Paracomplete Logic

The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxes. There is something very distinctive about ST: namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes. For example, the logic ST + (ST for a language with a transparent truth predicate and self-referential sentences) is a conservative extension of classical logic. That is, ST + is not only non-trivial, but it has exactly the same valid inferences as classical logic for the T-free fragment. How is this possible? Well, because ST + preserves all classically valid inferences but not some classical metainferences. The question then arises of exactly which are the metainferences of ST +. In a recent paper, Eduardo Barrio, Lucas Rosenblatt and Diego Tajer show that ST + metainferences are closely related to LP inferences. In this paper we review their result and put the connection in a broader context.

The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxes. There is something very distinctive about ST: namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes. For example, the logic ST + (ST for a language with a transparent truth predicate and self-referential sentences) is a conservative extension of classical logic. That is, ST + is not only non-trivial, but it has exactly the same valid inferences as classical logic for the T-free fragment. How is this possible? Well, because ST + preserves all classically valid inferences but not some classical metainferences. The question then arises of exactly which are the metainferences of ST +. In a recent paper, Eduardo Barrio, Lucas Rosenblatt and Diego Tajer show that ST + metainferences are closely related to LP inferences. In this paper we review their result and put the connection in a broader context.

El sistema LBVF se construye como una extensión de la lógica clásica positiva, al incluir operadores de afirmación alterna, negación alterna y completez, además, se definen a partir de ellos los operadores de negación clásica y buena fundamentación. El sistema es caracterizado por una semántica de valuaciones tradicionales con la cual se prueba que, respecto a los operadores de afirmación y de negación alterna el sistema es para completo. En el sistema se caracterizan las definiciones de verdad y falsedad presentadas por Aristóteles, representando la falsedad aristotélica con el operador de negación alterna y la verdad aristotélica con el operador de afirmación alterna, lográndose con esta interpretación dar solución a una variante de la paradoja del mentiroso.

MSC: 03BXX, 03B53, 03B42

It is commonly held that the ascription of truth to a sentence is intersubstitutable with that very sentence. However, the simplest subclassical logics available to proponents of this view, namely K3 and LP, are hopelessly weak for many purposes. In this paper, I argue that this is much more of a problem for proponents of LP than for proponents of K3. The strategies for recapturing classicality oered by proponents of LP are far less promising than those available to proponents of K3. This undermines the ability of proponents LP to engage in public reasoning in classical domains.

Jamie Tappenden was one of the first authors to entertain the possibility of a common treatment for the Liar and the Sorites paradoxes. In order to deal with these two paradoxes he proposed using the Strong Kleene semantic scheme. This strategy left unexplained our tendency to regard as true certain sentences which, according to this semantic scheme, should lack truth value. Tappenden tried to solve this problem by using a new speech act, articulation. Unlike assertion, which implies truth, articulation only implies non-falsity. In this paper I argue that Tappenden’s strategy cannot be successfully applied to truth and the Liar.

This paper contributes to the study of paracompleteness and paraconsistency. We present two logics that address the following questions in novel ways. How can the paracomplete theorist characterize the formulas that defy excluded middle while maintaining that not all formulas are of this kind? How can the paraconsistent theorist characterize the formulas that obey explosion while still maintaining that there are some formulas not of this kind?

We present a refinement of Anderson’s reduction of deontic
logic to modal logic with only alethic modalities. The refined proposal contextualizes the Andersonian sanction constant s by replacing it with a unary sanction operator S that is dependent on the concrete normative requirement that is violated. A formula SB is then for instance interpreted as “B causes a sanction” or as “B provides a reason for (the applicability of) a sanction”. Due to its modified sanction operator, the resulting logic DSL invalidates some instances of the inheritance principle.
This gives rise to new interesting features. For instance, DSL consistently allows for the presence of conflicting obligations. Moreover, it provides novel insights in various central ‘paradoxes’ in deontic logic such as the Ross paradox, the paradox of the good Samaritan, and Forrester’s ‘gentle murderer’ paradox.

(Resumo - Português)
Poderia haver um único sistema de lógica que permita trabalharmos simultaneamente com negações clássica, paraconsistente e paracompleta? Essas três negações foram separadamente estudadas em lógicas cujas negações levam seus nomes. Inicialmente iremos restringir nossa análise às lógicas proposicionais, analisando a negação clássica, ¬c, tal como tratada pela Lógica Proposicional Clássica LPC); a negação paraconsistente, ¬p como tratada através da hierarquia de Cálculos Proposicionais Paraconsistentes Cn (0 ≤ n ≤ ω); e a negação paracompleta, ¬q, como tratada pela hierarquia de Cálculos Proposicionais Paracompletos Pn (0 ≤ n ≤ ω). Ainda que possamos chamar os três conectivos de “negações”, suas propriedades são distintas de acordo com cada sistema (que, por sua vez, também preservam características metateóricas distintas). Em “Logics that are both paraconsistent and paracomplete” (1989), Newton da Costa propôs um sistema com características aproximadas ao que buscamos. Na hierarquia de Cálculos Proposicionais Não-Aléticos Nn (0 ≤ n ≤ ω), apenas uma negação é introduzida (como primitiva), chamada de “não-alética” (¬n), cujo funcionamento pode preservar as propriedades da negação clássica, ou paraconsistente ou paracompleta -- dependendo do bom ou mau comportamento da fórmula a ela conectada. No entanto, como veremos, na hierarquia Nn não podemos reiterar negações com comportamentos diferentes a uma mesma fórmula (e.g., ¬p¬cα ou ¬q¬c¬pα), ou mesmo analisar uma fórmula como ¬cα → ¬pα. Em vista desses problemas, podemos realmente dizer que a hierarquia Nn nos permite compreender as relações e interações dos três tipos de negações? Para tratarmos disso, posto o problema inicial, apresentaremos quatro sistemas axiomáticos (KG) nos quais, diferente de Nn, as três negações são diretamente introduzidas -- oferecendo também uma semântica e um método de provas por tableaux analíticos. Através dos Sistemas KG mostraremos como as negações interagem, obtendo teoremas não demonstráveis em LPC Cn, Pn e Nn (0 ≤ n ≤ ω). Por fim, ofereceremos também uma extensão em primeira ordem para os Sistemas KG.

(Abstract - Inglês)
Could there be a single logical system that would allow us to work simultaneously with classical, paraconsistent, and paracomplete negations? These three negations were separately studied in logics whose negations bear their names. Initially we will restrict our analysis to propositional logics by analyzing classical negation, ¬c, as treated by Classical Propositional Logic (LPC); the paraconsistent negation, ¬p, as treated through the hierarchy of Paraconsistent Propositional Calculi Cn (0 ≤ n ≤ ω); and the paracomplete negation, ¬q, as treated by the hierarchy of Paracomplete Propositional Calculi Pn (0 ≤ n ≤ ω). In “Logics that are both paraconsistent and paracomplete” (1989), Newton da Costa proposed a system with approximate characteristics to what we are looking for. In the hierarchy of Non-Alethical Propositional Calculi Nn (0 ≤ n ≤ ω), only one negation is introduced (as primitive), called a “non-alethic” (¬n), whose operation preserves the properties of classical, or paraconsistent or paracomplete negation -- depending on the well or ill behavior of the formula connected to it. However, as we shall see, in the hierarchy Nn we can not reiterate negations with different behaviors in a same formula (e.g., ¬p¬cα or ¬q¬c¬p α), or even analyze a formula like ¬cα → ¬pα. In view of these problems, can we really say that the hierarchy Nn allows us to understand the relationships and interactions of the three types of negations? In order to deal with this, given the initial problem, we will present four axiomatic systems (KG) in which, unlike Nn, the three negations are directly introduced -- offering a semantics and a method of proofs by analytic tableaux. Through the KG Systems we will show how the negations interact, obtaining non-demonstrable theorems in LPC, Cn, Pn, and Nn (0 ≤ n ≤ ω). Finally, we will also offer a first-order extension for the KG Systems.