Joaquín Santiago Toranzo Calderón

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 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.

According to the Adoption Problem certain basic logical principles cannot be adopted. Drawing on the AP, Suki Finn presents an argument against logical pluralism: Modus Ponens and Universal Instantiation both govern a general structure shared by every logical rule. As such, analogues of these two rules must be present in every meta-logic for any logical system L, effectively imposing a restriction to logical pluralism at the meta-level through their presence constituting a “meta-logical monism”. We find a tension in the dual role that the “unadoptable rules” must play in Finn’s “meta-logical monism” rendering it ineffective to restrict logical theories and systems. Consequently, we argue they cannot be both analogues of MP and UI and inferentially productive. We conclude with a series of suggestions regarding where a more satisfying and robust interpretation of the AP could lie.

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.

En este trabajo presentaré una forma de evitar los problemas más recurrentes en cierta versión del pluralismo lógico, aquella que defiende que incluso considerando un lenguaje fijo existen múltiples sistemas lógicos legítimos. Para ello, será necesario considerar los puntos de partida del programa pluralista y explicitar los problemas que de ellos surgen, principalmente el Desafío de Quine y el Problema del Colapso. Luego, propondré una modificación respecto de lo que se entiende por consecuencia lógica, para poder considerar una familia de sistemas lógicos, las lógicas mixtas, que abarcan tanto a las lógicas puras como a las impuras. Finalmente, mostraré que con una interpretación razonable del formalismo se puede eludir aquellos problemas a la vez que se respeta el espíritu del programa pluralista.