Book of Abstracts

Camila Gallovich

Title: "The Argument from Proof-Theoretic Strength "

Abstract: Arguably, the most echoed argument against non-classical theories is that they are deductively weaker than the corresponding classical theories. A typical example that illustrates the point is given by the axiomatic theories of truth KF and PKF, both formulated over Peano arithmetic. KF and PKF yield axiomatizations of Kripke’s model-theoretic characterization of truth in classical and non-classical logic, respectively. However, these theories are not on a par. One reason for this disparity is their difference in proof-theoretic strength: KF is much stronger than PKF. In this work, we provide two different answers to the argument from strength, focusing on KF and PKF as our test case. Our first answer is that, despite appearances, the difference in deductive power between KF and PKF is not detrimental to the non-classical theory. In particular, we contend that evaluating whether KF is superior to PKF (or vice versa) requires adopting not only a specific conception of truth but also a specific conception of arithmetic. We review some alternatives offered in the literature and conclude that PKF is in a better position as an axiomatization of most of these conceptions. The second response is based on an argumentative strategy usually known as classical recapture. There is a natural way to improve the proof-theoretical power of PKF by means of additional axioms. In particular, we shall extend PKF with a subset of the grounded instances of the principle of excluded middle and we will show that the resulting theory has the same proof-theoretical power as KF.

This is joint work with Camillo Fiore and Lucas Rosenblatt.

Camillo Fiore

Title: "Identifying Logical Theories in the Substructural Era"

Abstract: Often, two logical systems differ in what seems a merely superficial way; for example, in the symbols they choose for a given operation. When this happens, we say that the systems are notational variants of one another and that, as a consequence, they are formalisations of the same logical theory. Now, in the last years, we have witnessed the emergence of various logical systems that are radically substructural, in the sense that they abandon the reflexivity and/or transitivity of consequence. I claim that these systems pose serious challenges to our extant criteria for deciding when two systems are notational variants. Then, I propose new criteria that overcome those challenges.

Eduardo Barrio

Title: "ST, Deductive Closure and Entailment"

Abstract: The aim of this project is to represent the non-transitive consequence relation of ST within a Tarskian framework. While ST’s external consequence operator fails to be idempotent—and thus transitive—we assume that only internal notions of consequence (such as entailment) may legitimately depart from Tarskian constraints. To model this, I propose extending the language of ST with a new conditional that expresses its valid entailments while invalidating Hypothetical Syllogism. This strategy seeks to internalize the non-transitivity of ST at the level of meta-inference. However, standard many-valued conditionals prove inadequate, as they typically fail to validate Modus Ponens. I therefore turn to Kripke-style semantics to develop an alternative.

Rashed Ahmad

Title: "A Logic of Groundedness"

Abstract: In this talk, we argue that ungroundedness should not only be contagious at the level of formulas, but rather its infectiousness runs much deeper, affecting entire sequents, metasequents, and proofs in which they appear. This, in turn, motivates the view that proofs ought to be grounded. Consequently, validity is defined as groundedness preservation in addition to truth preservation. Thus, we present a logic of groundedness, LG. LG is a substructural logic where Weakening and meta(n)Weakening fail, since they enable us to go from grounded states to ungrounded ones. Additionally, Reflexivity also fails, as it is a form of Meta-Weakening.

In this context, we will explore the implications of introducing a predicate for groundedness into the language. This introduction provides a suitable means to regain classical strength. Interestingly, this corresponding theory avoids trivialization while enabling us to establish that sentences lacking grounding are indeed ungrounded.

This is a joint work with Jonathan Erenfryd.

Bas Kortenbach

Title: "The Incoherency of Mixed Logics"

Abstract: Mixed logics use different assessment standards for the premises as opposed to the conclusions of arguments—be it at the usual inferential level, and/or at some higher metainferential one. Some such logics have recently seen extensive application to for instance the semantic paradoxes. However, a family of worries has emerged; these logics are, in various senses, not ‘self-obeying’ or ‘closed under their own laws’, and there seems to be something incoherent about this. These phenomena are clearly tightly connected to the fact that the logics are mixed, so that the objection threatens to affect mixed logics in general. The first purpose of this talk is to distinguish various subtly different notions of self-obedience, and investigate how exactly they relate to each other as well as to the mixed nature of logics. While not all mixed logics are necessarily self-disobedient, we will show that under weak conditions, every mixed logic is either self-disobedient or non-reflexive. Secondly, I will argue that none of the responses to the incoherency objection offered by proponents of mixed logics thus far is satisfactory, and explore a more promising defense strategy.

Luca Tranchini

Title: "Bilateralism and classical logic: some comments from the intensional perspective"

Abstract:

According to bilateralism, classical logic can be justified through the adoption of natural deduction systems comprising rules governing denials alongside the usual rules governing assertions. The aim of this talk is to stress the philosophical significance of some results from the literature on functional programming languages that (via the Curry-Howard isomorphism) apply to the bilateral system for classical logic first proposed by Rumfitt. The lack of confluence of the reduction relation associated with this system suggests that the classically-minded bilateralist cannot treat assertion and denial on a par, on pain of trivializing the notion of identity of proofs. I will suggest that the bilateralist approach seems better suited to justify logical systems weaker than classical logic such as Bellin's polarized bi-intuitionistic logic and Wansing's 2Int. The comparison between classical logic and Wansing's 2Int also offers a way to understand the defects of classical logic discussed in the talk as resulting from the conflation of different notions of proof and refutation.

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