Pilar Terrés Villalonga

The lottery (Kyburg, Probability and inductive logic. Macmillan, 1970) and the preface (Makinson, Analysis 25(6):205–207, 1965) paradoxes challenge a fundamental principle of rationality: if two propositions A and B are rationally acceptable, so is their conjunction, “A and B”. The paradoxes show that there are cases in which it seems legitimate to endorse each proposition of a finite list, but it is not rational to endorse them all together. A prominent solution to the paradoxes uses substructural logics to distinguish these two senses of the universal quantifier (Zardini, ∀\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall $$\end{document} and ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}. In Quantifiers, quantifiers, and quantifiers: Themes in logic, metaphysics, and language (pp. 489–526). Springer, 2015; Paoli, Philos Stud 124(3):313–330, 2005). This paper is twofold: first, we develop this idea by giving a detailed analysis of why a logic without contraction such as linear logic is suitable for analysing the paradoxes, identifying the elements present in the paradoxes that classical logic fails to capture, and linear logic nicely formalises in its vocabulary. Second, we reinterpret Paoli and Zardini’s solution pragmatically, arguing that the classical universal quantifier correctly captures the literal meaning of “all” but that linear logic captures two pragmatically enriched senses which are the default reading in some contexts, such as in the lottery of preface scenarios.

A detailed analysis of joint-contribution of premises and conclusions in classically valid sequents is presented in terms of hypergraphs. In (Saint-Germier, P., Verdée, P., & Villalonga, P. T. (2024). _Relevant entailment and logical ground. Philosophical Studies_ (pp. 1–43). https://doi.org/10.1007/s11098-024-02101-1 ), this idea of joint-contribution is introduced and motivated as a method for characterizing four kinds of relevant validity, in the sense of selecting the relevantly valid sequents among the classically valid sequents. The account in (Saint-Germier, P., Verdée, P., & Villalonga, P. T. (2024). _Relevant entailment and logical ground. Philosophical Studies_ (pp. 1–43). https://doi.org/10.1007/s11098-024-02101-1 ) is built on a calculus, called \(\textsf{GLK}^{\hbox {a}}\), which proves grounding claims for (enthymematically) valid sequents. In the present paper an adequate representation of \(\textsf{GLK}^{\hbox {a}}\) is given in terms of hypergraphs. The hypergraphs are a kind of diagrammatic proofs for Classical Propositional Logic, entirely based on the grounds of premises and conclusions. The hypergraphs and their visualization provide insights into the relations between premises and conclusions and into the way validity is produced by the binding of premises and conclusions via their partial grounds. They visualize the network of elements of the sequent that contribute to its logical validity. Non-contributing (i.e. irrelevant) premises and conclusions are then specified to be those that are disconnected from the network, however one constructs the graphs.

In Logical Pluralism and Logical Consequence (Cambridge University Press, 2023), Erik Stei argues for logical monism, the view that there is exactly one correct logic, in opposition to logical pluralism and logical nihilism. The present review aims to challenge two premisses in the main argument of the volume. First, Stei argues that no version of pluralism based on a plurality of senses of the logical connectives succeeds in proving that logical vocabulary is genuinely plural in the required sense. Second, he also argues that pluralism cannot account for the rivalry between logics, which makes the position less charitable than it claims. I will give arguments against the two premisses after presenting the details of the main argument for monism that we find in the book.

According to an intuitive picture of relevant entailment, an entailment is relevant if all the formulas it contains contribute to its validity. In this paper, we provide a ground-theoretic analysis of this notion of contribution, and as a result of relevant entailment. We build a system of bilateral logical grounding within which we can derive classical entailment and analyze the contribution of premises and conclusions, in terms of a certain type of connection between their respective logical grounds. The resulting framework is shown to have a strong unifying and explanatory power by bringing together the notions of grounding and relevance and explaining the commonalities and differences between alternative contribution-based accounts of relevance.

The aim of this paper is to give a general solution to the paradoxes of the material conditional, including the paradoxes generated by embedded conditionals. The solution consists in a pragmatic reinterpretation of the formal languages of classical logic LK and relevant logic LR as presented in Paoli. In particular I argue that the material conditional in the classical logic LK captures the truth conditions of “if...then”, but ignores certain pragmatic enrichments that are associated to it, while relevant logic LR can give a systematic diagnostic of the cases in which a conditional is pragmatically enriched and those cases in which it is not. This diagnostic shows the reason why the paradoxes seem unacceptable and why they are, nevertheless, truth-preserving. This reinterpretation will also cover the solution that Paoli gives to McGee’s paradoxes of the Modus Ponens.

ABSTRACT My aim in this paper is to present a pluralist thesis about the inferential role of logical constants, which embraces classical, relevant, linear and ordered logic. That is, I defend that a logical constant c has more than one correct inferential role. The thesis depends on a particular interpretation of substructural logics' vocabulary, according to which classical logic captures the literal meaning of logical constants and substructural logics encode a pragmatically enriched sense of those connectives. The paper is divided into four parts: first, I introduce the motivation for the pluralist thesis of the paper; second I introduce the calculus for the different logics endorsed in the pluralist thesis; third, I motivate how the different behaviors of the logical vocabulary of these logics can be pragmatically interpreted, and fourth, I motivate how the different inferential roles that each substructural logic attribute to logical constants can coexist, and how we should reason with a logical constant c given this plurality.

I argue for a version of logical pluralism based on the plurality of legitimate formalizations of the logical vocabulary. In particular, I argue that the apparent rivalry between classical and relevant logic can be resolved, given that both logics capture and formalize normative and legitimate senses of logical consequence: classical logic encodes “follows from” as truth preservation and captures the truth conditions of the logical constants, while relevant logic encodes a notion of “follows from” which, apart from preserving truth, avoids the violation of certain Gricean maxims and captures a different inferential role for the same logical constants, enriching their meaning pragmatically.