ENS Workshop in Philosophical Logic

 

Damian Szmuc (IIF-SADAF - CONICET), "Meaningless, Off-topic, and Partially True"

 

We explore different semantics for the first-degree entailment fragment of Angell’s logic of Analytic Containment, understood itself as a logical system in its own right. For this purpose, we compare its twist-product semantics obtained through the meaningless reading of the weak Kleene algebra, and its direct product semantics obtained through the off-topic and partial truth interpretations of the four-element involutive semilattice.

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

Allard Tamminga (University of Greifswald), "The Truth about Tonk, Identity, and Cut"

Joint work with Barteld Kooi (University of Groningen)

 

Arthur Prior introduced his binary connective ’tonk’, henceforth T, by way of an introduction rule and an elimination rule. Prior’s introduction rule for T is : "A, therefore A T B". His elimination rule for T is : "A T B, therefore B". Of course, Prior’s rules cannot be classically valid both, because if they were, "A, therefore B" would be classically valid, which it is not. To pinpoint what is wrong with Prior’s rules for T, we assess the exact semantic conditions under which the introduction rule is classically valid and the exact semantic conditions under which the elimination rule is classically valid. (Note that, taken on itself, each of the two rules is perfectly acceptable : the introduction rule for T matches an introduction rule for disjunction and the elimination rule for T matches an elimination rule for conjunction). Using an elementary Gentzen system, we study what is wrong with these exact semantic conditions for T. We show that T is both incomplete and incorrect. Because of the incompleteness, we do not have Identity, and because of incorrectness, we do not have Cut. In the remainder of the talk, we explore the repercussions of these findings for the proof-theoretical debate on *harmony*.

Joint work with Brian Hill (GREGHEC)

 

Explanations, and in particular explanations which provide the reasons why their conclusion is true, are a central object in a range of fields. On the other hand, there is a long and illustrious philosophical tradition, which starts from Aristotle, and passes through scholars as Leibniz, Bolzano and Frege, that give pride to this type of explanations, and is rich with brilliant and profound intuitions. Recently, Poggiolesi (2024) has formalized ideas coming from this tradition using the logical tools proper to proof theory. On the one hand, recent work has focused on Boolean circuits that compile some common machine learning classifiers and have the same input-output behavior. In this framework, Darwiche and Hirth (2023) have proposed a theory for unveiling the reasons behind the decisions made by Boolean classifiers, and they have studied their theoretical implications. In this talk we will show the deep links behind these two trends : in particular, we will demonstrate that the proof-theoretic tools introduced by Poggiolesi can be used to compute the complete reasons behind the decisions made by Boolean classifiers and we will illustrate them using examples.