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We examine the set of formulatoformula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the firstdegree entailment fragment of R. Epstein's Relatedness Logic, and that it is a nontransitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined (...) 

In their recent article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc present novel and striking results about metainferential validity in various three valued logics. In the process, they have thrown open the door to a hitherto unrecognized domain of nonclassical logics with surprising intrinsic properties, as well as subtle and interesting relations to various familiar logics, including classical logic. One such result is that, for each natural number n, there is a logic which (...) 

The recent development and exploration of mixed metainferential logics is a breakthrough in our understanding of nontransitive and nonreflexive logics. Moreover, this exploration poses a new challenge to theorists like me, who have appealed to similarities to classical logic in defending the logic ST, since some mixed metainferential logics seem to bear even more similarities to classical logic than ST does. There is a whole STbased hierarchy, of which ST itself is only the first step, that seems to become more (...) 

I show that it is not possible to uniquely characterize classical logic when working within classical set theory. By building on recent work by Eduardo Barrio, Federico Pailos, and Damian Szmuc, I show that for every inferential level, either classical logic is not unique at that level or there exist intuitively valid inferences of that level that are not definable in modern classical set theory. The classical logician is thereby faced with a threehorned dilemma: Give up uniqueness but preserve characterizability, (...) 

I argue against abductivism about logic, which is the view that rational theory choice in logic happens by abduction. Abduction cannot serve as a neutral arbiter in many foundational disputes in logic because, in order to use abduction, one must first identify the relevant data. Which data one deems relevant depends on what I call one's conception of logic. One's conception of logic is, however, not independent of one's views regarding many of the foundational disputes that one may hope to (...) 

The paper presents a truthmaker semantics for Strict/Tolerant Logic, which is the currently most popular logic among advocates of the nontransitive approach to paradoxes. Besides being interesting in itself, the truthmaker presentation of ST offers a new perspective on the recently discovered hierarchy of metainferences that, according to some, generalizes the idea behind ST. While fascinating from a mathematical perspective, there is no agreement on the philosophical significance of this hierarchy. I aim to show that there is no clear philosophical (...) 

The use of models to assign truth values to sentences and to counterexemplify invalid inferences is a basic feature of model theory. Yet sentences and inferences are not the only phenomena that model theory has to take care of. In particular, the development of sequent calculi raises the question of how metainferences are to be accounted for from a modeltheoretic perspective. Unfortunately there is no agreement on this matter. Rather, one can find in the literature two competing modeltheoretic notions of (...) 

Substructural solutions to the semantic paradoxes have been broadly discussed in recent years. In particular, according to the nontransitive solution, we have to give up the metarule of Cut, whose role is to guarantee that the consequence relation is transitive. This concession—giving up a meta rule—allows us to maintain the entire consequence relation of classical logic. The nontransitive solution has been generalized in recent works into a hierarchy of logics where classicality is maintained at more and more metainferential levels. All (...) 

I explore, from a prooftheoretic perspective, the hierarchy of classical and paraconsistent logics introduced by Barrio, Pailos and Szmuc in. First, I provide sequent rules and axioms for all the logics in the hierarchy, for all inferential levels, and establish soundness and completeness results. Second, I show how to extend those systems with a corresponding hierarchy of validity predicates, each one of which is meant to capture “validity” at a different inferential level. Then, I point out two potential philosophical implications (...) 

Against the backdrop of the frequent comparison of theories of truth in the literature on semantic paradoxes with regard to which inferences and metainferences are deemed valid, this paper develops a novel approach to defining a binary predicate for representing the valid inferences and metainferences of a theory within the theory itself under the assumption that the theory is defined with a classical metatheory. The aim with the approach is to obtain a tool which facilitates the comparison between a theory (...) 

Strict/tolerant logic, ST, evaluates the premises and the consequences of its consequence relation differently, with the premises held to stricter standards while consequences are treated more tolerantly. More specifically, ST is a threevalued logic with left sides of sequents understood as if in Kleene’s Strong Three Valued Logic, and right sides as if in Priest’s Logic of Paradox. Surprisingly, this hybrid validates the same sequents that classical logic does. A version of this result has been extended to meta, metameta, … (...) 

A metainference is usually understood as a pair consisting of a collection of inferences, called premises, and a single inference, called conclusion. In the last few years, much attention has been paid to the study of metainferences—and, in particular, to the question of what are the valid metainferences of a given logic. So far, however, this study has been done in quite a poor language. Our usual sequent calculi have no way to represent, e.g. negations, disjunctions or conjunctions of inferences. (...) 

Intratheoretical logical pluralism is a form of meaninginvariant pluralism about logic, articulated recently by Hjortland :355–373, 2013). This version of pluralism relies on it being possible to define several distinct notions of provability relative to the same logical calculus. The present paper picks up and explores this theme: How can a single logical calculus express several different consequence relations? The main hypothesis articulated here is that the divide between the internal and external consequence relations in Gentzen systems generates a form (...) 

Logical nihilism is the view that the relation of logical consequence is empty: there are counterexamples to any putative logical law. In this paper, I argue that the nihilist threat is illusory. The nihilistic arguments do not work. Moreover, the entire project is based on a misguided interpretation of the generality of logic. 

In recent years, some theorists have argued that the clogics are not only defined by their inferences, but also by their metainferences. In this sense, logics that coincide in their inferences, but not in their metainferences were considered to be different. In this vein, some metainferential logics have been developed, as logics with metainferences of any level, built as hierarchies over known logics, such as \, and \. What is distinctive of these metainferential logics is that they are mixed, i.e. (...) 

Dentro del conjunto de las lógicas no clásicas, las lógicas paraconsistentes han suscitado de manera particular el interés de diversos filósofos. Además de las definiciones tradicionales, en los últimos años, se han propuesto nuevas maneras de caracterizar a la paraconsistencia. Lo que tienen en común todas estas definiciones es que alguna forma de la regla o de la metarregla de explosión debe ser rechazada. En este artículo, presentaré dichas definiciones y evaluaré el rol que juegan la negación y la transitividad (...) 

In this article, our aim is to take a step towards a full understanding of the notion of paraconsistency in the context of metainferential logics. Following the work initiated by Barrio et al. [2018], we will consider a metainferential logic to be paraconsistent whenever the metainferential version of Explosion is invalid. However, our contribution consists in modifying the definition of metaExplosion by extending the standard framework and introducing a negation for inferences and metainferences. From this new perspective, Tarskian paraconsistent logics (...) 

The aim of this article is to discuss the extent to which certain substructural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collect... 

The aim of this article is to study the notion of derivability and its semantic counterpart in the context of nontransitive and nonreflexive substructural logics. For this purpose we focus on the study cases of the logics ST and TS. In this respect, we show that this notion doesn’t coincide, in general, with a nowadays broadly used semantic approach towards metainferential validity: the notion of local validity. Following this, and building on some previous work by Humberstone, we prove that in (...) 

In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences holding in the stricttolerant logic of transparent truth ST+ and inferences holding in the logic of paradox LP+. They argue that LP+ is ST+’s external logic and they question whether ST+’s solution to the semantic paradoxes is fundamentally different from LP+’s. Here we establish that by parity of reasoning, ST+ can be related to LP+’s dual logic K3+. We clarify the distinction between internal and external logic and (...) 

Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are (...) 

This paper analyses the notion of ‘interpretation’, which is often tied to the semantic approach to logic, where it is used when referring to truthvalue assignments, for instance. There are, however, other uses of the notion that raise interesting problems. These are the cases in which interpreting a logic is closely related to its justification for a given application. The paper aims to present an understanding of interpretations that supports the modeltheoretic characterization of validity to the detriment of the prooftheoretic (...) 

It is widely accepted that classical logic is trivialized in the presence of a transparent truthpredicate. In this paper, we will explain why this point of view must be given up. The hierarchy of metainferential logics defined in Barrio et al. and Pailos recovers classical logic, either in the sense that every classical inferential validity is valid at some point in the hierarchy ), or because a logic of a transfinite level defined in terms of the hierarchy shares its validities (...) 

When discussing Logical Pluralism several critics argue that such an openminded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a clear (...) 

In this paper we discuss the extent to which the very existence of substructural logics puts the Tarskian conception of logical systems in jeopardy. In order to do this, we highlight the importance of the presence of different levels of entailment in a given logic, looking not only at inferences between collections of formulae but also at inferences between collections of inferences—and more. We discuss appropriate refinements or modifications of the usual Tarskian identity criterion for logical systems, and propose an (...) 

Antiexceptionalism about logic states that logical theories have no special epistemological status. Such theories are continuous with scientific theories. Contemporary antiexceptionalists include the semantic paradoxes as a part of the elements to accept a logical theory. Exploring the Buenos Aires Plan, the recent development of the metainferential hierarchy of ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$${\mathbf {ST}}$$\end{document}logics shows that there are multiple options to deal with such paradoxes. There is a whole ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) 

In this paper, we present a way to translate the metainferences of a mixed metainferential system into formulae of an extendedlanguage system, called its associated σsystem. To do this, the σsystem will contain new operators (one for each standard), called the σ operators, which represent the notions of "belonging to a (given) standard". We first prove, in a modeltheoretic way, that these translations preserve (in)validity. That is, that a metainference is valid in the base system if and only if its (...) 

For semantic inferentialists, the basic semantic concept is validity. An inferentialist theory of meaning should offer an account of the meaning of "valid." If one tries to add a validity predicate to one's object language, however, one runs into problems like the vCurry paradox. In previous work, I presented a validity predicate for a nontransitive logic that can adequately capture its own metainferences. Unfortunately, in that system, one cannot show of any inference that it is invalid. Here I extend the (...) 