Conditionals 2024

Title: A Semantic Framework for Modelling Conditional Reasoning

Abstract: We introduce and investigate a very basic semantics for conditionals that can be used to define a broad class of conditional reasoning, and we show that it encompasses the most popular kinds of conditional reasoning developed in logic-based KR. Also, the semantics we propose is appropriate for a structural analysis of those conditionals that do not satisfy closure properties typically associated with classical reasoning, like, for example, Right Weakening.

Title: Probabilistic consequence relations

Abstract: This paper, joint work with David Ripley (Monash), investigates logical consequence defined in terms of probability distributions, for a classical propositional language using a standard notion of probability. We examine three distinct probabilistic consequence notions, which we call material con- sequence, preservation consequence, and symmetric consequence. While material consequence is fully classical for any threshold, preservation consequence and symmetric consequence are subclassical, with only symmetric consequence gradually approaching classical logic at the limit threshold equal to 1. Our results extend earlier results obtained by J. Paris in a Set-Fmla setting to the Set-Set setting, and consider open thresholds beside closed ones. In the Set-Set setting, in particular, they reveal that probability 1 preservation does not yield classical logic, but supervaluationism, and conversely positive probability preservation yields subvaluationism.

Title: Conditionals , 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. This characterization represents the probability of a counterfactual  as the value given to its consequent by a belief function imaged upon its antecedent.

Such result hinges upon Lewis-Ganderfors notion of imaging and upon a proposal put forward by Dubois and Prade 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.

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Title: Proof theory of conditional logics

Abstract: Conditional logics, as introduced by David Lewis in 1973, enrich the language of classical propositional logic with a two-places modal operator, the conditional, suitable to represent fine-grained notions of conditionality. The proof theory of conditional logics relies on proof-theoretic techniques similar to those employed to define proof systems for modal logics: either the language of sequent calculus is enriched, giving rise to labelled calculi, or additional structural connectives are employed, thus defining various kinds of structured sequents (e.g., nested sequents). In this talk I will present sequent calculi for conditional logics belonging both to the labelled and to the structured approach. 

After introducing conditional logics and their semantics, that I will define in terms of neighborhood models, I will present a labelled sequent calculus, modularly capturing a large family of systems, and a nested-style sequent calculus, featuring a structural connective representing neighborhoods of the model. Other than the conditional operator, I will discuss the comparative plausibility operator, also introduced by Lewis, which expresses comparisons between states or concepts. I will show how this approach provides an uniform model-theoretic and proof-theoretic treatment of this operator as well.

This talk is based on joint work with: Tiziano Dalmonte, Bjoern Lellmann, Sara Negri, Nicola Olivetti and Gian Luca Pozzato.

Title: Relating the algebraic and the random/possibilistic variables approaches to conditionals: two sides of a same coin

Abstract: The aim of this talk is to highlight the strong connections between two apparently different approaches to (compound) conditionals: the algebraic approach by means the so-called Boolean algebras of conditionals and the canonical extension of probabilities to these algebras in accordance to Stalnaker thesis, and de Finetti’s notion of conditional as a three-valued object, with betting-based semantics, and its related approach of modelling (compound) conditionals as random quantities. It can be shown that, by means of a natural procedure to explicitly attach conditional random quantities to arbitrary compound conditionals, they can be endowed with a Boolean algebra structure, isomorphic to the one of the first approach. And moreover, one can show that the previsions of these random quantities coincide with the probabilities of their associated conditionals computed through the canonical extension approach.

In the last part of the talk, we show that this strong relationship within the probabilistic-based approach to conditionals can also be developed within the possibilistic framework, where conditionals are attached with suitable possibilistic variables instead of random quantities. The possibilistic expectation of these variables (a Generalized Sugeno integral) now provides a means of extending the original possibility distribution on events to (compound) conditional objects. And this possibilistic approach eventually leads to exactly the same underlying Boolean algebraic structure for the set of conditionals. 

Title: The Relevance of Conditionals for Cognitive Logics

Abstract: Classical logics like propositional or predicate logic have been considered as the gold standard for rational human reasoning, and hence as a solid, desirable norm on which all human knowledge and decision making should be based, ideally. For instance, Boolean logic was set up as kind of an algebraic framework that should help make rational reasoning computable in an objective way, similar to the arithmetics of numbers. Computer scientists adopted this view to (literally) implement objective knowledge and rational deduction, in particular for AI applications. Psychologists have used classical logics as norms to assess the rationality of human commonsense reasoning. However, both disciplines could not ignore the severe limitations of classical logics, e.g., computational complexity and undecidedness, failures of logic-based AI systems in practice, and lots of psychological paradoxes. Many of these problems are caused by the inability of classical logics to deal with uncertainty in an adequate way. Both disciplines have used probabilities as a way out of this dilemma, hoping that numbers and the Kolmogoroff axioms would do the job (somehow). However, psychologists have been observing also lots of paradoxes here (maybe even more).

So then, are humans hopelessly irrational? Is human reasoning incompatible with formal, axiomatic logics? In the end, should computer-based knowledge and information processing be considered as superior to human reasoning regarding objectivity and rationality?

Cognitive logics aim at overcoming the limitations of classical logics and resolving the observed paradoxes by proposing logic-based approaches that can model human reasoning consistently and coherently in benchmark examples. The basic idea is to reverse the normative way of assessing human reasoning in terms of logics resp. probabilities, and to use typical human reasoning patterns as norms for assessing the cognitive quality of logics. Cognitive logics explore the broad field of logic-based approaches between the extreme points marked by classical logics and probability theory with the goal to find more suitable logics for AI applications, on the one hand, and to gain more insights into the structures of human rationality, on the other.  This talk features conditionals and preferential nonmonotonic reasoning as a powerful framework to explore characteristics of human rational reasoning. We show that interpreting common-sense rules in terms of conditionals and processing them with basic techniques of nonmonotonic logics provides a key to formalize human rationality in a much broader and more adequate way,  resolving in particular lots of paradoxes in psychology.

Title: Inferentialism and Walrus conditionals  

Abstract: We will briefly review psychological results that confirm the conditional probability hypothesis (CPH): that the probability of the natural language conditional, P(if p then q), is the conditional probability of q given p, P(q|p). Edgington (1995) suggested that this hypothesis could fail for conditionals like, "If Napolean is dead then Oxford is in England". We will call such examples Walrus conditionals. Later research has supported Edgington's prediction, with a tendency for P(if p then q) judged to be lower than P(q|p) for Walrus conditionals. What is the explanation of this limitation to the CPH? Researchers who take a strong inferentialist view claim that a "standard" conditional if p then q is acceptable if and only if P(q|p) > P(q|not-p), i.e., p raises the probability of q. Truth condition inferentialism holds that if p then q is true if and only if there is a sufficiently strong relation (deductive, inductive, ...) between p and q. Note that there are many acceptable independence conditionals for which P(q|p) = P(q|not-p). Further, it is often highly informative to use the conjunction if p then q and if not-p then q, from which q necessarily follows in a perfectly acceptable dilemma inference. We will also consider the de Finetti normal form for if p then q, which is if p then p & q, and point out that, generally, P((p & q)|p) > P((p & q)|not-p). But if p then p & q is intuitively unacceptable when if p then q is a Walrus conditional. We will argue that these points support the argument that the problem with Walrus conditionals is pragmatic, and we will refer to work by Lassiter (2023) and others that provides strong support for this conclusion.

Title: Experimental formal philosophy of conditionals

Abstract: The normative side of conditionals is one of the key topics of formal epistemology, while the descriptive side is a key topic in experimental philosophy. In my talk, I build a bridge between both disciplines by presenting selected experiments on how people interpret conditionals within coherence-based probability logic which serves as a unified rationality framework for reasoning about conditionals. In particular, I discuss recent results on a generalised version of the probabilistic truth table task, which shows that people interpret conditionals as conditional probability statements.

The data speak also against inferentialist accounts of conditionals. Additionally, I present two probabilistic approaches to connexivity (i.e., logical principles that catch the intuition that conditionals, whose antecedents and consequents contradict each other, appear intuitively false) and show experimentally the psychological plausibility of the proposed approach.

Title: Conditionals in Explainable AI

Abstract: In this talk, we will discuss the role of conditionals in Explainable AI. I will focus on two classes of conditionals that align well with my research interests, namely probabilistic conditionals and counterfactual conditionals. The literature is currently mostly driven by ideas from machine learning and may benefit from insights about reasoning with conditionals to improve analytical guarantees of explanations and to design better informed algorithms. I think that this may be an interesting application area for the workshop audience and may spark interesting discussions and collaborations.

The role of probabilistic conditionals in Explainable AI emerged from rule-based explanations. While rule-based explanations have a long tradition in machine learning, recently an interesting direction evolved that applies propositional-logical reasoning technology to infer provably correct rules from machine learning models. While deterministic rules are interesting, they are unlikely to explain much of what a machine learning model learnt because most rules come with exceptions. This strand of work has already been expanded to probabilistic rules, but mostly makes use of classical-logical workarounds or purely numerical ideas. I will give an introduction to the area and discuss how some ideas from the literature on reasoning about (probabilistic) conditionals can enrich the current Landscape.

Another interesting direction in Explainable AI are counterfactual explanations. Here, explanations take the form of counterfactual conditionals of the form “If it had not been for X, then the decision would not have been Y”. For example, in a loan application scenario, such a counterfactual explanation may explain the reasons for a denied application to an applicant. For example, the conditional could take the form “If your debt-to-income ratio would be lower, then your application would not have been rejected.” An obvious problem here is that there is typically not a unique explanation, which causes multiple problems. I will again introduce the area and discuss some ideas how methods for reasoning about conditionals may be helpful to balance completeness and the number of counterfactual explanations, and to improve the robustness of counterfactual explainers.

Title: Strictly Strict Conditionals

Abstract: This work aims at developing a novel framework for conditional logic that addresses the limitations of both material and strict implication. While the strict conditional improves upon material implication by resolving certain paradoxes, it remains inadequate for capturing non-monotonic conditionals like counterfactuals. We argue that this limitation stems from the material implication component of the strict conditional. Building on previous precursory works (Rosella, Flaminio, Bonzio, 2023; Rosella, Sprenger, 2024), we propose a strict conditional analysis based on necessitated non-material implications. Specifically, we explore the consequences of replacing material implication with Lewis's conditional(s) and other kinds of connectives in the definition of strict conditionals. This leads to a new semantic framework capable of unifying various types of conditionals, traditionally treated with disparate semantics. We investigate the resulting logics, which we call Strictly Strict Conditional Logics, and demonstrate their expressive power by proving some embeddability of well-known conditional logics within this framework. On a conceptual level, we argue that our approach offers a more comprehensive, explanatory, and uniform account of conditionals, defined using basic primitive connectives and providing novel insights into their truth conditions. 

Title: Compound Conditionals as Conditional Random Quantities

Abstract: The problem of  how to assign degree of beliefs to conjunctions or disjunctions of conditionals, or to conditionals with conditionals in their antecedents or consequents, has been largely studied in literature. 

As an example of a conjoined conditional consider two soccer matches. For each (uncancelled) match the possible outcomes are:  home win, draw, and away win. 

Then, the conjunction sentence

The outcome of the 1st match is home win (if the 1st match is uncancelled)

and

the outcome of the 2nd is draw  (if the 2nd match is uncancelled)

is a conjoined conditional, because each conjunct is itself a conditional.

Tipically, a conditional event is viewed as a three-valued object (with possible values true,  false,  void) and compound  conditionals  have also been  defined  within trivalent logics.

We start by  reviewing  de Finetti's trivalent analysis of conditionals and  discussing the  equivalence between conditional bets and  bets on conditionals. We  examine selected trivalent logics and show that none satisfies  all the  basic logical and probabilistic properties valid for unconditional events. Then, we  illustrate the notions of compound and iterated conditionals introduced, in recent papers, as suitable conditional random quantities in the setting of coherence.

We show that in this framework all the basic logic and probabilistic properties valid for unconditional events are preserved. We discuss the notion of iterated conditional and the  invalidity of the Import-Export principle, which allow us to avoid Lewis' triviality results.

Next, we illustrate how compound and iterated conditionals can effectively characterize the p-validity of inference rules in nonmonotonic reasoning. We then explore possible  applications of compound conditionals in the  psychology of uncertain reasoning, to connexive  logic,  to non-monotonic reasoning, and to fuzzy logic. We also provide an overview of a possible generalization. Finally, we observe  that  the approach to compound conditionals as conditional random quantities  is in agreement with the theory of Boolean algebras of conditionals. 

Title: Conditionals in algebraic logic

Abstract: The role of algebra has been pivotal in the formalization and understanding of reasoning; indeed, modern logic really flourishes with the rise of the formal methods of mathematical logic, which moves its first steps with George Boole's intuition of using the symbolic language of algebra as a mean to formalize how sentences connect together via logical connectives. More recently, the advancements of the discipline of (abstract) algebraic logic have been one of the main drivers behind the surge of systems of nonclassical logics in the 20th century. Conditional statements are certainly pivotal in the representation of knowledge and reasoning, and there is a vast literature that analyses them also from a logical perspective, essentially rooted in the work of Lewis and Stalnaker; nonetheless, a logico-algebraic treatment of conditionals is currently lacking. 

The purpose of this talk is to start filling this gap by utilising the well-developed logico-algebraic machinery to study conditional statements, specifically, by expanding the language of classical logic by a binary operator a/b that reads as "a given b". First, we will carry Lewis’s hierarchy of logics within the realm of (abstract) algebraic logic, which entails considering such logics as consequence relations, instead of sets of theorems. We will then introduce the corresponding algebraic semantics, and deepen the theoretical understanding of the logics in Lewis’s framework, particularly by drawing connections and import results and techniques from modal logic. In the last part of the talk, we will “start over”, and rethink conditional statements starting from a purely logico-algebraic intuition which sees the operator / as a quotient operator in Boolean algebras. Interestingly, if not surprisingly, we will show that this intuitive idea leads us back to Lewis/Stalnaker-like models. This talk is based on joint works with Tommaso Flaminio, Francesco Manfucci, and Giuliano Rosella.

Title: Markov Graph Models for Conditionals

Abstract: The aim is to describe a general model for computing probabilities of conditionals. The mathematical environment is the theory of Markov chains (we need only rudimentary facts). The general idea is to represent the conditional as a game and the probability of the conditional is the probability of winning the game – i.e. the absorption probability in the special winning state in the graph. We use the term “Markov graph”, as the presented construction offers a natural and convenient graphical illustration of how the game evolves. We restrict our attention to conditionals where the antecedents have a positive probability. The presented models comprise conditionals of arbitrary complexity. We will discuss two implementations of these general idea.

“Dedicated graphs approach”. We present an inductive construction of a family of graphs G(α) – such that the graph G(α) is designed to model a particular conditional α. Each graph G(α) generates a canonical probability space S(α)=(Ωα, Σα, Pα), where α is given an interpretation as an event [α]⊆Ωα so that the probability of α can be computed ad Pα([α]). Elementary events in Ωα can represent the game scenarios. The graph allows to compute this probability by solving a system of linear equations (being the standard system for absorption probabilities for a Markov chain). In order to give a general inductive definition of the graph G(α) we define three operations on graphs which correspond to the negation, conjunction and the conditional →.

“General probability space approach”. We first identify an ascending chain of languages L0⊆L1⊆..., comprising all conditionals (with L0 being the factual language). Ln+1 contains all conditionals α→β with α,β∈Ln and their Boolean combinations. Every conditional is represented in this chain. Then we construct a corresponding sequence of probability spaces Sn, with S0 being the factual sample space, and Sn is suitable to interpret Ln. Informally speaking – each Sn allows to interpret all conditionals with a bounded degree of “nestedness” i.e. from the ascending chain of languages L0⊆L1⊆..., comprising all conditionals. In this way, all conditionals will have an interpretation in one of the spaces. The crucial step consists in defining (for a set of simple conditionals A1→B1; A2→B2; ... An→Bn) a Markov graph (with a corresponding probability space) which allows one to interpret all the conditionals and to compute the probabilities of all their Boolean combinations. This procedure can be iterated. The model allows one to show that the probability assignment concerning the initial, factual beliefs has a unique, well-defined extension to the probabilities of all conditionals definable in the language. The simplicity of the model makes it a more convenient tool than, for example, Stalnaker Bernoulli spaces.

The first method is directed as presenting a quick and efficient method of computing probabilities of certain conditionals. The second method is directed at providing a more general theoretical model, offering a natural semantics (endowed with a probabilistic structure).

9:00 - 9:30 Registration and Opening

9:30 - 10:30 Giovanni Casini - A Semantic Framework for Modelling Conditional Reasoning

10:30 - 11:00 Coffee Break

11:00 - 12:00 Marianna Girlando - Proof theory of conditional logics

12:00 - 12:10 Short Break

12:10 - 13:10 Nico Potyka - Conditionals in Explainable AI

13:10 - 15:00 Lunch Break

15:00 - 16:00 Gabriele Kern-Isberner - The Relevance of Conditionals for Cognitive Logics

16:00 - 16:30 Coffee Break

16:30 - 17:30 Anna and Krzysztof Wójtowicz - Markov Graph Models for Conditionals

9:00 - 10:00 David Over and Simone Sebben - Inferentialism and Walrus conditionals

10:00 - 10:30 Coffee Break

10:30 - 11:30 Niki Pfeifer - Experimental formal philosophy of conditionals

11:30 - 11:45 Short Break

11:45 - 12:45 Sara Ugolini - Conditionals in algebraic logic

12:45 - 15:00 Lunch Break

15:00 - 16:00 Giuliano Rosella - Strictly Strict Conditionals

16:00 - 16:30 Coffee Break

16:30 - 17:30 Tommaso Flaminio - Conditionals, Counterfactuals and Their Probability

9:00 - 10:00 Paul Égré - Probabilistic consequence relations

10:00 - 10:30 Coffee Break

10:30 - 11:30 Giuseppe Sanfilippo - Compound Conditionals as Conditional Random Quantities

11:30 - 11:45 Short Break

11:45 - 12:45 Lluis Godo - Relating the algebraic and the random/possibilistic variables approaches to conditionals: two sides of a same coin

12:45 - 15:00 Lunch Break