Algebraic axiomatization of tense intuitionistic logic

Ivan Chajda

Open Mathematics (2011)

  • Volume: 9, Issue: 5, page 1185-1191
  • ISSN: 2391-5455

Abstract

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We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.

How to cite

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Ivan Chajda. "Algebraic axiomatization of tense intuitionistic logic." Open Mathematics 9.5 (2011): 1185-1191. <http://eudml.org/doc/269762>.

@article{IvanChajda2011,
abstract = {We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.},
author = {Ivan Chajda},
journal = {Open Mathematics},
keywords = {Brouwerian lattice; Heyting algebra; Complete lattice; Relative pseudocomplementation; Tense operators; Intuitionistic logic; tense intuitionistic logic; algebraic semantics; tense operator; complete lattice; relative pseudocomplementation},
language = {eng},
number = {5},
pages = {1185-1191},
title = {Algebraic axiomatization of tense intuitionistic logic},
url = {http://eudml.org/doc/269762},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Ivan Chajda
TI - Algebraic axiomatization of tense intuitionistic logic
JO - Open Mathematics
PY - 2011
VL - 9
IS - 5
SP - 1185
EP - 1191
AB - We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.
LA - eng
KW - Brouwerian lattice; Heyting algebra; Complete lattice; Relative pseudocomplementation; Tense operators; Intuitionistic logic; tense intuitionistic logic; algebraic semantics; tense operator; complete lattice; relative pseudocomplementation
UR - http://eudml.org/doc/269762
ER -

References

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  3. [3] Brouwer L.E.J., Intuitionism and formalism, Bull. Amer. Math. Soc., 1913, 20(2), 81–96 http://dx.doi.org/10.1090/S0002-9904-1913-02440-6 
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  5. [5] Chajda I., Halaš R., Kühr J., Semilattice Structures, Res. Exp. Math., 30, Heldermann, Lemgo, 2007 Zbl1117.06001
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  8. [8] Chiriţă C., Tense ϑ-valued Łukasiewicz-Moisil algebras, J. Mult.-Valued Logic Soft Comput., 2011, 17(1), 1–24 Zbl1236.03046
  9. [9] Diaconescu D., Georgescu G., Tense operators on MV-algebras and Łukasiewicz-Moisil algebras, Fund. Inform., 2007, 81(4), 379–408 Zbl1136.03045
  10. [10] Ewald W.B., Intuitionistic tense and modal logic, J. Symbolic Logic, 1986, 51(1), 166–179 http://dx.doi.org/10.2307/2273953 Zbl0618.03004
  11. [11] Heyting A., Intuitionism. An Introduction, North-Holland, Amsterdam, 1956 
  12. [12] Rasiowa H., Sikorski R., The Mathematics of Metamathematics, Monogr. Mat., 41, PWN, Warszawa, 1963 Zbl0122.24311
  13. [13] Turunen E., Mathematics Behind Fuzzy Logic, Adv. Soft Comput., Physica-Verlag, Heidelberg, 1999 Zbl0940.03029
  14. [14] Wijesekera D., Constructive modal logics. I, Ann. Pure Appl. Logic, 1990, 50(3), 271–301 http://dx.doi.org/10.1016/0168-0072(90)90059-B 

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