Federico Pailos

The hierarchy of metainferences $\textbf{ST }_{\omega }$ has garnered significant attention in the literature due to its ability to recover as many classical metainferences as possible. In this study, we introduce a fixed point validity predicate for $\textbf{ST }_{\omega }$. Subsequently, we demonstrate not only that this predicate captures all valid metainferences according to $\textbf{ST }_{\omega }$ logic but also the naïve principles governing validity. The resulting theory of validity is non-trivial, and it can internalize its valid metainferences.

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 some Strong-Kleene type of mixed and inferentially connexive logics that cannot be interpreted as truth-preserving, but that can receive other interpretations in terms to falsity-preservation and related features. 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.

Validity is usually taken to be a bivalent property: every inference is either valid or invalid, and never both. We argue for the controversial thesis that, if one endorses a many-valued semantics for the object language, then one likely has good reasons to also endorse a manyvalued notion of validity. We present several logical systems (based on Belnap’s algebra 4) whose notion of validity is non-bivalent: there are inferences that are both valid and invalid, and/or inferences that are neither valid nor invalid. We show that, under natural assumptions, the validity of metainferences is also non-bivalent in many of these systems. Lastly, we claim that our framework has fruitful applications to philosophical issues, such as the link between logical consequence and the conditional, and the link between logic and metalogic.

We present naïve non-substructural solutions to the Validity Paradox. The main idea behind these solutions is to break the link between a valuation satisfying an inference and a valuation not being a counterexample to that inference. A philosophical consequence of this technical move is that, in some of these theories, Validity Detachment will be valid while at the same time invalid, while in others Validity Detachment won’t be invalid, though it will not be valid either. No matter what path is chosen, it will not be possible to have the problematic instances of Validity Detachment as axioms of the derivations that lead to the paradoxical result. The first cannot have them as axioms because the problematic instances are not valid, and the second cannot have them either because they are invalid. Given the details of these proposals, new forms of both (i) naïve validity, and (ii) non-substructurality arise.

The goal of this paper is to offer an initial taxonomy of the LFIs (logics of formal inconsistency) and that can be non-trivially extended with a transparent truth predicate. To fulfill this task, we take a semantic point of view. We define two families of logics: infectious logics (obtained as a generalization of the semantic behavior present in the {¬,∧,∨}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ \neg, \wedge, \vee \}$$\end{document}-fragment of logics of nonsense) and non-infectious logics (i.e. logics that are not infectious). Using two paradigmatic logics of each kind, we generate an extensive collection of non-classical logics –some of which have been discussed in the literature and many that have not– and consider the prospect of extending them with a consistency operator and a truth predicate. We show that, when extended in this way, most of the non-infectious systems are trivial and that all the infectious ones are non-trivial. To complete the analysis, we provide philosophical reasons for being interested in the logics of formal inconsistency and truth introduced in this paper.

Much of the interest that substructural logics held over during the last years revolves around how they deal with metainferences. Most notably, the non-transitive—and, therefore, substructural—logic ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ST }$$\end{document} has sometimes been described as a theory that validates every classical inference while making invalid some classical metainferences. But, what exactly is a metainference?

The validity of an inference has traditionally been associated with the preservation of truth from premises to conclusions. Mixed logics call against this way of understanding this relation: impure mixed logics cannot be understood as preserving some set of designated values from premises to conclusions. Also, some of them have proved to be extremely interesting, and have multiple applications, despite lacking this feature of designated-values preservation. ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ST }$$\end{document}, in particular, shares all the inferential validities of classical logic, but, unlike it, can be expanded in a non-trivial way with a transparent truth predicate. Thus, if a logic is just a set of inferential validities, then it is no longer true that classical logic cannot be non-trivially extended with a transparent truth predicate.

In this chapter, we will introduce the most studied family of pure and impure mixed metainferential logics of level 1: the Strong Kleene metainferential logics. We will only focus on some of them: the ones that can be characterized through the inferential logics K3,LP,ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K3}, \textbf{LP}, \textbf{ST }$$\end{document}, and TS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{TS }$$\end{document}, which will be introduced below.

In this book, as the title suggests, we undertake the task of giving a general account of several aspects of metainferences and metainferential logics. Thus, we need to start by providing a precise definition of what a metainference is. As is commonly assumed, we will take an inference to be an ordered pair of collections of formulas of some formal language. Moreover, logics are usually identified with a certain set of inferences, which are the valid ones. Therefore, inferential consequence relations are defined over collections of formulas. In the traditional picture, metainferences appear as properties that emerge from the set of valid inferences. This is what we call the notion of metainferences as properties.

As we have mentioned in Chap. 3, in [1], PailosPailos, Federico introduces twelve impure metainferential logics defined using two different inferential Tarskian or non-Tarskian (e.g., substructural) logics: K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K3}$$\end{document}, LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{LP}$$\end{document}, ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ST }$$\end{document} or TS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{TS }$$\end{document}. The standard for premises and conclusions is different in each case. Five of these logics—i.e., TS/ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {TS/ST }$$\end{document}, TS/LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {TS/LP }$$\end{document}, TS/K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {TS/K3 }$$\end{document}, LP/ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP/ST }$$\end{document}, K3/ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K3/ST }$$\end{document}—are Tarskian logics–i.e., they validate Cut, Identity and Weakening. Moreover, they also validate Meta-Identity, Contraction, and Exchange. Nevertheless, they are defined using substructural inferential logics. In fact, TS/ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {TS/ST }$$\end{document} is built entirely with substructural inferential standards. More significantly, TS/ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {TS/ST }$$\end{document} collapses with CL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{CL }$$\end{document} at the metainferential level, proving to be the only one of these twelve logics with this feature. So, one could wonder what happens when at least one of these standards is not defined through the Strong Kleene schema.

As we have seen, there is a whole family of Strong Kleene mixed and impure metainferential logics. We have already presented the twelve metainferential mixed and impure consequence relations that can be characterized through four inferential logics with the previously mentioned—and well-known—three-valued semantics based on the Strong Kleene schema: ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ST }$$\end{document}, TS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{TS }$$\end{document}, K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K3}$$\end{document}, and LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{LP}$$\end{document}.

The aim of this chapter is to apply to some metainferential logics the notions of meta-validity presented in Chapter 2. First, we will explore the global2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document} notion (since it is a bit simpler to handle from a technical point of view), and we will show that the pure metainferential logic ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ST }$$\end{document} also recovers classical logic by substructural means, just as the ST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ST }$$\end{document}-hierarchy does locally.

In this paper, we present two ways of modelling every epistemic formal conditional commitment that involves (at most) three key epistemic attitudes: acceptance, rejection and neither acceptance nor rejection. The first one consists of adopting the plurality of every mixed Strong Kleene logic (along with an epistemic reading of the truth-values), and the second one involves the use of a unified system of six-sided inferences, named 6SK, that recovers the validities of each mixed Strong Kleene logic. We also introduce a sequent calculus that is sound and complete with respect to both approaches. We compare both accounts, and finally, we suggest that the plurality of Strong Kleene logic as well as the general framework 6SK are linked to formal epistemic norms via bridge principles.

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of ‘increasingly classical’ logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by taking a somehow different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician.

This book is the first to present a comprehensive investigation of the technical features of the metainferential logics developed in the last years, with their most relevant results and applications. It provides some new paths to define and investigate metainferential logics and offers a thorough study of the semantics and the proof-theories of this new and exciting variety of families of logics. This volume examines the hierarchies of metainferential logics and gives a general and systematic theory of them, and of the truth theories based on these logics. This book puts forward the prospects for truth-theories based on the metainferential logics of the TS/ST hierarchy and argues for its promise noting that each of these logics can be safely expanded with a transparent truth predicate. It also goes onto to explore new developments in three fields related to logics – namely metainferential logics built by means of the Weak Kleene schema and combining them with logics defined through the Strong Kleene schema, proof-theoretic presentations, and those with a with a global or an absolutely global validity standard, instead of a local one. This book is of interest to scholars in formal logic.

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. In this paper we tackle this expressive issue. We assume some background sentential language as given and define what we call an inferential language, that is, a language whose atomic formulas are inferences. We provide a model-theoretic characterization of validity for this language—relative to some given characterization of validity for the background sentential language—and provide a proof-theoretic analysis of validity. We argue that our novel language has fruitful philosophical applications. Lastly, we generalize some of our definitions and results to arbitrary metainferential levels.

In order to define some interesting consequence relations, certain generalizations have been proposed in a many-valued semantic setting that have been useful for defining what have been called pure, mixed and ordertheoretic consequence relations. But these generalizations are insufficient to capture some other interesting relations, like other intersective mixed relations (a relation that cannot be defined as a mixed relation, but only as the intersection of two mixed relations) or relations with a conjunctive (or, better, “universal”) interpretation for multiple conclusions. We propose a broader framework to define these cases, and many others, and to set a common background that allows for a direct compared analysis. At the end of the work, we illustrate some of these comparisons.

In this paper, we present a way to translate the metainferences of a mixed metainferential system into formulae of an extended-language system, called its associated σ-system. To do this, the σ-system 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 model-theoretic way, that these translations preserve (in)validity. That is, that a metainference is valid in the base system if and only if its translation is a tautology of its corresponding σ-system. We then use these results to obtain other key advantages. Most interestingly, we provide a recipe for building unlabeled sequent calculi for σ-systems. We then exemplify this with a σ-system useful for logics of the ST family, and prove soundness and completeness for it, which indirectly gives us a calculus for the metainferences of all those mixed systems. Finally, we respond to some possible objections and show how our σ-framework can shed light on the “obeying” discussion within mixed metainferential contexts

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. the standard for the premises and the conclusion is not necessarily the same. However, so far, all of these systems have been presented following a semantical standpoint, in terms of valuations based on the Strong Kleene truth-tables. In this article, we provide sound and complete sequent-calculi for the valid inferences and the invalid inferences of the logics \ and \, and introduce an algorithm that allows obtaining sound and complete sequent-calculi for the global validities and the global invalidities of any metainferential logic of any level.

Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems.