Conservative translations of four-valued logics in modal logic

Kubyshkina, Ekaterina

doi:10.1007/s11229-019-02139-3

Cited by 5 publications

(5 citation statements)

References 8 publications

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“…That is, the observable-dependent logic fills a gap between logic and quantum mechanics. In addition, from a logical viewpoint, we can utilize some notions of quantum mechanics for proving logical property 10 if the observable-dependent logic is employed.…”

Section: Resultsmentioning

confidence: 99%

“…As suggested in [9] for the non-quantum threevalued logic, the result can extend to n-valued logics. In particular, a translation from the four-valued logic to a kind of modal logic was studied in detail by [10].…”

Section: The Definition Of 3-equivalence For Quantum Logicmentioning

confidence: 99%

“…Notice that P(α(|ψ ) | O A ) and P(O A ) are in the closed interval [0, 1] by the definition of probability 10. For example, Lemma 1 (and as a result Theorem 5) is proved by utilizing the completeness relation in quantum mechanics.…”

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confidence: 99%

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Translation from Three-Valued Quantum Logic to Modal Logic

Takagi

2021

Int J Theor Phys

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We translate the three-valued quantum logic into modal logic, and prove 3-equivalence between the valuation of the three-valued logic and a kind of Kripke model in regard to this translation. To prove 3-equivalence, we introduce an observable-dependent logic, which is a fragment of the many-valued quantum logic. Compared to the Birkhoff and von Neumann’s quantum logic, some notions about observables, the completeness relation for example, in quantum mechanics can be utilized if the observable-dependent logic is employed.

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Section: Resultsmentioning

confidence: 99%

Section: The Definition Of 3-equivalence For Quantum Logicmentioning

confidence: 99%

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Translation from Three-Valued Quantum Logic to Modal Logic

Takagi

2021

Int J Theor Phys

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“…Let us mention Kubyshkina's recent paper [17], where fifteen-valued logic GLRA, a generalization of LRA, is presented. The entailment relation is defined via the preservation of the only designated value.…”

Section: Discussionmentioning

confidence: 99%

The Logic of Internal Rational Agent

Petrukhin

2021

AJL

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In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev's Logic of Rational Agent \textbf{LRA} \cite{LRA}. We call our logic $ \bf LIRA$ (Logic of Internal Rational Agency). In contrast to \textbf{LRA}, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko's bi-facial truth logic \cite{ZS}. This logic may be useful in a situation when according to an agent's point of view (i.e. internal point of view) her/his reasoning is rational, while from the external one it might be not the case. One may use \textbf{LIRA}, if one wants to reconstruct an agent's way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev's necessity and possibility operators for \textbf{LRA} definable by means of four-valued Kripke-style semantics and show that, due to two negations (as well as their combination) of \textbf{LRA}, two more possibility operators for \textbf{LRA} can be defined. Then we slightly modify all these modalities to be appropriate for $\bf LIRA$. Finally, we formalize all the truth-functional $ n $-ary extensions of the negation fragment of $\bf LIRA$ (including $\bf LIRA$ itself) as well as their basic modal extension via linear-type natural deduction systems.

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“…Third, it could be fruitful to study connection of non-distributive ETL and NFL logics with modal logics as it has been recently done in [22] for first degree entailment.…”

Section: Discussionmentioning

confidence: 99%

Non-distributive Relatives of ETL and NFL

Kozhemiachenko

2020

Stud Logica

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In this paper, we devise non-distributive relatives of Exactly True Logic (ETL) by Pietz and Riveccio and its dual (NFL) Non-Falsity Logic by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL's and NFL's entailment relations on the De Morgan lattice 4. We generalise these pre-orders and determine which distributive properties that hold on 4 are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics, we also devise a truth table semantics which uses non-distributive lattice M3 as their lattice of truth values. We also provide analytic tableaux systems which work with sequents of the form φ ⊢ χ. We also prove correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL-and NFL-like logics built over a certain family of non-distributive modular lattices.

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Conservative translations of four-valued logics in modal logic

Cited by 5 publications

References 8 publications

Translation from Three-Valued Quantum Logic to Modal Logic

Translation from Three-Valued Quantum Logic to Modal Logic

The Logic of Internal Rational Agent

Non-distributive Relatives of ETL and NFL

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