Martina Zirattu

We explore certain algebraic structures that naturally emerge within the framework of logics of synonymy, analytic containment, and hyperintensionality. In particular, we argue that Angell’s logic $$\textrm{AC}$$, one of the earliest and most successful attempts to analyse the properties of logical constants with a topic-transformative character, can be better understood through a direct algebraic study of De Morgan bisemilattices. Inter alia, we study and compare the quasivarieties of De Morgan bisemilattices generated by certain finite algebras considered in the literature, viewed as formalising different approaches to the validity of inferences among statements of synonymy.

We explore certain algebraic structures that naturally emerge within the framework of logics of synonymy, analytic containment, and hyperintensionality. In particular, we argue that Angell's logic AC, one of the earliest and most successful attempts to analyse the properties of logical constants with a topic-transformative character, can be better understood through a direct algebraic study of De Morgan bisemilattices. Inter alia, we study and compare the quasivarieties of De Morgan bisemilattices generated by certain finite algebras considered in the literature, viewed as formalising different approaches to the validity of inferences among statements of synonymy.

This paper examines a range of logical systems within the family of variable inclusion logics—also known as containment logics. We focus on those logics that restrict classically valid inferences to ones meeting specific variable inclusion constraints, hence called variable inclusion companions of classical logic. These constraints can be seen as enforcing varying degrees of relevance between premises and conclusions, placing these systems within the broader tradition of relevance logics. We review established companions of Classical Logic, including Weak Kleene logics (Bochvar, 1938, Halldén, 1949), pure variable inclusion logics (Paoli et al., 2021), uniform Weak Kleene logics (Borzi and Zirattu, forth.), and their intersection—pure uniform logics. We then introduce two new systems named analytic and synthetic companions of Classical Logic. We characterize their consequence relations and interpret them, as their names suggest, as validating only analytic and synthetic inferences. Lastly, we argue that these systems more effectively address the irrelevance cases that motivated earlier proposals, by correctly tracing them to the Monotonicity of the consequence relation.

This chapter is about non-reflexive logics in the “tolerant-strict” family. We present them both from a proof-theoretic and a model theoretic point of view. For the first, we define new sequent calculi in which identity is not admissible nor derivable, that are obtained by removing the identity axiom from the calculus and possibly adding or removing other rules. For the second, we build on some notions already introduced in the previous chapter and present different semantics, which are then related to the calculi through soundness and completeness results both at the inferential and metainferential level. The level of metainferences is further discussed showing that the non-reflexive calculi are structurally incomplete and then proving some results regarding their global, local and absolute global ranges. Next, the non-reflexive logic TS is related to other logics introduced in this and the previous chapters, namely CL, ST, LP, and K3. In particular, several notions of translation and duality which can be found in the literature are presented and compared. Some space is also devoted to the introduction of metainferential mixed logics. Finally, we build on the notion of anti-inference central to the discussion on dualities and show that, given a Tarskian logic, one can construct a non-reflexive logic with the same anti-inferences. We conclude by briefly presenting some three-sided sequent calculi used in the literature to characterize some of the logics discussed here and in the previous chapters.

In a multiple-premise and multiple-conclusion setting, logicians typically use the metalinguistic expression ‘,’ to gather premises and conclusions. In most logical systems, this comma can be freely interchanged with the object-language conjunction on the premise side and with the objectlanguage disjunction on the conclusion side. However, this is not the case for Weak Kleene Logics (WK logics), which include Paraconsistent Weak Kleene (PWK) and its paracomplete counterpart (Kw3). The aim of this work is to define two new substructural systems called uniform Weak Kleene logics (uWK logics), uPWK and uKw3, which reestablish what we call ‘Uniformity’: the correspondence between the structural comma and the object-language WK operators. We begin by providing some philosophical arguments advocating for it as a desirable logical feature. We then characterize their consequence relations and introduce a novel two-sided sequent calculus free of linguistic restrictions for PWK and for Kw3 and show that, with minor adjustments, we can achieve sound and complete sequent calculi for uPWK and uKw3. Finally, we generalize our results for WK logics and their uniform counterparts to other infectious Tarskian logics that meet specific criteria.

We provide non-deterministic semantics for some content inclusion logics standing between the first-degree entailment fragments of Parry's logic PAI and Angell's logic of analytic implication AC. Our semantics is inspired by two-address semantics developed following ideas introduced by Herzberger and Woodruff, suggesting to independently evaluate formulas on their alethic and topical status. Building on this, we explore the results of allowing negation to be non-deterministic on either of these independent aspects. For this purpose, we emulate the presence of truth-value gaps and gluts by letting negation work non-deterministically on the alethic coordinate. At the same time, we emulate the presence of topic-transformed negated formulas by letting negation work non-deterministically on the topical coordinate. The outcome is a unifying framework for classical logic and a vast collection of non-classical systems that characterize them according to how negation and the set of designated values are defined.

In his 2016 article _On All Strong Kleene Generalizations of Classical Logic_, Stefan Wintein provides a detailed and comprehensive semantic and tableau-based analysis of the consequence relations that can be defined over the four-valued Belnap–Dunn semantics. These include familiar consequence relations like $$\textsf{FDE}$$, which takes $$\{t, b\}$$ as the set of designated values, but also much less familiar relations that don’t follow the designated-value strategy (i.e. defining logical consequence as preservation of a set of values). It turns out that many of the interesting features of these relations are made evident at the level of metainferences and, although Wintein discusses some aspects of metainferences, his work does not provide a systematic way to decide on the validity of metainferences for these logics. In this paper, we extend Wintein’s tableaux from inferences to metainferences. The paper ends with a discussion about different ways in which we can understand the notion of metainference validity.