In 1999, Jeffrey Ketland published a paper which posed a series of technical problems for deflationary theories of truth. Ketland argued that deflationism is incompatible with standard mathematical formalizations of truth, and he claimed that alternate deflationary formalizations are unable to explain some central uses of the truth predicate in mathematics. He also used Beth’s definability theorem to argue that, contrary to deflationists’ claims, the T-schema cannot provide an ‘implicit definition’ of truth. In this article, I want to challenge (...) this final argument. Whatever other faults deflationism may have, the T-schema does provide an implicit definition of the truth predicate. Or so, at any rate, I shall argue. (shrink)

Skolem's Paradox involves a seeming conflict between two theorems from classical logic. The Löwenheim Skolem theorem says that if a first order theory has infinite models, then it has models whose domains are only countable. Cantor's theorem says that some sets are uncountable. Skolem's Paradox arises when we notice that the basic principles of Cantorian set theory—i.e., the very principles used to prove Cantor's theorem on the existence of uncountable sets—can themselves be formulated as a collection of (...) first order sentences. How can the very principles which prove the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first order sentence which says that there are uncountably many mathematical objects—e.g., uncountably many real numbers? (shrink)

The Lowenheim-Skolem theorems say that if a first-order theory has infinite models, then it has models which are only countably infinite. Cantor's theorem says that some sets are uncountable. Together, these theorems induce a puzzle known as Skolem's Paradox: the very axioms of set theory which prove the existence of uncountable sets can be satisfied by a merely countable model. ;This dissertation examines Skolem's Paradox from three perspectives. After a brief introduction, chapters two and three examine several formulations of (...) Skolem's Paradox in order to disentangle the roles which set theory, model theory, and philosophy play in these formulations. In these chapters, I accomplish three things. First, I clear up some of the mathematical ambiguities which have all too often infected discussions of Skolem's Paradox. Second, I isolate a key assumption upon which Skolem's Paradox rests, and I show why this assumption has to be false. Finally, I argue that there is no single explanation as to how a countable model can satisfy the axioms of set theory ;In chapter four, I turn to a second puzzle. Why, even though philosophers have known since the early 1920's that Skolem's Paradox has a relatively simple technical solution, have they continued to find this paradox so troubling? I argue that philosophers' attitudes towards Skolem's Paradox have been shaped by the acceptance of certain, fairly specific, claims in the philosophy of language. I then tackle these philosophical claims head on. In some cases, I argue that the claims depend on an incoherent account of mathematical language. In other cases, I argue that the claims are so powerful that they render Skolem's Paradox trivial. In either case, though, examination of the philosophical underpinnings of Skolem's Paradox renders that paradox decidedly unparadoxical. ;Finally, in chapter five, I turn away from "generic" formulations of Skolem's Paradox to examine Hilary Putnam's "model-theoretic argument against realism." I show that Putnam's argument involves mistakes of both the mathematical and the philosophical variety, and that these two types of mistake are closely related. Along the way, I clear up some of the mutual charges of question begging which have characterized discussions between Putnam and his critics. (shrink)

odel’s theorem than he has often been credited with. Substantively, they find in Wittgenstein’s remarks “a philosophical claim of great interest,” and they argue that, when this claim is properly assessed, it helps to vindicate some of Wittgenstein’s broader views on G¨.

Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes' Theorem is central to these enterprises both because it simplifies the calculation of conditional probabilities and because it clarifies significant features (...) of subjectivist position. Indeed, the Theorem's central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology. (shrink)

Richard Swinburne: Introduction Elliott Sober: Bayesianism - its scopes and limits Colin Howson: Bayesianism in Statistics A P Dawid: Bayes's Theorem and Weighing Evidence by Juries John Earman: Bayes, Hume, Price, and Miracles David Miller: Propensities May Satisfy Bayes's Theorem 'An Essay Towards Solving a Problem in the Doctrine of Chances' by Thomas Bayes, presented to the Royal Society by Richard Price. Preceded by a historical introduction by G A Barnard.

A fundamental problem in science is how to make logical inferences from scientific data. Mere data does not suffice since additional information is necessary to select a domain of models or hypotheses and thus determine the likelihood of each model or hypothesis. Thomas Bayes’ Theorem relates the data and prior information to posterior probabilities associated with differing models or hypotheses and thus is useful in identifying the roles played by the known data and the assumed prior information when (...) making inferences. Scientists, philosophers, and theologians accumulate knowledge when analyzing different aspects of reality and search for particular hypotheses or models to fit their respective subject matters. Of course, a main goal is then to integrate all kinds of knowledge into an all-encompassing worldview that would describe the whole of reality. A generous description of the whole of reality would span, in the order of complexity, from the purely physical to the supernatural. These two extreme aspects of reality are bridged by a nonphysical realm, which would include elements of life, man, consciousness, rationality, mental and mathematical abstractions, etc. An urgent problem in the theory of knowledge is what science is and what it is not. Albert Einstein’s notion of science in terms of sense perception is refined by defining operationally the data that makes up the subject matter of science. It is shown, for instance, that theological considerations included in the prior information assumed by Isaac Newton is irrelevant in relating the data logically to the model or hypothesis. In addition, the concepts of naturalism, intelligent design, and evolutionary theory are critically analyzed. Finally, Eugene P. Wigner’s suggestions concerning the nature of human consciousness, life, and the success of mathematics in the natural sciences is considered in the context of the creative power endowed in humans by God. (shrink)

Bell's theorem is expounded as an analysis in Bayesian probabilistic inference. Assume that the result of a spin measurement on a spin-1/2 particle is governed by a variable internal to the particle (local, “hidden”), and examine pairs of particles having zero combined angular momentum so that their internal variables are correlated: knowing something about the internal variable of one tells us something about that of the other. By measuring the spin of one particle, we infer something about its internal (...) variable; through the correlation, about the internal variable of the second particle, which may be arbitrarily distant and is by hypothesis unchanged by this measurement (locality); and make (probabilistic) prediction of spin observations on the second particle. Each link in this chain has a counterpart in the Bayesian analysis of the situation. Irrespective of the details of the internal variable description, such prediction is violated by measurements on many particle pairs, so that locality—effectively the only physics invoked—fails. The time ordering of the two measurements is not Lorentz-invariant, implying acausality. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of the statistics of the results imply it has a nonlocal—acausal interpretation; one such, the “transactional” interpretation, is presented to demonstrable advantage, and some misconceptions about quantum theory are pursued. The “unobservability” loophole in photonic Bell experiments is proven to be closed. It is shown that this mechanism cannot be used for signalling; signalling would become possible only if the hidden variables, which we insist must underlie the statistical character of the observations (the alternative is to give up), are uncovered in deviations from quantum predictions. Their reticence is understood as a consequence of their nonlocality: it is not easy to isolate and measure something nonlocal. Once the hidden variables are found, all the problems of quantum field theory and of quantum gravity might melt away. (shrink)

Legal reasoning on the requirements and application of law has been studied for centuries, but in this subject area the legal profession maintains predominantly the same stance it did in the time of the Ancient Greeks. There is a gap between the standards of proof, one which has been always demonstrated by percentages and in terms of the evaluation of these standards by percentages by mathematical or statistical methods. One method to fill the gap is Bayes theorem that (...) describes an event’s probability based on conditions that might be related to an event. Bayes theorem can help to establish or confirm a relation between facts and rules if there is sufficient other evidence that connect a party in a procedure with a considered legal action. (shrink)

This article examines the benefits of incorporating religious reflection into the psychology of religion and vice versa. By applying Bayes’ theorem, we discover that scientists and theologians can collaborate without sharing prior beliefs. Instead, rationality requires updating our beliefs before data collection in response to the degree of surprise generated by the data. Moreover, although people who start with different beliefs may become more aligned after data collection, rationality does not entail a convergence to identical beliefs. To illustrate (...) the potential for growth in understanding from greater collaboration between theologians and scientists, I examine a longitudinal investigation of religion after a natural disaster. This case study illustrates how conversations between theological and psychological perspectives on religion can lead to a more comprehensive understanding of virtue cultivation, benefiting both science and theology. (shrink)

This article discusses the use of Bayes’ Theorem in medical diagnosis with a view to examining the epistemological problems of interpreting the concept of pre-test probability value. It is generally maintained that pre-test probability values are determined subjectively. Accordingly, this paper investigates three main philosophical interpretations of probability (the “classic” one, based on the principle of non-sufficient reason, the frequentist one, and the personalistic one). This study argues that using Bayes’ Theorem in medical diagnosis does not (...) require accepting the radical personalistic interpretation. It will be shown that what distinguishes radical and moderate personalist interpretations is the criterion of conditional inter-subjectivity which applies only to the moderate account of personalist interpretation. (shrink)

Bayes's theorem is a tool for assessing how probable evidence makes some hypothesis. The papers in this volume consider the worth and applicability of the theorem. Richard Swinburne sets out the philosophical issues. Elliott Sober argues that there are other criteria for assessing hypotheses. Colin Howson, Philip Dawid and John Earman consider how the theorem can be used in statistical science, in weighing evidence in criminal trials, and in assessing evidence for the occurrence of miracles. David (...) Miller argues for the worth of the probability calculus as a tool for measuring propensities in nature rather than the strength of evidence. The volume ends with the original paper containing the theorem, presented to the Royal Society in 1763. (shrink)

In introducing the papers of the symposiasts, I distinguish between statistical, physical, and evidential probability. The axioms of the probability calculus and so Bayes’s theorem can be expressed in terms of any of these kinds of probability. Sober questions the general utility of the theorem. Howson, Dawid, and Earman agree that it applies to the fields they discuss--statistics, assessment of guilt by juries, and miracles. Dawid and Earman consider that prior probabilities need to be supplied by empirical (...) evidence, while Howson considers that there are no objective constraints on prior probabilities. I argue that simplicity is a crucial determinant of prior probability. Miller discussed how Bayes’s theorem can be interpreted so as to apply to physical probability. (shrink)

This is a high quality, concise collection of articles on the foundations of probability and statistics. Its editor, Richard Swinburne, has collected five papers by contemporary leaders in the field, written a pretty thorough and even-handed introductory essay, and placed a very clean and accessible version of Reverend Thomas Bayes’s famous essay (“An Essay Towards the Solving a Problem in the Doctrine of Chances”) at the end, as an Appendix (with a brief historical introduction by the noted statistician G.A. (...) Barnard). I will briefly discuss each of the five papers in the volume, with an emphasis on certain issues arising from the use of probability as a tool for thinking about evidence. (shrink)

About a British Academy collection of papers on Bayes' famous theorem.

This is an introduction to a collected volume. It distinguishes between evidential, statistical, and physical probability, and between objective and subjective understandings of evidential probability, in the use of Bayes’s theorem. If Bayes’s theorem is to be used to assess an objective evidential probability, a priori criteria--mainly the criterion of simplicity--are required to determine prior probability. The five main contributors to the volume discuss the use of Bayes’s theorem to assess the evidential probability of (...) scientific theories, statistical hypotheses, criminal guilt, and miracles; and also its value for assessing physical probability. (shrink)

In this paper I identify a fallacy. The fallacy is worth noting for practical and theoretical reasons. First, the rampant occurrences ofthis fallacy-especially at moments calling for careful thought-indicate that it is more pernicious to clear thinking than many of those found in standard logic texts. Second, the fallacy stands apart from most others in that it contains multiple kinds oflogical error (i.e., fallacious and non-fallacious logical errors) that are themselves committed in abnormal ways, and thus it presents a two-tiered (...) challenge to oversimplified accounts of how an argument can go bad. (shrink)

Is the restricted, consistent, version of the T-scheme sufficient for an ‘implicit definition’ of truth? In a sense, the answer is yes (Haack 1978 , Quine 1953 ). Section 4 of Ketland 1999 mentions this but gives a result saying that the T-scheme does not implicitly define truth in the stronger sense relevant for Beth’s Definability Theorem. This insinuates that the T-scheme fares worse than the compositional truth theory as an implicit definition. However, the insinuation is mistaken. For, as (...) Bays rightly points out, the result given extends to the compositional truth theory also. So, as regards implicit definability, both kinds of truth theory are equivalent. Some further discussion of this topic is mentioned (Gupta 2008 , Ketland 2003 , McGee 1991 ), all in agreement with Bays’s analysis. (shrink)

Carter and Leslie (1996) have argued, using Bayes's theorem, that our being alive now supports the hypothesis of an early 'Doomsday'. Unlike some critics (Eckhardt 1997), we accept their argument in part: given that we exist, our existence now indeed favors 'Doom sooner' over 'Doom later'. The very fact of our existence, however, favors 'Doom later'. In simple cases, a hypothetical approach to the problem of 'old evidence' shows that these two effects cancel out: our existence now yields (...) no information about the coming of Doom. More complex cases suggest a move from countably additive to non-standard probability measures. (shrink)

This chapter discusses the Bayesian analysis of miracles. It is set in the context of the eighteenth-century debate on miracles. The discussion is focused on the probable response of Thomas Bayes to David Hume's celebrated argument against miracles. The chapter presents the claim that the criticisms Richard Price made against Hume's argument against miracles were largely solid.

The central problem with Bayesian philosophy of science is that it cannot take account of the relevance of simplicity and unification to confirmation, induction, and scientific inference. The standard Bayesian folklore about factoring simplicity into the priors, and convergence theorems as a way of grounding their objectivity are some of the myths that Earman's book does not address adequately. 1Review of John Earman: Bayes or Bust?, Cambridge, MA. MIT Press, 1992, £33.75cloth.

This paper claims that adoption of Bayes's theorem as the schema for the appraisal of scientific theories can greatly reduce the distance between Kuhnians and logical empiricists. It is argued that plausibility considerations, which Kuhn considered outside of the logic of science, can be construed as prior probabilities, which play an indispensable role in the logic of science. Problems concerning likelihoods, especially the likelihood on the "catchall," are also considered. Severe difficulties concerning the significance of this probability arise (...) in the evaluation of individual theories, but they can be avoided by restricting our judgments to comparative assessments of competing theories. (shrink)