Interpolation in 16-Valued Trilattice Logics

Reinhard Muskens, Stefan Wintein

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Abstract

In a recent paper we have defined an analytic tableau calculus PL16 for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice SIXTEEN3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations , , and that each correspond to a lattice order in SIXTEEN3; and , the intersection of and . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that , when restricted to Ltf, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.
Original languageEnglish
Pages (from-to)1-26
JournalStudia Logica
DOIs
Publication statusPublished - 2017

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Interpolate
Logic
Calculus
Intersection
Tableau
Interpolation
Calculi
Language
Entailment
Range of data
Semantics
Syntax
Expressive Language
Semantic Relations

Keywords

  • Interpolation
  • 16-Valued logic
  • Trilattice SIXTEEN3
  • Multiple tree calculus

Cite this

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title = "Interpolation in 16-Valued Trilattice Logics",
abstract = "In a recent paper we have defined an analytic tableau calculus PL16 for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice SIXTEEN3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations , , and that each correspond to a lattice order in SIXTEEN3; and , the intersection of and . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that , when restricted to Ltf, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.",
keywords = "Interpolation, 16-Valued logic, Trilattice SIXTEEN3, Multiple tree calculus",
author = "Reinhard Muskens and Stefan Wintein",
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doi = "10.1007/s11225-017-9742-z",
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journal = "Studia Logica",
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}

Interpolation in 16-Valued Trilattice Logics. / Muskens, Reinhard; Wintein, Stefan.

In: Studia Logica, 2017, p. 1-26.

Research output: Contribution to journalArticleScientificpeer-review

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T1 - Interpolation in 16-Valued Trilattice Logics

AU - Muskens, Reinhard

AU - Wintein, Stefan

PY - 2017

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N2 - In a recent paper we have defined an analytic tableau calculus PL16 for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice SIXTEEN3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations , , and that each correspond to a lattice order in SIXTEEN3; and , the intersection of and . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that , when restricted to Ltf, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.

AB - In a recent paper we have defined an analytic tableau calculus PL16 for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice SIXTEEN3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations , , and that each correspond to a lattice order in SIXTEEN3; and , the intersection of and . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that , when restricted to Ltf, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.

KW - Interpolation

KW - 16-Valued logic

KW - Trilattice SIXTEEN3

KW - Multiple tree calculus

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