Reasoning under Uncertainty: Subjective Probability and Logic

This interdisciplinary workshop brings together renowned international researchers from philosophy, logic, formal epistemology, and mathematics to discuss the major issues, contemporary methodologies that have arisen in the study of reasoning under uncertainty, subjective probability and logic. We will investigate a specific perspective concerning drawing inferences from uncertain premises. Topics include, but are not limited to reasoning with and about coherence, conditionals, counterfactuals, fuzzy sets, quantum probability, subjective and objective probability, trivalent logics, and maximum entropy.

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The Workshop will start on Monday, August 26, at 2 p.m. and finish on Wednesday, August 28, at 2:20 p.m. The venue is the campus of the University of Regensburg (Room VG 1.36 of the "Vielberth-Gebäude").

Monday, August 26
2.00 p.m. - 2.55 p.m.	Gert de Cooman: A common framework for conservative inference in classical and quantum probability
2.55 p.m. - 3.05 p.m.	Short break
3.05 p.m. - 4.00 p.m.	Nicole Cruz: Disentangling conditional dependencies
4.00 p.m. - 4.20 p.m.	Coffee break
4.20 p.m. - 5.15 p.m.	Sabine Frittella: Probabilistic reasoning with incomplete and contradictory information

Tuesday, August 27
10.00 a.m. - 10.55 a.m.	Giuseppe Sanfilippo & Niki Pfeifer: On Trivalent Logics, Probabilistic Weak Deduction Theorems, and a General Import-Export Principle (Part 1)
10.55 a.m. - 11.05 a.m.	Short break
11.05 a.m. - 12.00 p.m.	Niki Pfeifer & Giuseppe Sanfilippo: On Trivalent Logics, Probabilistic Weak Deduction Theorems, and a General Import-Export Principle (Part 2)
12.00 p.m. - 12.20 p.m.	Coffee break
12.20 p.m. - 1.15 p.m.	Tommaso Flaminio: Conditionals, Counterfactuals, and Their Probability
1.15 p.m. - 3.15 p.m.	Lunch break
3.15 p.m. - 4.10 p.m.	Lydia Castronovo: A compound conditionals approach to fuzzy sets
4.10 p.m. - 4.30 p.m.	Coffee break
4.30 p.m. - 5.25 p.m.	Jon Williamson: Where do we stand on maximal entropy?

Wednesday, August 28
10.00 a.m. - 10.55 a.m.	Hans Rott: Conditionals, Support and Connexivity
10.55 a.m. - 11.05 a.m.	Short break
11.05 a.m. - 12.00 p.m.	Paul Égré: Probability for trivalent conditionals
12.00 p.m. - 12.20 p.m.	Coffee break
12.20 p.m. - 1.15 p.m.	Gianluigi Oliveri: Knowledge and Uncertainty in Contemporary Mathematics
1.15 p.m. - 1.25 p.m.	Short break
1.25 p.m. - 2.20 p.m.	Holger Leuz: Objective Chance and Teleology

Timetable

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Lydia Castronovo (University of Palermo, Italy)
Gert de Cooman (Ghent University, Belgium)
Nicole Cruz (University of Potsdam, Germany)
Paul Égré (Institut Jean-Nicod, France)
Tommaso Flaminio (Artificial Intelligence Research Institute IIIA-CSIC, 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)

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Lydia Castronovo

A compound conditionals approach to fuzzy sets

A fuzzy set is a set characterized by a membership function which assigns to each object a grade of membership. The probabilistic approach to fuzzy set theory has usually been addressed as not-flexible enough to deal with fuzzy concepts. Coletti and Scozzafava, agreeing that a “classical” probability approach is not adequate to handle fuzzy logic structures, proposed a different approach to fuzzy set theory based on coherent conditional probability. Given a property φ of a random quantity X with range C_x, the authors focused on the conditional events E_φ|A_x=“You claim (that X has) the property φ, knowing that (X = x)”, for each X ∈ C_x, and they define fuzzy sets based on them. In our work, following their approach, we slightly modify this interpretation of fuzzy sets, looking at them in terms of conditional events and coherent conditional probabilities within the framework of the betting scheme. A revised definition of fuzzy sets is proposed and we introduce operations such as complement and intersection. To achieve this, we employ logical operations among conditional events in the framework of conditional random quantities. In particular in our interpretation, the membership function of a fuzzy set is seen as a coherent conditional prevision assessment on suitable compound conditionals. Furthermore, we provide examples and examine the class of Frank t-norms to describe coherent conditional previsions.

Gert de Cooman

A common framework for conservative inference in classical and quantum probability

Over the past 30 years, methods have been developed for representing, and performing conservative inference based on, partial probabilistic information, such as bounds on probabilities and expectations, and weakened structural (independence, exchangeability, symmetry, …) assessments. This has led to the development of efficient methods for dealing with partial information in stochastic processes and probabilistic graphical networks, amongst other fields. While these developments have largely focused on classical probabilities, recent developments have also started to consider quantum probabilities. The talk, based on my recent research, will present the basic ideas and tools for, as well as a number of interesting aspects of, a unified approach to dealing with conservative inference in both areas of probability theory.

Nicole Cruz

Disentangling conditional dependencies

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

Paul Égré

Probability for trivalent conditionals

This paper develops a trivalent semantics for indicative conditionals and extends it to a probabilistic theory of valid inference and inductive learning with conditionals. On this account, (i) all complex conditionals can be rephrased as simple conditionals, connecting our account to Adams's theory of p-valid inference; (ii) we obtain Stalnaker's Thesis as a theorem while avoiding the well-known triviality results; (iii) we generalize Bayesian conditionalization to an updating principle for conditional sentences. The final result is a unified semantic and probabilistic theory of conditionals with attractive results and predictions.

Authors: Paul Égré, Lorenzo Rossi, Jan Sprenger

Tommaso Flaminio

Conditionals, Counterfactuals, and Their Probability

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.

Sabine Frittella

Probabilistic reasoning with incomplete and contradictory information

Abstract: Belnap-Dunn logic (BD) [1] was designed to reason about incomplete and contradictory information. It is a paracomplete and paraconsistent propositional logic, that is, BD is a propositional logic that satisfies the same axioms as classical propositional logic except for the principle of explosion and the law of excluded middle. In this talk, we will discuss probabilistic reasoning over BD. In the first part of the talk, we provide preliminaries on BD and probabilities over BD, and we present two-layered logics and how to use them to formalize probabilistic reasoning [2,3]. In the second part, we introduce belief functions (a generalization of probability measures) and discuss how to define and interpret them over BD [2,4].

[1] Belnap, N.D. (2019). How a Computer Should Think. In: Omori, H.,
Wansing, H. (eds) New Essays on Belnap-­Dunn Logic. Synthese Library,
vol 418. Springer, Cham.
[2] Marta Bílková, Sabine Frittella, Daniil Kozhemiachenko, Ondrej
Majer, Sajad Nazari: Reasoning with belief functions over Belnap–Dunn
logic, Annals of Pure and Applied Logic, 2023.
[3] Marta Bílková, Sabine Frittella, Daniil Kozhemiachenko, Ondrej
Majer: Two-Layered Logics for Paraconsistent Probabilities. WoLLIC 2023:
101-117
[4] Marta Bílková, Sabine Frittella, Daniil Kozhemiachenko, Ondrej
Majer, Krishna Manoorkar: Describing and quantifying contradiction
between pieces of evidence via Belnap Dunn logic and Dempster-Shafer
theory. ISIPTA 2023: 37-47

Holger Leuz

Objective Chance 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.

Niki Pfeifer & Giuseppe Sanfilippo

On Trivalent Logics, Probabilistic Weak Deduction Theorems, and a General Import-Export Principle (Part 2)

We show that the deduction theorem does not follow for conditional events when p-entailment is used. We give some probabilistic weak versions of the deduction theorem, with further results and examples. We also consider new inference
rules related to Aristotelian syllogisms. We focus on iterated conditionals and the invalidity of the Import-Export principle for the conditional event. In particular, we explain how the invalidity of the classical deduction theorem is related to the invalidity of the Import-Export principle for conditional events. Then, we introduce a General Import-Export principle for iterated conditionals, and we present a result which relates it to p-consistency and p-entailment. We also illustrate the validity of the General Import-Export principle for some inference rules of System P, where the Probabilistic Weak Deduction Theorem is not applicable.

Authors: Angelo Gilio, David E. Over, Niki Pfeifer, & Giuseppe Sanfilippo

Hans Rott

Conditionals, Support and Connexivity 

In natural language, conditionals are frequently used for giving explanations. Thus the antecedent of a conditional is typically understood as being connected to, being relevant for, or providing evidential support for the conditional’s consequent. This aspect has not been adequately mirrored by the logics that are usually offered for the reasoning with conditionals: neither in the logic of the material conditional or the strict conditional, nor in the plethora of logics for suppositional conditionals that have been produced over the past 50 years. In this paper I survey some recent attempts to come to terms with the problem of encoding evidential support or relevance in the logic of conditionals. I present models in a qualitative-modal and in a quantitative-probabilistic setting. Focusing on some particular examples, I show that no perfect match between the two kinds of settings has been achieved yet.

Giuseppe Sanfilippo & Niki Pfeifer

On Trivalent Logics, Probabilistic Weak Deduction Theorems, and a General Import-Export Principle (Part 1) 

We recall selected results on coherence for conditional events and conditional random quantities. We show that, when a conditional probability is evaluated, the following schemes are equivalent: the scheme for betting on a conditional, the scheme of a conditional bet, and a third scheme. We recall the notions of p-consistency and p-entailment under coherence. Then, we illustrate results on compound and iterated conditionals in the framework of conditional random quantities. We introduce the generalized Equation or conditional prevision hypothesis. We review de Finetti’s trivalent analysis of conditionals. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity, and compare these notions with with our apprach to compound and iterated conditionals. In particular, we discuss the notions of assertability for conditionals and SS-TT-validity. We illustrate our and related approaches by discussing several examples and counterexamples.

Authors: Angelo Gilio, David E. Over, Niki Pfeifer, & Giuseppe Sanfilippo

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.

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