scholarly journals Revisitando a Lógica de Dunn-Belnap

Manuscrito ◽  
2017 ◽  
Vol 40 (2) ◽  
pp. 99-126 ◽  
Author(s):  
Carolina Blasio
Keyword(s):  
First Degree Entailment

RESUMO O presente artigo apresenta uma semântica baseada nas atitudes cognitivas de aceitação e rejeição por uma sociedade de agentes para lógicas inspiradas no First Degree Entailment (E) de Dunn e Belnap. Diferente das situações epistêmicas originalmente usadas em E, as atitudes cognitivas não coincidem com valores-de-verdade e parecem mais adequadas para as lógicas que pretendem considerar o conteúdo informacional de proposições “ditas verdadeiras” tanto quanto as proposições “ditas falsas” como determinantes da noção de validade das inferências. Após analisar algumas lógicas associadas à semântica proposta, introduzimos a lógica E B cuja relação de consequência semântica subjacente - o B-entailment - é capaz de expressar diversos tipos de raciocínio em relação às atitudes cognitivas de aceitação e rejeição. Apresentamos também um cálculo de sequentes correto e completo para E B .

scholarly journals Hybrid Deduction-Refutation Systems for FDE-Based Logics

2021 ◽  
Vol 18 (6) ◽  
pp. 599-615
Author(s):  
Eoin Moore
Keyword(s):  
Hybrid Systems ◽  
Hybrid System ◽  
Classical Logic ◽  
First Degree Entailment ◽  
Refutation Systems

Hybrid deduction-refuation systems are presented for four first-degree entailment based logics. The hybrid systems are shown to be deductively and refutationally sound with respect to their logics. The proofs of completeness are presented in a uniform way. The paper builds on work by Goranko, who presented a deductively and refutationally sound and complete hybrid system for classical logic.

First-Degree Entailment and its Relatives

Studia Logica ◽  
2017 ◽  
Vol 105 (6) ◽  
pp. 1291-1317 ◽  
Author(s):  
Yaroslav Shramko ◽  
Dmitry Zaitsev ◽  
Alexander Belikov
Keyword(s):  
First Degree Entailment

scholarly journals Kripke-Style Models for Logics of Evidence and Truth

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 100 ◽  
Author(s):  
Henrique Antunes ◽  
Walter Carnielli ◽  
Andreas Kapsner ◽  
Abilio Rodrigues
Keyword(s):  
Classical Logic ◽  
First Degree Entailment ◽  
Intended Interpretation

In this paper, we propose Kripke-style models for the logics of evidence and truth LETJ and LETF. These logics extend, respectively, Nelson’s logic N4 and the logic of first-degree entailment (FDE) with a classicality operator ∘ that recovers classical logic for formulas in its scope. According to the intended interpretation here proposed, these models represent a database that receives information as time passes, and such information can be positive, negative, non-reliable, or reliable, while a formula ∘A means that the information about A, either positive or negative, is reliable. This proposal is in line with the interpretation of N4 and FDE as information-based logics, but adds to the four scenarios expressed by them two new scenarios: reliable (or conclusive) information (i) for the truth and (ii) for the falsity of a given proposition.

scholarly journals Parity, Relevance, and Gentle Explosiveness in the Context of Sylvan's Mate Function

2018 ◽  
Vol 15 (2) ◽  
pp. 381
Author(s):  
Thomas Macaulay Ferguson
Keyword(s):  
Infinite Number ◽  
Proof Theory ◽  
Possible Worlds ◽  
Consequence Relations ◽  
Deductive Systems ◽  
First Degree Entailment ◽  
Broad Field ◽  
Soundness And Completeness ◽  
Set Up ◽  
Hallmark Feature

The Routley star, an involutive function between possible worlds or set-ups against which negation is evaluated, is a hallmark feature of Richard Sylvan and Val Plumwood's set-up semantics for the logic of first-degree entailment. Less frequently acknowledged is the weaker mate function described by Sylvan and his collaborators, which results from stripping the requirement of involutivity from the Routley star. Between the mate function and the Routley star, however, lies an broad field of intermediate semantical conditions characterizing an infinite number of consequence relations closely related to first-degree entailment. In this paper, we consider the semantics and proof theory for deductive systems corresponding to set-up models in which the mate function is cyclical. We describe modifications to Anderson and Belnap's consecution calculus LE_fde2 that correspond to these constraints, for which we prove soundness and completeness with respect to the set-up semantics. Finally, we show that a number of familiar metalogical properties are coordinated with the parity of a mate function's period, including refined versions of the variable-sharing property and the property of gentle explosiveness.

scholarly journals Game Theoretical Semantic for Relevant Logic

2015 ◽  
Vol 21 (2) ◽  
pp. 42-52
Author(s):  
В. Л. Васюков
Keyword(s):  
Relevant Logic ◽  
Game Semantics ◽  
First Degree Entailment

In 1979 D.E. Over proposed game theoretical semantics for first-degree entailment formulated by Anderson and Belnap. In order to extend this approach to include other systems of relevant logc (e.g., $\boldsymbol{R}$) we have two promoting facts. Firstly, there is Routley- Meyer’s situational semantic for system$\boldsymbol{R}$ of relevant logic. Secondly, this semantics shows some resemblance with W__ojcicki’s situational semantic of non-fregean logic for which the situational game semantics was developed by author exploiting essentially the notion of non-fregean games. In the paper an attempt is done to give a partial account of these results and some conception of situational games developed which laid down into foundation of the game theoretical semantics of relevant logic $\boldsymbol{R}$.

scholarly journals Аналитические таблицы для интуиционистского аналога FDE

2018 ◽  
Vol 24 (2) ◽  
pp. 116-122
Author(s):  
Я. И. Петрухин
Keyword(s):  
First Degree Entailment

Н. Д. Белнап сформулировал релевантную логику первоуровневого следования $\textbf{FDE}$(First Degree Entailment), избегающую так называемых парадоксов классического следования: и . В $\textbf{FDE}$ рассматриваются формулы, главным знаком которых является импликация, антецедент и консеквент которой содержат только отрицание, дизъюнкцию и конъюнкцию. В связи с тем, что интуиционистское следование имеет те же парадоксы, что и классическое, возникла проблема построения интуиционистского аналога $\textbf{FDE}$, избегающего парадоксов интуиционистского следования. Я.В. Шрамко удалось решить эту проблему, построив логику $\bf IE_{fde}$. В $\bf IE_{fde}$ наряду с релевантной импликацией рассматривается интуиционистская, поскольку, в отличие от классической, она не выражается через отрицание, конъюнкцию и дизъюнкцию. Я. В. Шрамко сформулировал интуиционистскую версию разработанной Е. К. Войшвилло семантики обобщенных описаний состояний для $\textbf{FDE}$. В этой работе мы предлагаем адекватные аналитические таблицы в стиле М. Фиттинга для $\bf IE_{fde}$, опираясь на семантику этой логики, разработанную Я.В. Шрамко. Мы модифицируем аналитические таблицы М. Фиттинга для интуиционистской логики, добавив два новых типа отмеченных формул ($\overline{T}A$ (не-истинно $A$) и $\overline{F}A$ (не-ложно $A$)), правила редукции для них, адаптировав соответствующим образом определения, а также правила для $TA$ и $FA$. Множество отмеченных формул $ S $ называется замкнутым, если оно одновременно содержит отмеченные формулы вида $ TA $ и $ \overline{T}A $ или $ FA $ и $ \overline{F}A $. Замкнутая таблица для $ \{TA, \overline{T}B\} $ называется доказательством формулы $A\rightarrow B $. В тех правилах, в которых в интуиционистской логике вычеркиваются отмеченные формулы вида $FA$, в $\bf IE_{fde}$ вычеркиваются также отмеченные формулы вида $\overline{T}A$. Кроме того, построенные нами аналитические таблицы для $\bf IE_{fde}$ являются разрешающей процедурой для этой логики. DOI: 10.21146/2074-1472-2018-24-2-116-122

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