Rafael Ongaratto

Logics of Formal Inconsistency (LFIs) are usually considered a philosophically neutral logicwith respect to paraconsistency, in the sense that they provide a good basis to think in termsof dialetheias, i.e. true contradictions, or in terms of conflicting information, a notion weakerthan truth. In this article, I will show how this claim of neutrality fails in the face of the LiarParadox: in classical logic, the sentence A: “A is false.” implies a contradiction. By adoptinga paraconsistent logic, we avoid that contradictions lead to triviality. LFIs, however, add aconsistency operator that recovers classical logic to propositions to which it applies. In thiscontext, the Liar Paradox arises once again by a type of Liar’s Revenge: now, we can constructthe sentence B: “B is only false.”, and that gives us triviality once again. With this result, weconclude that LFIs are not suitable logics to deal with semantic paradoxes.

In the Remarks on the Foundations of Mathematics, Wittgenstein discusses in detail the significance of the Gödelian “true but unprovable” proposition. Despite the negative reviews of early commentators, there have been recent attempts to provide a more positive assessment of these remarks. However, no consensus has emerged. Based on an analysis of Wittgenstein’s distinction between proof and prose and its connections with his views on indirect and inductive proofs, we approach this issue afresh, and provide a reading which not only brings to light these pivotal aspects, but shows how they are necessary for an accurate exegesis of the text and clarifies the relationship between Wittgenstein’s commentaries and Gödel’s incompleteness proof.

In the Remarks on the Foundations of Mathematics, Wittgenstein discusses in detail the significance of the Gödelian “true but unprovable” proposition. Despite the negative reviews of early commentators, there have been recent attempts to provide a more positive assessment of these remarks. However, no consensus has emerged. Based on an analysis of Wittgenstein’s distinction between proof and prose and its connections with his views on indirect and inductive proofs, we approach this issue afresh, and provide a reading which not only brings to light these pivotal aspects, but shows how they are necessary for an accurate exegesis of the text and clarifies the relationship between Wittgenstein’s commentaries and Gödel’s incompleteness proof.

In this article, the logic PALFI1 is introduced, a paraconsistent dynamic epistemic logic based on LFI1. Within an epistemic approach of paraconsistency, the themes of change and knowledge can be discussed in the framework of paraconsistent logic. This engenders new ways to think about epistemic states in epistemic logics, and it also illuminates the intricate connection between paraconsistency, change, and knowledge. Using PALFI1, it is possible to give a satisfactory reply to Kripke’s Paradox of Knowledge, and also to represent the evolution of informational environments with multiple agents.

Este artigo explora o uso de metodologias lúdicas, como gamificação e narrativas, para tornar o ensino de Lógica mais dinâmico e acessível no Ensino Médio. Através de enigmas como os do tipo Cavaleiros e Patifes, abordam-se tanto a lógica proposicional clássica quanto as não-clássicas (notadamente, as lógicas paraconsistentes e modal),proporcionando um aprendizado ativo e colaborativo. O artigo oferece ferramentas práticas para educadores, com ênfase no desenvolvimento de competências críticas e na criação de um ambiente inclusivo, acessível a todos os alunos.

In the Appendix I of his "Remarks on the Foundations of Mathematics", Wittgenstein elaborates a different interpretation of Gödel’s First Incompleteness Theorem, which we have come to refer to as "Gödel’s Theorem" or "Incompleteness Theorem". This nomenclature arises from the recognition that the so-called "Second Incompleteness Theorem" is essentially a corollary of the primary theorem. Wittgenstein aims to reassess Gödel’s conclusion that there exist true formulas not demonstrable within formal systems capable of representing a sufficient amount of arithmetic theory. Gödel’s initial reaction, as well as other commentators, was that Wittgenstein had not understood the proof. Nevertheless, recent commentators view worthy commentaries in wittgensteinian writings: some commentators, such as Juliet Floyd and Hilary Putnam, distinguish between mathematical proof and philosophical prose that surrounds the theorem, making it possible to understand Wittgenstein’s remarks. Ultimately, Wittgenstein’s observations serve as a lens through which Gödel’s theorem can be reconsidered within the realm of non-classical logics, such as paraconsistent logic.

The article makes an analysis of a paraconsistent solution to the liar paradox, namely, Priest’s solution. Paraconsistent logics are characterized, in opposition to classical logic, as rejecting the Principle of Explosion, which says that “from a contradiction everything follows”. Priest, in turn, is a dialetheist, an interpretation of paraconsistency that admits the truth of contradictions. His answer to the liar paradox, therefore, accepts the liar sentence as being a truly paradoxical sentence using LP, the “Logic of Paradox. Then, we propose some critiques of his position. Finally, we evaluate Priest’s solution to the paradox in comparison with Tarski and Kripke canonical solutions.