Logic@LLC: Seminars

Giuliano Rosella (University of Turin): Trivial Pursuit: A journey through triviality results and probability of conditionals.

Tuesday 17 December 2024, 12:00-13:00, Palazzo Nuovo, Aula 23

In his seminal 1976 paper, David Lewis demonstrated that a straightforward interpretation of conditional probability as the probability of a conditional event leads to a trivialization of the probability calculus. This result, often referred to as Lewis's Triviality Result, has profound implications for our understanding of conditional probability and its role in various fields, including philosophy, logic, and artificial intelligence.

This talk aims to explore extensions and generalizations of Lewis's Triviality Result to other uncertainty measures and updating procedures. By doing so, we will establish a hierarchy of triviality results that delineate the boundaries of probabilistic reasoning for conditionals. One significant consequence of these results is that the probabilistic behavior of a wide range of conditionals diverges from standard Bayesian revision and updating procedures.

These findings raise intriguing philosophical questions about the nature of conditional probability, the mechanisms of belief revision and updating, and the methodological soundness of probabilistic approaches to conditionals. We will argue that many existing probabilistic accounts of conditionals may be fundamentally flawed, as they fail to accurately capture the probability of conditionals in various contexts.

Mariela Rubin (University of Buenos Aires):  A substructural route to Gibbard’s collapse result for conditionals

Tuesday 29 October, 12-13h, Aula di Antica, Palazzo Nuovo, second floor

In 1980, Gibbard proved that when one extends classical logic with an indicative conditional that is supraclassical, that validates Import-Export and that is at least as strong as the material conditional, then they collapse. Later on, Fitelson showed that it is not necessary to assume classical logic, rather intuitionistic logic suffices for the collapse to happen. Both results assume the structurality of the consequence relation.

In this work, I will show that the collapse can happen in even weaker logics. As a consequence of this result, several non-monotonic conditionals will also collapse to the indicative. Following Belnap (1963)’s arguments about tonk, I will reflect on how the consequence relation influences the meaning of the connectives one is defining and I will argue that if one thinks the meaning of a conditional in terms of the rules it validates, then some of these conditionals are good candidates to model indicatives.

Francesco Genco (University of Turin): A Logical Theory of Computational Trustworthiness in a Calculus with Dependent Types 

Tuesday 22 October, 12-13h, Aula di Antica, Palazzo Nuovo, second floor

Probabilistic programs—that is, programs featuring stochastic computational steps—are used extensively and fruitfully in many areas of computer science. These programs have radically changed our perspective on well-established computational notions such as correctness. Because of their nondeterministic behaviour, indeed, typical probabilistic programs cannot be said to always compute the correct output. In practice, nevertheless, we often have quite strong expectations about the frequency with which certain outputs should be returned by a given probabilistic program. A suitable weakening of the notion of correctness, called trustworthiness, has been defined in order to formalise this intuition. We present a type-theoretical system that enables us to encode the computational procedures required to establish whether a program is trustworthy or not and to logically reason about the behaviour of programs with respect to trustworthiness.