Interpolation Methods for Dunn Logics and Their Extensions

Stefan Wintein, Reinhard Muskens

Research output: Contribution to journalArticleScientificpeer-review

Abstract

The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the Dunncalculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the Dunn calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.
Original language English 1319–1347 Studia Logica 105 6 https://doi.org/10.1007/s11225-017-9720-5 Published - 2017

Fingerprint

Interpolation Method
Logic
Tableau
Interpolate
Interpolation
Calculus
Classical Logic
Interpolants
Signed
Valuation
Mirror

Keywords

• Interpolation methods
• Dunn logics
• First degree entailment
• Logic of paradox
• Strong Kleene logic
• Tableau calculus

Cite this

Wintein, Stefan ; Muskens, Reinhard. / Interpolation Methods for Dunn Logics and Their Extensions. In: Studia Logica. 2017 ; Vol. 105, No. 6. pp. 1319–1347.
@article{7602dd9b2e4f452c887a46d379e4a834,
title = "Interpolation Methods for Dunn Logics and Their Extensions",
abstract = "The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the Dunncalculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the Dunn calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.",
keywords = "Interpolation methods, Dunn logics, First degree entailment, Logic of paradox, Strong Kleene logic, Tableau calculus",
author = "Stefan Wintein and Reinhard Muskens",
year = "2017",
doi = "10.1007/s11225-017-9720-5",
language = "English",
volume = "105",
pages = "1319–1347",
journal = "Studia Logica",
issn = "0039-3215",
publisher = "Springer Netherlands",
number = "6",

}

In: Studia Logica, Vol. 105, No. 6, 2017, p. 1319–1347.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Interpolation Methods for Dunn Logics and Their Extensions

AU - Wintein, Stefan

AU - Muskens, Reinhard

PY - 2017

Y1 - 2017

N2 - The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the Dunncalculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the Dunn calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.

AB - The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the Dunncalculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the Dunn calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.

KW - Interpolation methods

KW - Dunn logics

KW - First degree entailment

KW - Logic of paradox

KW - Strong Kleene logic

KW - Tableau calculus

U2 - 10.1007/s11225-017-9720-5

DO - 10.1007/s11225-017-9720-5

M3 - Article

VL - 105

SP - 1319

EP - 1347

JO - Studia Logica

JF - Studia Logica

SN - 0039-3215

IS - 6

ER -