XII Workshop on Philosophical Logic:
Satellite (@ SADAF): Symposium on Philosophical Logic
July 31, August 1, 4, 2023
Argentinean Society of Philosophical Analysis
Buenos Aires, Argentina
14.00. Dave Ripley (University of Melbourne): “Meaning containment and analytic containment”
15.10. Federico Pailos (IIF-SADAF-CONICET/UBA): “Inferential-connexive mixed logics”
16.10. Coffee break
16.50. Carlos Olarte (Université Sorbonne Paris Nord): “What is the cost of cut?”
17:50. Coffee break
18:20. Sara Uckelman (Durham University, UK): The role of History of Logic / Historical Logic in the study of modern logics
19.30. Luis Estrada González (UNAM): “You can get no cancellation?”
14:00. Joao Marcos (UFSC): “Restoring the lost perfection of classically-defective connectives”
15.10. Bruno Jacinto (University of Lisbon): “Iteration and Logicism”
16.10. Coffee break
16:50. Luca Castaldo (University of Warsaw): “A simultaneous axiomatization of truth and paradoxicality”
17:50. Coffee break
18:20. Edson Bezerra (IIF-SADAF-CONICET) “Labelled Tableaux for many-valued modal logics”
19:30. Bogdan Dicher (University of Lisbon): “Two ST conditionals?”
14:00. Elaine Pimentel (University College London): “A tour on ecumenical systems”
15.00. Yll Buzoku (University College London):: “Proof-theoretic semantics for substructural logics”
16.00. Luis Estrada González (UNAM): “A plea for relevance”
17.00. Coffee break
17:30. Quentin Blomet (IJN): “Mixed Interpolations”
18:30. Miguel Álvarez Lisboa (IIF-SADAF-CONICET): “Towards an abstract calculus of structures”
19:30. Mariela Rubin (IIF-SADAF-CONICET/UBA): “No Import, only Export”
Luca Castaldo: “A simultaneous axiomatization of truth and paradoxicality”
Truth and paradoxicality interact in many different ways. In particular, several truth theories have been developed in order to solve semantic paradoxes. However, very few theories of truth say something about paradoxicality itself, and the extent to which a unified treatment of these two notions is possible remains unclear. In the talk I move some steps towards filling this gap by presenting a simultaneous axiomatization of truth and paradoxicality. I prove the theory to be consistent, and I discuss some of its limitations and some of its virtues.
Bruno Jacinto: “Iteration and Logicism”
According to the iterative conception of set, sets are “built in stages” in a process to be pursued “as far as possible”. It is notoriously difficult to make sense of the “building” analogy unless the mind-dependence of sets is accepted. In this talk I will sketch the sets as properties view, according to which set theory is nothing but a part of cumulative type theory, and sets are just a specific kind of properties. I will offer some reasons for thinking that the sets as properties view, broadly based on (Linnebo & Rayo 2012, Degen & Johannsen 2000) delivers a more satisfactory and realist understanding of the iterative conception than those presently available. In addition, I will address Button and Trueman’s (2022) “no bootstrapping” objection to the sets as properties view. Finally, I will show how this reply paves the way to a modal and neoRussellian form of logicism which is thoroughly contingentist.
Elaine Pimentel: “A tour on ecumenical systems”
Ecumenism can be understood as a pursuit of unity, where diverse thoughts, ideas, or points of view coexist harmoniously. In logic, ecumenical systems refer, in a broad sense, to proof systems for combining logics. One captivating area of research over the past few decades has been the exploration of seamlessly merging classical and intuitionistic connectives, allowing them to coexist peacefully.
In this talk, we will embark on a journey through ecumenical systems, drawing inspiration from Prawitz’ seminal work. We will begin by elucidating Prawitz’ concept of “ecumenism” and present a pure sequent calculus version of his system. Building upon this foundation, we will expand our discussion to incorporate alethic modalities, leveraging Simpson’s meta-logical characterization. This will enable us to propose several proof systems for ecumenical modal logics.
We will conclude our tour with some discussion towards an ecumenical view of proof-theoretic semantics.
This is a joint work with Luiz Carlos Pereira, Sonia Marin, Emerson Sales, Valeria de Paiva and Victor Nascimento.
Carlos Olarte: “What is the cost of cut?”
Various kinds of game semantics have been introduced to characterize computational features of substructural logics, in particular fragments and variants of linear logic (LL). Our particular view of game semantics is that it is not just a technical tool for characterizing provability in certain calculi, but rather a playground for illuminating specific semantic intuitions underlying certain proof systems. Especially, we aim at a better understanding of resource conscious reasoning, which is often cited as a motivation for substructural logics.
In this talk, we will look at substructural calculi from a game semantic point of view, guided by certain intuitions about resource conscious and, more specifically, cost conscious reasoning. In particular, we will show a game where cost labels are added to assumptions, and games are played under the correspondent budget.
This will lead to a new kind of labelled calculus, which can be seen as a fragment of subexponential linear logic, and allows for an elegant interpretation of the dereliction rule. We will show, using examples, how this concept of costs in proofs can be generalised by using a semiring structure.
We will finish with a discussion on the proof theoretical effect of costs in the cut-elimination process.
Edson Bezerra: “Labelled tableaux for many-valued modal logics”
The connection between many-valued logics and modal logics is old. Although modal and multivalued logics constitute autonomous fields of investigation, their interactions are very prolific from a philosophical point of view. As widely known, the expressivity of the many-valued semantics demands expressive proof systems. This becomes more evident when we work with many-valued modal logics where modal sentences □A and ◇A can receive only true or false, since the language of the logical system may not be expressive to separate their truth-values. In this talk, we generalize Carnielli’s labelled tableaux systems for many-valued logics to the modal case, where the modal sentences are bivalent. We will show tableaux for the many-valued counterparts of the logics K, KT e S4. As we will show, this generalization covers the standard many- valued logics, including some substructural logics, such as the Strict Tolerant Logic, ST.
Federico Pailos: “Inferential-connexive mixed logics”
In this article, we present a new type of connexive-like logics. We call them inferential-connexive mixed logics, as they satisfy inferential versions of the main four connexive principles: AT, AT’, BT and BT’. We focus first on the Strong-Kleene type of mixed and inferentially connexive logics, the inferential-connexive Strong-Kleene logics, and we provide a philosophical interpretation for these logics in terms of formal inferential commitments. We introduce afterwards the notion of inferential-connexive Boolean logic, and present some logics that fall in this ballpark. We also explain the relation between inferentially-connexive Boolean logics and some inferentially-connexive Strong-Kleene logics, and show how they can also be interpreted in epistemic (or attitudinal)-related terms.
Miguel Álvarez Lisboa: “Towards and Abstract Calculus of Structures”
The recent literature on substructural logics understands the word ‘substructural’ in two senses: one refers to logics studied through sequent calculi where some of the rules called (by Gentzen) ‘structural’ fail; the other refers to logics that fail to define a consequence relation in the usual (Tarskian) way. (The logics of the second kind may have sub-Gentzenian sequent calculi, but the rules dropped will not always belong to the ‘structural’ group.) It can be argued that nothing interesting is happening here: surely the second group was called ‘substructural’ for marketing purposes only, and “the rose by any other name / would be just as sweet”. But I want to explore a different approach: I will propose an explication of the concept of ‘structure’ that 1) encompasses these two uses of the word ‘substrcutural’ and 2) provides a framework for the study of potentially new (sub-)(super-)structural logics. I call this framework Abstract Calculus of Structures.
Dave Ripley: “Meaning containment and analytic containment”
Sara Uckelman: “The role of History of Logic / Historical Logic in the study of modern logics”
Quentin Blomet: “Mixed Interpolations”
Which choice of truth-tables and consequence relation for two logics X and Y guarantees that the following interpolation property hold : If a formula A is classically satisfiable, a formula B is classically falsifiable, and A classically entails B, then there is a formula C sharing all its atoms with both A and B, such that A entails C in the logic X and C entails B in the logic Y? We answer this question for any two propositional logics based on the same three-valued boolean normal monotonic scheme for the connectives and two (possibly identical) monotonic consequence relations. Given that the resulting logics will be subclassical, any positive answer to the question must be seen as a particular fine-tuning of Craig’s interpolation theorem.
Luis Estrada González: “You can get no cancellation?”
According to Wansing and Skurt in “Negation as cancellation, connexive logic, and qLPm”, “[s]ince the cancellation view of negation is problematic, it is better not seen as a conceptual basis of anything, and should therefore m[u]st definitely not be used as a conceptual basis of connexive logic.” We partially agree: connexive logic need not be based on a cancellation view of negation. On the other hand, many of the problems plaguing the cancellation view —both the originally reported by the Routleys and the new ones detected by Wansing and Skurt— can be solved relatively easily, so it is not necessary to detach connexive logic from the cancellation account, either.
We work here with three simple accounts of content, namely, the content of a proposition A is (1) its amount of falsity, or (2) its amount of truth, or (3) its having a value or its lacking a value. Accordingly, a cancellation of A’s content does one of these three things: (1) it removes falsity in A, or (2) it removes truth in A, or (3) it removes any values of A or put them in it when they are missing. We explore with some detail each of these options and evaluate whether they live up to the expectations of the Routleys and Wansing and Skurt. (Joint work with Fernando Cano-Jorge.)
Joao Marcos: “Restoring the lost perfection of classically-defective connectives”
Where does a truth-functional designation-based approach to the meaning of a logical constant originate from? How could the meaning of a two-valued connective be fully captured by rules and axiomatic constraints? Further, in case a given classical connective fails to fulfill its expected inferential rôle, behaving instead in a non-deterministic fashion, how can its intended meaning be restored with the help of sufficiently expressive auxiliary connectives that assure that the associated notion of consequence allows for the necessary derivability adjustments? The present contribution will present a general recipe for providing legitimacy conditions for the characterization of arbitrary classic-like connectives as well as for restoring the lost perfection of defective versions of those very connectives.
Luis Estrada González: “A plea for relevance”
Alison Duncan Kerr, in “A plea for KR”, argued that even if both positive and negative paradoxes —A→(B→A) and A→(˜A→B), respectively— are indeed paradoxes of relevance, other implicative theorems of classical logic are not, contrary to what relevance logicians have claimed for scores. She evaluates whether it is a mistake to believe the antecedent and not to believe the consequent of a given implicational theorem of classical logic: if it is a mistake to do so, the implication is not a paradox; if it is not a mistake to the believe the antecedent and not to believe the consequent, the implication is a paradox.
In this talk we will argue that her conclusion is no wonder, since she allows herself to go from the mistake of believing A to the mistake of believing A and not believing B (for any B whatsoever) and then to the admissibility of believing that A implies B. This, inadvertently, is reasoning by negative paradox, which, by her own lights, is bad reasoning. (Joint work with Fernando Cano-Jorge.)
Yll Buzoku: “Proof-theoretic semantics for substructural logics”
Mariela Rubin: “No Import, only Export”
In 1980, Gibbard proved that when one extends classical logic with an indicative conditional and that conditional validates the equivalence known as Import-Export, then if one assumes that the indicative conditional is at least as strong as the material conditional, both conditionals collapse into the material one. In 2013, Fitelson gives an axiomatic route to Gibbard’s collapse theorem, where the main idea is to assume as little as possible for the derivation to take place. He presents a proof that hinges on eight axioms that involve assumptions regarding the conjunction, the conditional of the base logic, the indicative conditional and a few reasonable things about the notion of logical consequence and theoremhood. He shows that it is not necessary to assume classical logic, rather intuitionistic logic suffices for the collapse to happen. Therefore, what Gibbard proved, says Fitelson, was that if one assumes IE and that the indicative conditional implies any conditional at least as strong as the intuitionist conditional, then both conditionals collapse into the same conditional.
The main goal of this work is to show that Import is not playing any role in the derivation of collapse. In particular, it is not only that you do not need to apply the Import Rule at any step of the derivation, but that you do not need the strength of assuming the validity of Import for the base logic. I will also show that depending on how one formulates the rules of the calculus it is possible to derive the collapse without assuming anything regarding conjunctions and without assuming weakening. The corollary of this result is that the collapse will take place for even weaker logics than the intuitionistic one.
Eduardo Barrio (UBA/IIF-SADAF-CONICET)
Federico Matías Pailos (UBA/IIF-SADAF-CONICET)
Lucas Rosenblatt (UBA/IIF-SADAF-CONICET)
Damián Szmuc (UBA/IIF-SADAF-CONICET)
Aylén Bavosa Castro (IIF-SADAF-CONICET)
Miguel Álvarez Lisboa (IIF-SADAF-CONICET)
We are thankful for the support provided by CONICET, MINCYT (Project 01-RC-2022-01-00028) and the PLEXUS (Grant Agreement no 101086295) a Marie Sklodowska-Curie action funded by the EU under the Horizon Europe Research and Innovation Programme.