In this paper we study a new operator of strong modality ⊞, related to the non-contingency operator ∆. We then provide soundness and completeness theorems for the minimal logic of the ⊞-operator.

We argue that many-valued logics can be useful in analysing informational conflicts by using society semantics. This work concentrates on four-valued Łukasiewicz logic. SSs were proposed by Carnielli and Lima-Marques to deal with conflicts of information involving rational agents that make judgements about propositions according to a given logic within a society, where a society is understood as a collection $\mathcal{A}$ of agents. The interesting point of such semantics is that a new logic can be obtained by combining the logic (...) of the agents under some appropriate rules. Carnielli and Lima-Marques defined SSs for the three-valued logics $I^{1}$ and $P^{1}$. In this kind of semantics, all the agents reason according to classical logic and the molecular formulas behave in the same way as in CL. Marcos provided SSs with classical agents for the three-valued Łukasiewicz logic Ł$_{3}$, but in this case, the molecular formulas do not behave classically. We prove here that one can characterize Ł$_{4}^{\prime}$, a conservative extension of Ł$_{4}$ obtained by adding a connective $\blacktriangledown$, by means of a closed society where the agents reason according to Ł$_{3}$. We shall emphasize the importance of recovery operators in the construction of this class of societies. Moreover, we shall relate this semantics to Suszko’s view on the ‘two-valuedness’ of logic. (shrink)

Semantic paradoxes pose a real threat to logics that attempt to be capable of expressing their own semantic concepts. Particularly, Curry paradoxes seem to show that many solutions must change our intuitive concepts of truth or validity or impose limits on certain inferences that are intuitively valid. In this way, the logic of a universal language would have serious problems. In this paper, we explore a different solution that tries to avoid both limitations as much as possible. Thus, we argue (...) that it is possible to capture the naive concepts of truth and validity without losing any of the valid inferences of classical logic. This approach is called the Buenos Aires plan. We present the logic of truth and validity, $$\mathsf {STTV}_{\omega }$$ based on the hierarchy of logics $$\textsf{ST}_{\omega }$$, whose validity predicate has the same semantic conditions as the material conditional. We argue that $$\mathsf {STTV}_{\omega }$$ is capable of blocking the problematic results while keeping the deductive power of classical logic as much as possible and offering an adequate semantic theory. On the other hand, one could object that it is not possible to reason with $$\mathsf {STTV}_{\omega }$$ because it is not closed under its logical principles. We respond to this objection and argue that the local characterization of validity shows how to make inferences using the logic $$\textsf{ST}_{\omega }$$. (shrink)

In this paper, we argue that the systems Basic Logic of Evidence (BLE) and Logic of Evidence and Truth (LETJ) suffer a kind of semantic incompleteness with respect to the informal notion of evidence. More especifically, we argue that the connective o of the logic LETJ fails to validate intuitive principles about conclusive evidence.

Las paradojas semánticas muestran que las teorías semánticas que internalizan sus propios conceptos semánticos, como la verdad y la validez, no pueden validar toda la lógica clásica. Es decir, es necesario debilitar algún conectivo del lenguaje objeto, tomado como culpable de las paradojas, o renunciar a alguna propiedad de la relación de consecuencia de la teoría lógica. Ambas estrategias pueden alejarnos de la lógica clásica, que es la lógica comúnmente utilizada en nuestras teorías matemáticas actuales. Por tanto, una solución deseable (...) a las paradojas semánticas no debería alejarnos de la lógica clásica. En este trabajo analizamos dos propuestas interesantes que pretenden mantener la lógica clásica al máximo posible. La primera estrategia, Barrio & Pailos & Szmuc-approach (2017) (BPS-approach), propone la lógica paraconsistente MSC que contiene en su lenguaje objeto un conectivo capaz de recuperar la inferencia clásica siempre que las oraciones en cuestión sean consistentes. Así, muestran que es posible construir una teoría semántica sobre esta lógica que sea inmune a las paradojas semánticas. El segundo enfoque se basa en la jerarquia STω de sistemas no transitivos STn, propuesta por Pailos (2020a). Esta jerarquía recupera tantas metainferencias clásicas como sea posible en los niveles superiores de la jerarquía. Argumentamos a favor del segundo enfoque, argumentando que la primera estrategia tiene que adoptar procedimientos autorreferenciales débiles para evitar las paradojas. (shrink)

Many semantic theories become trivial when extended with a naïve validity predicate due to the validity paradoxes. The non-classical semantic theories are the ones that allegedly preserve the naïveté of the validity predicate while being capable of avoiding the validity paradoxes. This blocking, on the other hand, usually comes at a high cost. In this paper, we argue that the pre-theoretical notion of validity that the naïve validity predicate intends to capture is unattainable.

In this paper, we investigate the family LS0.5 of many-valued modal logics LS0.5's. We prove that the modalities of necessity and possibility of the logics LS0.5's capture well-defined bivalent concepts of logical validity and logical consistency. We also show that these modalities can be used as recovery operators.