Bioethics, Volume 36, Issue 1, Page 25-41, January 2022.

Discussing the relations between logic and probability, this book compares classical 17th- and 18th-century theories of probability with contemporary theories, explores recent logical theories of probability, and offers a new account of probability as a part of logic.

We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, (...) if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

We consider the following question of Kunen: Does Con imply Con? We use core model theory to investigate consequences of the existence of such a j : M $\rightarrow$ V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the (...) question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$ , provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$ . There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.

Ramsey's “Facts and Propositions” is terse, allusive, and dense. The paper is far from easy to understand. The present essay is an effort, largely following Brian Loar's account,1 to say what Ramsey's goal is, to spell out what he took to be the means to accomplish it, and to show how those means, at least in the terms of F&P, cannot accomplish that end. I also contrast Loar's own account of judgment, explicitly modeled on Ramsey's view, with the latter. The (...) exercise is not at all academic. Loar makes clear the striking depth and originality of Ramsey's insights. (shrink)

How does deductive logic constrain probability? This question is difficult for subjectivistic approaches, according to which probability is just strength of (prudent) partial belief, for this presumes logical omniscience. This paper proposes that the way in which probability lies always between possibility and necessity can be made precise by exploiting a minor theorem of de Finetti: In any finite set of propositions the expected number of truths is the sum of the probabilities over the set. This is generalized to apply (...) to denumerable languages. It entails that the sum of probabilities can neither exceed nor be exceeded by the cardinalities of all consistent and closed (within the set) subsets. In general any numerical function on sentences is said to be logically coherent if it satisfies this condition. Logical coherence allows the relativization of necessity: A function p on a language is coherent with respect to the concept T of necessity iff there is no set of sentences on which the sum of p exceeds or is exceeded by the cardinality of every T-consistent and T-closed (within the set) subset of the set. Coherence is easily applied as well to sets on which the sum of p does not converge. Probability should also be relativized by necessity: A T-probability assigns one to every T-necessary sentence and is additive over disjunctions of pairwise T-incompatible sentences. Logical T-coherence is then equivalent to T-probability: All and only T-coherent functions are T-probabilities. (shrink)

behaviour from the rival manufacturer. We consider the case where franchise fees can be used to extract retailers' surplus. We show that vertical separation is in the collective, as well as individual, interest of manufacturers, and hence facilitates some collusion in the simple setting..

The main sources here are Hume, Husserl, and De Finetti. The problem is how phenomenological investigation has to do with partial or probabilistic judgment. Behavioristic, frequentist and subjectivistic views are briefly surveyed. A variant of Hume's account of the probability of chances is developed with the help of De Finetti's concept of exchangeability. The question of transcendental elements in or behind partial judgment is considered in the light of understanding disagreement and error in partial judgment.

In this paper I consider the question of the reality of chance. This is not what divides contemporary probabilists into the objective and subjective schools. That division is accomplished by the question whether there are objective grounds for the correctness of probability judgments. The subjectivists say that there need not be such grounds, and that probability judgments thus need not be empirically meaningful in the verificationist sense, or perhaps that they are not judgments at all, but rather expressions of attitude. (...) (See [11], pp. 156-198). The objectivists say that the truth conditions for probability judgments are such as the frequencies of traits in populations or the existence of propensities. Objectivists and subjectivists are, however, agreed that there are no objective chances. (See [2], pp. 141, 142; and [12], §16).There is a Humean argument against the reality of chance which depends mainly upon the premise that we have no experience of it. (shrink)

Ramsey's “Facts and Propositions” is terse, allusive, and dense. The paper is far from easy to understand. The present essay is an effort, largely following Brian Loar's account,1 to say what Ramsey's goal is, to spell out what he took to be the means to accomplish it, and to show how those means, at least in the terms of F&P, cannot accomplish that end. I also contrast Loar's own account of judgment, explicitly modeled on Ramsey's view, with the latter. The (...) exercise is not at all academic. Loar makes clear the striking depth and originality of Ramsey's insights. (shrink)

The interpretation of the calculus of probability as a logic of partial belief has at least two advantages: it makes the assignment of probabilities plausible in cases where classical frequentist interpretations must find such assignments meaningless, and it gives a clear meaning to partial belief and to consistency of partial belief.

This fine book is a comprehensive and careful survey of the current situation in the methodology of economics. It is directed primarily at economists and students of economics. Indeed, the economist who reads it with the care it deserves will have a better grip on matters of methodology in economics than most philosophers of science, but philosophers and historians of science will also find the work rewarding and interesting. Though a few examples may be beyond the economically untutored reader, they (...) are not essential to the exposition, and other examples are accessible and enlightening: the brief discussions of Paul Samuelson's revealed preference approach and Julie Nelson's criticism of Gary Becker's work on economics of the family are good instances. Chapter 2, “The Methodological Tradition in Economics,” is a calm and brief introduction to its subject.Until quite recently practicing scientists, economists among them, and philosophers of science presupposed several strong theses about science and knowledge: that all genuine knowledge is scientific; that the objects of this knowledge are eternal and free of context; that all knowledge and all science is one, unified at least by a common scientific method; that, though not infallible, this method is progressive—it approaches truth as a limit—and operates in independence of the nonscientific beliefs and values of its practitioners. These principles cannot be presupposed today. Those who would accept any of them must be ready to meet serious criticisms. The import of this contemporary Methodenstreit as it affects economics is the dominant theme of Reflection without Rules. The reader is left with the conviction that the science of economics and the study of its methodology are inseparable and that it was only dedicated and myopic attachment to The Legend that kept this messy business at bay. Nor can there be any return to the age of innocent division of labor—economists to economics, philosophers to methodology.There are too many topics treated in the book even to list in a brief review. There are calm and clear accounts of Thomas Kuhn's work; of logical positivism ; of the deep effects of the collapse of The Legend on Walrasian equilibrium models; of Daniel Hausman's work on the nature of economics; of the strong program, which strives to reduce scientific activity to the effects of the social interests of scientists; of Alvin Goldman's reliabilism and social epistemology ; and of the important differences between pre‐ and post‐Legend views of science and knowledge.The question of values in science arises for economics in a special and pointed way, for economics studies many sorts of values. It also, at least since Adam Smith, deals in the unintended collective consequences of individual intentional actions. Economics evidently consists in the largely unintended consequences of the intentional actions of individual economists, and, once The Legend is discredited, economics is seen to navigate the very currents that it studies. This opens the door to the virtuous possibility of self‐criticism in tune with the strong naturalism of contemporary epistemology.It is not the least virtue of Reflection without Rules that its author wears his impressive erudition gracefully and well. The book is sophisticated and at the same time naive in the best sense. It will fortunately be the standard reference for work in the methodology of economics for some time to come. (shrink)