WAYS IN WHICH CONCEPTUAL INEXACTNESS OR PRACTICAL DIFFICULTY MAY PRODUCE UNCERTAINTY ARE DISTINGUISHED. NEITHER TERMINOLOGICAL VAGUENESS, NOR PRACTICAL DIFFICULTY IN DETERMINING THE APPLICABILITY OF A CONCEPT ARE NECESSARY FEATURES OF EVERY LANGUAGE IN EVERY PHYSICAL WORLD, BUT CONCEPTUAL (...) class='Hi'>INEXACTNESS IS A NECESSARY FEATURE OF EMPIRICAL CONCEPTS. THESE RESULTS ARE UTILIZED TO DETERMINE THE SOURCES AND NECESSITY OF IMPRECISION IN MEASUREMENT. (shrink)

Kit Fine distinguishes between inexact and exact truthmaking. He argues that the former can be defined from the latter, but not vice versa, and so concludes that truthmaker semanticists should treat the exact variety of truthmaking as primitive. I argue that this gets things backwards. We can define exact truthmaking in terms of inexact truthmaking and we can’t define inexact truthmaking in terms of exact truthmaking. I conclude that it’s inexact truthmaking, rather than exact truthmaking, that truthmaker semanticists should treat (...) as the primitive semantic relation. (shrink)

Timothy Williamson claims that margin for error principles govern all cases of inexact knowledge. I show that this claim is unfounded: there are cases of inexact knowledge where Williamson’s argument for margin for error principles does not go through. The problematic cases are those where the value of the relevant parameter is fixed across close cases. I explore and reject two responses to my objection, before concluding that Williamson’s account of inexact knowledge is not compelling.

The differential aging effect is shown to be valid in any physically reasonable extension of the special theory of relativity which includes a description of accelerating observers. Einstein's controversial assumption—the clock hypothesis—is avoided. Instead, it is sufficient to assume accessibility—that it is possible to travel from one inertial observer to another and then return to the first in a reasonable manner. Since Minkowski space-time displays this accessibility property, there must be an error in Sachs's quaternion development of general relativity. No (...) assumptions concerning measurements—proper time—of accelerating observers are necessary, nor are assumptions concerning special features of their motion—uniform acceleration. (shrink)

Philosophic inquiry into the mental states of elite athletes during skilled motor performance continues to grow. In contrast to the bulk of these works that focus almost exclusively on skillful performance, this paper examines athletic motor behavior from a point of inexactness – or even failure – in athletic performance. Utilizing the works of Michael Polanyi, who believed that both ideas of achievement and failure were equally necessary to understand the behavior of living things and their physical actions, I (...) examine the notion of failure as a framework to scrutinize the cognitive processes occurring during the development and performance of skilled motor behavior. After reviewing Polanyi’s conceptions of personal knowing to locate the source of inaccuracy in human activity, I present Polanyi’s distinction between two kinds of mistakes and apply each to inaccurate sport performance. I then suggest that mistakes in sport should be re-conceptualized beyond their current negative connotations. Instead, conceptions of mistakes should also include respect for ‘man’s most distinguished act’ – that being the production of knowledge. From this expanded perspective, the value of inexact motor performance can be found in addition to notions of uncertainty and skill development in what Polanyi calls ‘metaphysical implication of a groping for reality’. In some final thoughts, I will suggest future implications of the value of the inexact on broader sport issues. (shrink)

The ontologies of scientific theories include a variety of objects: point-mass particles, rigid rods, frictionless planes, flat and curved spacetimes, perfectly spherical planets, continuous fluids, ideal gases, nonidentical but indistinguishable electrons, atoms, quarks and gluons, strong and weak nuclear forces, ideally rational agents, and so on. But the scientific community currently regards only some of these objects as real. According to Paul Teller, a group sometimes can be justified in regarding competing ontologies as real and the ontologies we are justified (...) in regarding as real are inexact, because the theories that give those ontologies characterize what things are like rather than what they are. In this paper, I argue that Teller's view is incomplete and suggest that one way to remove this incompleteness is to adopt a criterion for when we are justified in regarding a theory's ontology as real that is based upon a theory's comparative degree of confirmation. I argue that this criterion is prima-facie plausible and that Teller's view is false if this criterion is correct. (shrink)

I argue that though a satisfactory semantics for the logic of inexact reference may assign no truth value to some statements, it should not assign truth (or falsity) of various degrees. Well-formed assertions are simply true or not. Inexactness does not “ramify.” I distinguish inexactness from other sorts of vagueness, including nonspecificity. I show that arguments from (i) use of quantifiers, (ii) the existence of properties which can be construed as a series of properties (as, e. g., red (...) can be construed as a set of shades of red), (iii) the constructability of apparently paradoxical sorites arguments, and (iv) the presence of prototypes in the extension of a predicate do not show that there are degrees of truth.Much of the alleged evidence that inexactness ramifies is, in fact, a misreading of the undeniable evidence that there may be uncertainty about the truth value of a claim. In support of my claims, I discuss how cases of deeming that a predicate applies relate to its actually applying. A distinction between predicates of “pure” and “impure” function is essential to this. (shrink)

I argue that though a satisfactory semantics for the logic of inexact reference may assign no truth value to some statements, it should not assign truth (or falsity) of various degrees. Well-formed assertions are simply true or not. Inexactness does not “ramify.” I distinguish inexactness from other sorts of vagueness, including nonspecificity. I show that arguments from (i) use of quantifiers, (ii) the existence of properties which can be construed as a series of properties (as, e. g., red (...) can be construed as a set of shades of red), (iii) the constructability of apparently paradoxical sorites arguments, and (iv) the presence of prototypes in the extension of a predicate do not show that there are degrees of truth.Much of the alleged evidence that inexactness ramifies is, in fact, a misreading of the undeniable evidence that there may be uncertainty about the truth value of a claim. In support of my claims, I discuss how cases of deeming that a predicate applies relate to its actually applying. A distinction between predicates of “pure” and “impure” function is essential to this. (shrink)

The Inexact and Separate Science of Economics offers an overview of standard microeconomics and general equilibrium theory. These are not the whole of orthodox economics, and orthodox economics is not the whole of economics. But orthodox economics dominates the profession, and the theoretical core of microeconomics and general equilibrium theory – what I called ‘equilibrium theory’ – is central to most orthodox economics. Unlike many methodological works, which focus almost exclusively on the empirical problems of equilibrium theory and its applications, (...) ISSE is also concerned with the structure, strategy and heuristics of equilibrium theorizing, and it attempts to link questions about theory appraisal to questions about structure and strategy. It is addressed both to philosophers interested in epistemological questions posed by the social sciences and to economists interested in reflecting on and improving their discipline. Its inspiration lies in the work of John Stuart Mill. (shrink)

This chapter contains section titled: Pragmatism Redirected From Knowledge Approximated to Approximate Knowledge Technology, Standardization, and Experimental Science Measurement and Orders of Magnitude Non‐Reductionism, Hierarchy, and Complexity Action, Progress, and Moving Beyond Cartesian Intellectualism.

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.

Exact sciences are described as sciences whose theories are formalized. These are contrasted to inexact sciences, whose theories are not formalized. Formalization is described as a broader category than mathematization, involving any form/content distinction allowing forms, e.g., as represented in theoretical models, to be studied independently of the empirical content of a subject-matter domain. Exactness is a practice depending on the use of theories to control subject-matter domains and to align theoretical with empirical models and not merely a state of (...) a science. Inexact biological sciences tolerate a degree of “mismatch” between theoretical and empirical models and concepts. Three illustrations from biological sciences are discussed in which formalization is achieved by various means: Mendelism, Weismannism, and Darwinism. Frege’s idea of a “conceptual notation” is used to further characterize the notion of a form/content distinction. (shrink)

The paper attempts to give an account of the fallibility of medical judgments without recourse to the incompleteness of scientific knowledge. It is argued that because of the inexactness of observations and thus the existence of borderline cases any theory applied for explanation and predicition will produce some false results. This state of affairs is independent of the nature of a theory, i.e., it applies both for non-probabilistic and for probabilistic theories. Some epistemological issues and consequences with regard to (...) a better understanding of clinical reasoning are discussed and the view is compared with the theory of medical fallibility of Gorovitz and MacIntyre.1. (shrink)

The Encyclopedia of the Inexact Sciences brings together Raymond Queneau's research into an idiosyncratic canon of nineteenth-century works, each written by what he variously refers to as a ‘litera...

The paper attempts to give an account of the fallibility of medical judgments without recourse to the incompleteness of scientific knowledge. It is argued that because of the inexactness of observations and thus the existence of borderline cases any theory applied for explanation and predicition will produce some false results. This state of affairs is independent of the nature of a theory, i.e., it applies both for non-probabilistic and for probabilistic theories. Some epistemological issues and consequences with regard to (...) a better understanding of clinical reasoning are discussed and the view is compared with the theory of medical fallibility of Gorovitz and MacIntyre.1. (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)