Hidden modalities in algebras with negation and implication
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2013
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Vysoké učení technické v Brně, Fakulta strojního inženýrství, Ústav matematiky
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Abstract
Lukasiewicz 3-valued logic may be seen as a logic with hidden truthfunctional
modalities de ned 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 de ned 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 Lukasiewicz-
Moisil algebras Ln, identities corresponding to (K), (T), and (D) hold for all n
3 and 1 = 1. We de ne 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
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Mathematics for Applications. 2013, 2, č. 1, s. 5-20. ISSN 1805-3629.
http://ma.fme.vutbr.cz/archiv/2_1/jarvinen_kondo_mattila_radeleczki_final.pdf
http://ma.fme.vutbr.cz/archiv/2_1/jarvinen_kondo_mattila_radeleczki_final.pdf
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en
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© Vysoké učení technické v Brně, Fakulta strojního inženýrství, Ústav matematiky