Many-Valuation, Modality, and Fuzziness

Mattila, Jorma K.

doi:10.1007/978-3-540-93802-6_13

Cited by 3 publications

(2 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Notice that and ♦ are mutually dual, that is, A ≡ ¬♦¬A and ♦A ≡ ¬ ¬A for all formulas A, where A ≡ B means that the formulas A and B are semantically equivalent, that is, v(A) = v(B) for all valuations v. A complete analysis of 3-valued valuations and evaluation rules in L 3 is given, e.g., in Mattila [13]. Alfred Tarski, being at that time Lukasiewicz' assistant, found out that the formula ¬A → A (1.1) has exactly the same truth table as the formula ♦A.…”

Section: As a Modal Logicmentioning

confidence: 99%

Hidden modalities in algebras with negation and implication

Kondo¹,

Järvinen²,

Mattila³

et al. 2013

Math. Appl.

Self Cite

View full text Add to dashboard Cite

Abstract. Lukasiewicz 3-valued logic may be seen as a logic with hidden truthfunctional modalities defined by ♦A := ¬A → A and A := ¬(A → ¬A). It is known that axioms (K), (T), (B), (D), (S4), (S5) are provable for these modalities, and rule (RN) is admissible. We show that, if analogously defined modalities are adopted in Lukasiewicz 4-valued logic, then (K), (T), (D), (B) are provable, and (RN) is admissible. In addition, we show that in the canonical n-valued LukasiewiczMoisil algebras Ln, identities corresponding to (K), (T), and (D) hold for all n ≥ 3 and 1 = 1. We define analogous operations in residuated lattices and show that residuated lattices determine modal systems in which axioms (K) and (D) are provable and 1 = 1 holds. Involutive residuated lattices satisfy also the identity corresponding to (T). We also show that involutive residuated lattices do not satisfy identities corresponding to (S4) nor (S5). Finally, we show that in Heyting algebras, and thus in intuitionistic logic, ♦ and are equal, and they correspond to the double negation ¬¬.

show abstract

Section: As a Modal Logicmentioning

confidence: 99%

Hidden modalities in algebras with negation and implication

Kondo¹,

Järvinen²,

Mattila³

et al. 2013

Math. Appl.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Two of these logics are Kleene's logic and Łukasiewicz logic. Some analysis about Łukasiewicz and Kleene's logic is given for example in Mattila (Mattila, 2009). Especially, connections to modal logic are considered in that paper.…”

Section: Introductionmentioning

confidence: 99%