• Korean Journal of Logic
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• v.8 no.2
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• pp.85-107
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• 2005

Kleene first investigated a three-valued system which follows the evaluations of the Lukasiewicz infinite-valued logic ${\L}C$ with respect to negation, conjunction, and disjunction, and treats $\rightarrow$ as material-like implication in the sense that A $\rightarrow$ B is defined as ${\sim}A{\vee}B$ in its evaluation. Diense and Rescher extended it to many-valued logic and infinite-valued logic, respectively. This paper investigates a variant of the infinite-valued Kleene-Diense logic KD, which we shall call strong Kleene-Diense logic (sKD): sKD has the same evaluations as KD except that sKD takes a variant of Kleene-Diense implication. Following the idea of Dunn [2], we provide algebraic completeness for sKD together with its deduction theorem.

• Korean Journal of Logic
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• v.9 no.2
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• pp.1-30
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• 2006

This paper investigates algebraic semantics for some weak Boolean (wB) logics, which may be regarded as left-continuous t-norm based logics (or monoidal t-norm based logics (MTLs)). We investigate as infinite-valued logics each of wB-LC and wB-sKD, and each corresponding first order extension $wB-LC\forall$ and $wB-sKD\forall$. We give algebraic completeness for each of them.

• Journal for History of Mathematics
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• v.13 no.1
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• pp.125-136
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• 2000

The notion of MV-algebra was introduced by C.C. Chang in 1958 to provide an algebraic proof of the completeness of Lukasiewicz axioms for infinite valued logic. These algebras appear in the literature under different names: Bricks, Wajsberg algebra, CN-algebra, bounded commutative BCK-algebras, etc. The purpose of this paper is to give a topological lattice completion of semisimple MV-algebras. To this end, we characterize the complete atomic center MV-algebras and semisimple algebras as subalgebras of a cube. Then we define the $\delta$-completion of semisimple MV-algebra and construct the $\delta$-completion. We also study some important properties and extension properties of $\delta$-completion.

• Korean Journal of Logic
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• v.9 no.1
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• pp.1-29
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• 2006

This paper investigates algebraic semantics for sKD and its extensions $sKD_\triangle$, $sKD\forall$, and $sKD\forall{_\triangle}$: sKD is a variant of the infinite -valued Kleene- Diense logic KD; $sKD_\triangle$ is the sKD with the Baaz's projection A; and $sKD\forall$ and $sKD\forall{_\triangle}$: are the first order extensions of sKD and $sKD_\triangle$, respectively. I first provide algebraic completeness for each of sKD and $sKD_\triangle$. Next I show that each $sKD\forall$ and $sKD\forall{_\triangle}$: is algebraically complete.