Logic@LLC: Seminars

Camila Gallovich (University of Buenos Aires & CONICET): On two aspects of ungroundedness 

Thursday 10 April, 16-18h, Meeting Room 1, Philosophy Library, Palazzo Nuovo (basement)

The process of grounding the statements of a first order language with truth and the characterization of the semantic notions associated with that process can be made precise in different ways. In his Outline of a theory of truth, Kripke provides a bottom-up explanation of the process that makes use of a fixed-point semantics. The fixed-point construction allows for a rigorous characterization of the notions of groundedness, paradoxicality, hypodoxicality, s-truth, and s-falsity. Alternatively, in Truth, Dependence, and Paradox, Yablo provides a top-down explanation of the process of grounding that makes use of the elements of graph theory. The dependence account yields characterizations of the notions of groundedness and paradoxicality that are coextensive with those provided by the fixed-point conception of truth. One limitation affecting Yablo's proposal is that it remains silent on the semantic status of statements that are neither grounded nor paradoxical.

To the extent that both theories are intended to serve as formal explanations of different aspects of the concepts of groundedness and ungroundedness, they can be naturally seen as motivating a form of theoretical pluralism. However, one may contend that the two approaches are not equally good since Yablo’s treatment of ungroundedness is not as fine-grained as one might expect. The aim of this talk is to build on the dependence account and argue that, at least to the extent that fine-grainedness is concerned, there is no principled reason to prefer one explanation over the other.

The talk will run as follows. In the first part, I make Kripke's fixed-point construction and Yablo’s dependence-based construction precise. In the second part, I provide dependence-style characterizations of the relevant semantic notions and show that the resulting characterizations extensionally coincide with those provided by the fixed-point conception.

Martina Zirattu (University of Turin): Non-deterministic semantics for logics of analytic implication. 

Thursday 13 March, 12:00--13:00, Meeting Room 1, Philosophy Library, Palazzo Nuovo

We provide novel non-deterministic semantics for some content inclusion logics located between the first-degree entailment fragment of Parry's logic PAI and that of Angell's logic of analytic implication AC. Our semantics is inspired by a family of two-address semantics recently developed by Song et al. following previous ideas introduced by Herzberger and Woodruff, that suggested independently evaluating formulas in what pertains to their alethic and topical status. Building on such a proposal, we explore the result of allowing negation to be non-deterministic on either of these independent aspects. Thus, we simulate the presence of truth-value gaps and gluts by letting negation work non-deterministically on the alethic coordinate. At the same time, we do as much with the presence of topic-transformed negated formulas by letting negation work non-deterministically on the topical coordinate. (The paper is based on joint work with Damian Szmuc.)

Lorenzo Rossi and Jan Sprenger (University of Turin) : Trivalent Semantics for Conditional Obligations

Wednesday 5 March, 12:00--13:00, Meeting Room 1, Philosophy Library, Palazzo Nuovo

This paper provides a new framework for formalizing conditional obligations in natural language by combining a trivalent semantics for the indicative conditional with a unary deontic operator. This means that we provide a fully compositional and modular theory of "if",  "ought", and the way they interact. Our account saves both Factual and Deontic Detachment (i.e., varieties of Modus Ponens) for deontic reasoning and treats "if'' as a proper sentential connective with a non-dynamic relation of logical consequence as preservation of designated values in a context. This combination of (arguably desirable) properties is novel in the literature, and it matches well with general theories of natural language reasoning. Specifically, our account makes sensible predictions for classical deontic puzzles such as Chisholm's quartet and the miners' paradox. (The paper is based on joint work with Paul Égré.)

Luca San Mauro (University of Bari): Argumentation and semantic paradoxes

Monday 27 January 2024, 12:00-13:00, Palazzo Nuovo, Aula 7

Abstract argumentation theory is a core research area in artificial intelligence, offering a robust framework for reasoning in the presence of conflicting information. While the study of finite argumentation frameworks (AFs) has received extensive attention, the exploration of infinite AFs remains largely underdeveloped. In this work, we take a significant step toward filling this gap by systematically investigating the algorithmic complexity of problems associated with infinite AFs. We will also explore intriguing connections between infinite AFs and truth theory. This is joint work with Uri Andrews.

Tommaso Moraschini (University of Barcelona): Measuring the expressivity of Kripke semantics: degrees of incompleteness and of the finite model property

Thursday 23 January, 16:00-18:00, Palazzo Nuovo, Aula di Antica

Despite the immense success of Kripke semantics, examples of modal logics that do not admit a completeness theorem in terms of Kripke frames have been known since the 70s. These logics are called Kripke incomplete. The degree of incompleteness of a logic L is the number of logics that share the same Kripke frames as L. Intuitively, this number measures to which extent L is surrounded by logics indistinguishable from it from the point of view of Kripke semantics.

A celebrated result in modal logic, known as Blok Dichotomy Theorem, states that the degree of incompleteness of a modal logic is either one or the continuum. However, Blok Dichotomy Theorem does not transfer immediately to the setting of intuitionistic logic. On the contrary, the problem of understanding incompleteness degrees for extensions of the intuitionistic propositional calculus IPC remains an outstanding problem in the area.

In this talk, we will show that the degree of incompleteness of an implicative logics (i.e., an extension of the implicative fragment of IPC) is either one or omega or the continuum. Furthemore, we will introduce a variant of the degree of incompleteness, called the degree of the finite model property, and show that, when applied to extensions of IPC, this degree is either a positive integer or omega or the continuum (with all these possibilities being witnessed by some concrete example).

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.

Bogdan Dicher (University of Witwatersrand): The Evans Counterpoint (joint work with G. Restall).

Monday 9 December 2024, 12:00-13:00, Palazzo Nuovo, Aula 25

How many conclusions does an argument have? Conventional wisdom and philosophical work both teach that an argument has at most one conclusion. A good reason for believing this is the co-called 'Evans point’: if an argument were to have multiple-conclusions, then these function as a disjunction; multiple-conclusions are disjunctions in disguise. In this talk we develop a different interpretation of multiple-conclusions, showing that they can be understood as structural resources for dealing with alternatives in proofs. We argue that, linguistic appearance notwithstanding, this interpretation does not ground multiple-conclusionedness in disjunctions. Hence an argument can have many conclusions—in a very peculiar way.

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.