The Workshop on Nondeterministic Semantics aims at gathering researchers to discuss recent topics on the field. 

Speakers 

Marcelo Esteban Coniglio (UNICAMP/CLE)

Thomas Macaulay Fergusson (RPI)

Stef Frijters (KU Leuven)

Pablo Rivas-Robledo (University of Amsterdam)

Daniel Skurt (RUB)

Mahan Vaz (UNICAMP/RUB)

Peter Verdée (UCLouvain)

Abstracts

Marcelo Esteban Coniglio (UNICAMP/CLE): On a new decision method for Intuitionistic Logic by 3-valued non-deterministic truth-tables

Abstract: In 1932 Kurt Gödel proved that it is not possible to characterize Intuitionistic Propositional Logic (IPL) by means of finite and deterministic truth-tables. Building on Gödel's proof, J. Dugundji proved in 1940 that no modal system between Lewis' S1 and S5 can be characterized by a single finite logical matrix. In 2022, L. Grätz, based on work by J. Kearns from 1981, obtained a decision procedure for KT and S4 using a 3-valued non-deterministic matrix (Nmatrix) with a restricted set of valuations  (i.e., a restricted Nmatrix, a.k.a. an RNmatrix) along with an algorithm to delete unsound rows. Thanks to the conservative (and computable) translation from IPL into S4 introduced by Gödel in 1933, composing the two algorithms yields a decision procedure for IPL. The definition of a pure decision procedure for IPL based on a finite-valued RNmatrix, independent of Gödel and Grätz's constructions, is a challenge that naturally arises.

In this talk, we present a solution to this challenge: after we prove that IPL also cannot be characterized by a single finite Nmatrix, we introduce a 3-valued RNmatrix for IPL, together with a simple method for deleting unsound rows. This constitutes a new and very simple decision procedure for IPL. The method can be seen as truth-tables in a broader sense, providing a way to overcome Gödel's limiting result.

This is a joint work with Renato Leme and Bruno Lopes

Thomas Macaulay Fergusson (RPI): Extending Suszko-Style Reductions to Many-Sided Calculi Through RNMatrices

Abstract: Many have worked on the famous Suszko-style reductions in which Tarskian, transitive, reflexive, and substructural consequence relations can be given two, three, or four-valued semantics. Recent work (joint with Jitka Kadlecikova and Pawel Pawlowski) has shown how to represent such reductions in the framework of restricted non-deterministic matrices. I intend to show work in progress (joint with Jitka Kadlecikova) that both shows how to frame these results in generalized RNmatrices for n-sided calculi, showing that every n-sided Tarskian calculus has an n-valued RNmatrix semantics, every monotonic n-sided calculus has a 2^n-valued RNmatrix semantics (and similar results for transitive and reflexive n-sided calculi). I either intend to provide proofs of new results or, at least, sketches of where we are on this project (in case work is still ongoing).

Stef Frijters (KU Leuven): Aristotelian Diagrams for many-valued and non-deterministic semantics

Abstract: Aristotelian relations and the diagrams that represent them have played an important role throughout the history of logic. Recently, there has been a revitalized interest in these relations and diagrams. They are being used in a number of different fields (e.g. philosophy, computer science, linguistics, psychology, …), and are also studied as objects of independent interest in the research program known as ``Logical Geometry’’. However, there is little work on their application to non-classical logics.

In this talk we propose a uniform way of defining Aristotelian relations and diagrams for logics using many-valued and non-deterministic semantics. We argue that this approach has advantages over possible alternatives, and we show diagrams for a number of logical systems. Through doing this, we illustrate that our approach can have interesting pedagogical, heuristical and meta-logical applications.

Pablo Rivas-Robledo (University of Amsterdam): Philosophical Aspects of Non-deterministic Semantics. Possible worlds, many valued logics and potentialism in set theory

Abstract: This talk advances a theory of modality based on non-deterministic semantics, offering a philosophically robust alternative to possible worlds semantics. On this account, necessity and possibility are treated as properties of propositions (or facts) grounded in the actual world's inherent structure. We thus posit that the world contains modal facts about what could or could not be the case. For example, the truth of ‘□P’ means that ‘P’ enjoys support within reality that guarantees its truth. Under this interpretation, non-deterministic semantics reflects a world with built-in modal structure rather than a plurality of worlds.

I will demonstrate how several influential philosophical theories (particularly the modal analysis of dispositions and potentialism in set theory) are problematically committed to a realist conception of possible worlds due to their reliance on possible worlds semantics. Finally, I will argue that the non-deterministic theory of modality can illuminate these debates by abandoning the notion of possible worlds altogether.

Daniel Skurt (RUB): Solving a New Paradox of Deontic Logic (and a dozen other paradoxes)

Abstract: In this presentation we will solve a seemingly hitherto unnoticed paradox of deontic logic (the Paradox of Obligatory Negated Conditionals) by employing a modal extension of the material connexive logic MC and making use of  RNmatrices.

Mahan Vaz (UNICAMP/RUB): Modal extensions of CLoN

Abstract: In this talk I present a neighborhood semantics approach to modal extensions of CLoN. Although the positive modal logics follow the script of positive normal modal logics, the lack of a strong negation in CLoN requires the stipulation of a notion of rejection sets to permit the definition of interaction axioms between two distinct modalities or to deal with negated modalities.  This is a joint collaboration with Daniel Skurt.

Peter Verdée (UCLouvain): Minimally Deficient Logics

Abstract: Using the framework of adaptive logic and non-deterministic semantics, we develop minimally deficient logics—a generic approach to handling formulas whose truth-functional behavior may exceptionally deviate from what a given logic (referred to as the ideal logic) prescribes. The ideal logic can be any system with a truth-functional semantics, whether deterministic or non-deterministic. Our aim is to formalize methods for addressing specific, isolated truth-value assignments that are deficient, in the sense that they do not align with the norms of a standard deductive system (i.e., a normal Tarskian truth-functional logic), the ideal logic. Such faulty assignments are treated as errors or bugs, potentially resulting from faulty processes of information storage, processing, or transmission. In our approach, sets of sentences are presumed to be non-erroneous unless it becomes evident that the only way to make sense of the information is by assuming that some sentences cannot be correctly interpreted using the ideal logic.

Let L be the ideal logic. For each such logic L, we construct an adaptive logic in which L serves as the Upper Limit Logic, the logic applied when no (disjunctions of) abnormalities follow from premises. The Lower Limit Logic (LLL), the logic that can always be applied unconditionally, of the minimally deficient adaptive logic had the same matrices as L, except that a deficient counterpart v′ for each truth value v of L  is added to the truth values of the LLL. These deficient values are treated identically to their non-deficient counterparts. Besides this duplication of the truth values, deficient values are added to the original ones in matrix entries where faulty assignments may occur, introducing an additional layer of indeterminism. A unary connective D is added to the language of L, which receives a designated value if and only if a formula has a deficient truth value. The abnormalities in the adaptive logic are formulas of the form Dφ.

In this talk, we will introduce the concept of minimal deficiency, define the generic adaptive logic that formalizes it, demonstrate how this logic prevents the propagation of faulty information, and show that several existing adaptive logics can be interpreted as minimally deficient logics.

This research has been partially conducted in collaboration with Hitoshi Omori

Location

Blandijnberg 2 , Campus Boekentoren

3rd floor - room 3.30 Camelot

UGent

(Find a building map here)

Organization

Pawel Pawlowski (UGent)

Mahan Vaz (UNICAMP/RUB)

Centre for Logic and Philosophy of Science