A Recovery Operator for Nontransitive Approaches

Abstract

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities.

Non-transitive Correspondence Analysis

“…Lastly, let us agree with the first referee's kind recommendation that it is instructive to apply (see (Barrio et al, 2019) for one of applications) the method of Frankowski (2004a, b) who underlines that his approach is closely related to the one of Malinowski (1990), to the logics discussed in this paper. With all due respect, both the first and the second points above together with the non-embryonic completeness proof below make applying the Frankowskian method a topic of a future paper.…”

A Hierarchy of Classical and Paraconsistent Logics

“…The aforementioned subtleties have their origin in recent debates in the philosophical logic literature, more particularly, in a number of observations revolving around a project carried out by the collective formed by Pablo Cobreros, Paul Egré, David Ripley and Robert van Rooij. With the goal of solving paradoxes coming from the semantic, set-theoretic and vagueness corners, these authors have put forward a number of conceptual and technical arguments that support them backing a nice, innovative and quite revolutionary stance towards these riddles: the strict-tolerant approach, which led them to entertain the non-transitive logic ST. 1 The main advantage of this choice lies, allegedly, in keeping Classical Logic as the underlying inferential framework, even in what pertains to the problematic phenomena referred above. Thus, these authors claim to have shown that Classical Logic-actually, ST, but both systems are identical, according to them-can non-trivially deal with these otherwise problematic phenomena.…”

“…5 Another way to fully recover Classical Logic, starting from ST, is through the introduction of a so-called consistency or recovery operator. This path was explored by us in detail in [1].…”

Expressing consistency consistently

“…3 3 Their view is mostly developed in Barrio, Pailos, and Szmuc (2017). See also Barrio, Pailos, and Szmuc (2020) and Pailos (2020) for other applications of the same idea.…”