Reasoning under Uncertainty: Subjective Probability and Logic

Lydia Castronovo (University of Palermo, Italy)
Gert de Cooman (Ghent University, Belgium)
Nicole Cruz (University of Potsdam, Germany)
Tommaso Flaminio (Autonomous University of Barcelona, Spain)
Sabine Frittella (Institut National des Sciences Appliquées Centre Val de Loire, France)
Holger Leuz (University of Regensburg, Germany)
Gianluigi Oliveri (University of Palermo, Italy)
Niki Pfeifer (University of Regensburg, Germany)
Hans Rott (University of Regensburg, Germany)
Giuseppe Sanfilippo (University of Palermo, Italy)
Jon Williamson (University of Kent, UK)

Nicole Cruz

Disentangling conditional dependencies

Abstract: People draw on event co-occurrences as a foundation for
causal and scientific inference, but in which ways can events
co-occur? Statistically, one can express a dependency between events A
and C as P(C|A) != P(C), but this relation can be further specified in
a variety of ways, particularly when A and/or C are themselves
conditional events. In the psychology of reasoning, the conditional
P(C|A) is often thought to become biconditional when people add the
converse, P(A|C), or inverse, P(not-C|not-A), or both, with the
effects of these additions largely treated as equivalent. In contrast,
from a coherence based logical perspective it makes a difference
whether the converse or the inverse is added, and in what way. In
particular, the addition can occur by forming the conjunction of two
conditionals, or by merely constraining their probabilities to be
equal. Here we outline four distinct ways of defining biconditional
relationships, illustrating their differences by how they constrain
the conclusion probabilities of a set of inference types. We present a
Bayesian latent-mixture model with which the biconditionals can be
dissociated from one another, and discuss implications for the
interpretation of empirical findings in the field.

Authors: Nicole Cruz & Michael Lee

Tommaso Flaminio

Conditional, Counterfactuals, and Their Probability

Abstract: The present contribution investigates the probability of counterfactuals and their associated updating procedures using a recent characterization that combines Dempster-Shafer belief functions with probabilities of modal conditionals. The characterization represents the probability of a counterfactual as the value given to its consequent by a belief function imaged upon its antecedent.

Our result hinges upon Lewis-Ganderfors notion of imaging and upon a proposal put forward by Dubois-Prades to extend imaging outside the borders of Bayesian probability theory, and precisely to the context of Dempster-Shafer belief function theory.

While the literature lacks a comprehensive account of imaging-type procedures beyond Bayesian settings, our work addresses this gap by exploring novel classes of imaged belief functions and their connections to counterfactuals. Specifically, we leverage the established characterization to explore how properties of Lewisian models for counterfactuals induce specific properties on the corresponding imaged belief functions.

Holger Leuz

Objective Change and Teleology

Not every probability can plausibly be interpreted as a subjective probability. Probabilities in quantum mechanics are a good example. So there exist objective probabilities, or objective chance. Following B. van Fraassen, a modal frequency interpretation of objective chances of indeterministic events will be proposed and defended. Then it will be shown that objective chance, under this interpretation, has teleological and holistic features. However, the specific teleology of objective chance cannot be used to predict, per impossibile, indeterministic events because this is prevented by the holistic features of objective chance.

It will be suggested that objective chance is hard to understand when teleology and holism are seen as outdated metaphysical concepts in a physicalist metaphysics based on hidden mechanist presumptions. But since such presumptions are not part of science but only of a certain metaphysical interpretation of science, there is no reason to reject teleology and holism tout court.

Gianluigi Oliveri

Knowledge and Uncertainty in Contemporary Mathematics

From the time of Plato to the first thirty years of the twentieth century mathematics represented, within Western culture, the prototype of a knowledge producing activity whose methods, and results, are certain. It was only with Logicism, Hilbert’s programme, and Intuitionism’s failure to secure the foundations of mathematics that a new image of this field of study started to unfold. In this talk I will endeavour to show that relatively recent discussions concerning the limits, and nature, of mathematical knowledge appear to be gesturing towards a picture of mathematics in which certainty plays a less important role than it did in the traditional view of this subject.

Jon Williamson

Where do we stand on maximal entropy?

Edwin Jaynes’ principle of maximum entropy holds that one should use the probability distribution with maximum entropy, from all those that fit the evidence, to draw inferences, because that is the distribution that is maximally non-committal with respect to propositions that are underdetermined by the evidence. The principle was widely applied in the years following its introduction in 1957, and in 1978 Jaynes took stock, writing the paper ‘Where do we stand on maximum entropy?’ to present his view of the state of the art. Jaynes’ principle needs to be generalised to a principle of maximal entropy if it is to be applied to first-order inductive logic, where there may be no unique maximum entropy function. The development of this objective Bayesian inductive logic has also been very fertile and it is the task of this paper to take stock. The paper provides an introduction to the logic and its motivation, explaining how it overcomes some problems with Carnap’s approach to inductive logic and with the subjective Bayesian approach. It also describes a range of recent results that shed light on features of the logic, its robustness and its decidability, as well as methods for performing inference in the logic.