Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is stochastically (...) independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. (shrink)

We investigate the relative probabilistic support afforded by the combination of two analogies based on possibly different, structural similarity (as opposed to e.g. shared predicates) within the context of Pure Inductive Logic and under the assumption of Language Invariance. We show that whilst repeated analogies grounded on the same structural similarity only strengthen the probabilistic support this need not be the case when combining analogies based on different structural similarities. That is, two analogies may provide less support than each would (...) individually. (shrink)

We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the language with just a single predicate.

We give an account of some relationships between the principles of Constant and Atom Exchangeability and various generalizations of the Principle of Instantial Relevance within the framework of Inductive Logic. In particular we demonstrate some surprising and somewhat counterintuitive dependencies of these relationships on ostensibly unimportant parameters, such as the number of predicates in the overlying language.

We give a unified account of some results in the development of Polyadic Inductive Logic in the last decade with particular reference to the Principle of Spectrum Exchangeability, its consequences for Instantial Relevance, Language Invariance and Johnson's Sufficientness Principle, and the corresponding de Finetti style representation theorems.

It is well known that Rudolf Carnap’s original system of inductive logic failed to provide an adequate account of analogical reasoning. Since this problem was identified, there has been no shortage of proposals for how to incorporate analogy into inductive inference. Most alternatives to Carnap’s system, unlike his original one, have not been derived from first principles; this makes it to some extent unclear what the epistemic situations are to which they apply. This paper derives a new analogical inductive logic (...) from a set of axioms which extend Carnap’s postulates in a natural way. The key insights come from Bruno de Finetti’s ideas about analogy. The axioms of the new system capture epistemic conditions that call for a strong kind of analogical reasoning. The new system has a number of merits, but is also subject to limitations. I shall discuss both, together with some possible ways to generalize the approach taken in this paper. (shrink)

We propose and investigate an Analogy Principle in the context of Unary Inductive Logic based on a notion of support by structural similarity which is often employed to motivate scientific conjectures.

We propose an Analogy Principle in the context of Unary Inductive Logic and characterize the probability functions which satisfy it. In particular in the case of a language with just two predicates the probability functions satisfying this principle correspond to solutions of Skyrmsʼ ‘Wheel of Fortune’.

Additionally, the text shows how to develop computationally feasible methods to mesh with this framework.

An inductive logic is a system of inference that describes the relation between propositions on data, and propositions that extend beyond the data, such as predictions over future data, and general conclusions on all possible data. Statistics, on the other hand, is a mathematical discipline that describes procedures for deriving results about a population from sample data. These results include predictions on future samples, decisions on rejecting or accepting a hypothesis about the population, the determination of probability assignments over such (...) hypotheses, the selection of a statistical model for studying the population, and so on. Both inductive logic and statistics are calculi for getting from the given data to propositions or results that transcend the data. (shrink)

On July 14, 1866 Franz Brentano stepped up to the pulpit to defend his thesis that “the true method of philosophy is none other than that of the natural sciences”. This thesis bound his first students to him and became the north star of his school, against the complex background of the progress and specialization of the natural sciences as well as the growth and professionalization of universities. I will discuss the project of the renewal of philosophy as science in (...) the School of Brentano and how this aimed to provide a scientific foundation for the humanities independently from the natural sciences, while preserving the unity of science. Through his well-known re-introduction of the concept of intentionality as criterion to distinguish internal and external perception, Brentano was able to supply an empirical foundation for the Geisteswissenschaften. While philosophy would use the method of natural science, its domain would not be nature, but consciousness: a full-blooded science of the mind that did not require a reduction to the physical in order to be scientific. Brentano’s science of consciousness was empirical, but not experimental, and relied on subjective methods, but was not introspective. Brentano’s students Carl Stumpf, Anton Marty, Alexius Meinong, Christian von Ehrenfels, Edmund Husserl and others came to occupy important chairs in philosophy throughout Europe. While they were certainly not all orthodox followers, they adapted and spread his theories far and wide in the schools and movements they founded and influenced: Gestalt psychology, Prague linguistics, phenomenology, etc.. Moreover, the 19th century idea of scientific research as a collaborative and collective achievement led to a division of labor in Brentano’s school. Each of his students was meant to work out a part of the greater whole: Stumpf, the philosophy of sound and music; Marty, language; Meinong the history of philosophy; Husserl, mathematics; etc. Yet all of them also contributed to the shared project of the renewal of philosophy as science and discussed the (foundational) relation of philosophy to other sciences in programmatic works. Though often forgotten and overlooked due to contingent historical circumstances, the scientific paradigm of the School of Brentano was very fruitful and highly influential in philosophy and the human sciences in general, throughout the second half of the 19th and into the 20th centuries. Yet it is relevant then as now to preserve the independent scientific dignity of the humanities. (shrink)