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Kripke Semantics for Intuitionistic Łukasiewicz Logic
Studia Logica ( IF 0.7 ) Pub Date : 2020-04-21 , DOI: 10.1007/s11225-020-09908-z
A. Lewis-Smith , P. Oliva , E. Robinson

This paper proposes a generalization of the Kripke semantics of intuitionistic logic IL appropriate for intuitionistic Łukasiewicz logic IŁL —a logic in the intersection between IL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on the poset sum construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009). to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that $$w \Vdash \psi $$ w ⊩ ψ —which for IL is a relation between worlds w and formulas $$\psi $$ ψ , and can be seen as a function taking values in the booleans $$(w \Vdash \psi ) \in {{\mathbb {B}}}$$ ( w ⊩ ψ ) ∈ B —becomes a function taking values in the unit interval $$(w \Vdash \psi ) \in [0,1]$$ ( w ⊩ ψ ) ∈ [ 0 , 1 ] . An appropriate monotonicity restriction (which we call sloping functions ) needs to be put on such functions in order to ensure soundness and completeness of the semantics.

中文翻译:

直觉主义 Łukasiewicz 逻辑的 Kripke 语义

本文提出了直觉逻辑 IL 的 Kripke 语义的概括,适用于直觉 Łukasiewicz 逻辑 IŁL——IL 和(经典)Łukasiewicz 逻辑之间的交叉逻辑。这种广义 Kripke 语义基于 Bova 和 Montagna (Theoret Comput Sci 410(12):1143–1158, 2009) 中使用的偏序和构造。显示可交换、积分和有界 GBL 代数的拟方程理论的可判定性(和 PSPACE 完备性)。主要思想是 $$w \Vdash \psi $$ w ⊩ ψ ——对于 IL 来说,它是世界 w 和公式 $$\psi $$ ψ 之间的关系,可以看作是一个函数,在布尔值 $ $(w \Vdash \psi ) \in {{\mathbb {B}}}$$ ( w ⊩ ψ ) ∈ B — 成为取单位区间 $$(w \Vdash \psi ) \in [ 0,1]$$ ( w ⊩ ψ ) ∈ [ 0 , 1 ] 。
更新日期:2020-04-21
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