# Algebraic axiomatization of tense intuitionistic logic

Open Mathematics (2011)

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

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topIvan 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

top- [1] Birkhoff G., Lattice Theory, Amer. Math. Soc. Colloq. Publ., 3rd ed., 25, American Mathematical Society, Providence, 1967
- [2] Botur M., Chajda I., Halaš R., Kolařík M., Tense operators on basic algebras, Internat. J. Theoret. Phys. (in press), DOI: 10.1007/s10773-011-0748-4 Zbl1246.81014
- [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
- [4] Burgess J.P., Basic tense logic, In: Handbook of Philosophical Logic II, Synthese Lib., 165, Reidel, Dordrecht, 1984, 89–133
- [5] Chajda I., Halaš R., Kühr J., Semilattice Structures, Res. Exp. Math., 30, Heldermann, Lemgo, 2007 Zbl1117.06001
- [6] Chajda I., Kolařík M., Dynamic effect algebras, Math. Slovaca (in press) Zbl1324.03026
- [7] Chiriţă C., Tense ϑ-valued Moisil propositional logic, International Journal of Computers, Communications & Control, 2010, 5(5), 642–653
- [8] Chiriţă C., Tense ϑ-valued Łukasiewicz-Moisil algebras, J. Mult.-Valued Logic Soft Comput., 2011, 17(1), 1–24 Zbl1236.03046
- [9] Diaconescu D., Georgescu G., Tense operators on MV-algebras and Łukasiewicz-Moisil algebras, Fund. Inform., 2007, 81(4), 379–408 Zbl1136.03045
- [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] Heyting A., Intuitionism. An Introduction, North-Holland, Amsterdam, 1956
- [12] Rasiowa H., Sikorski R., The Mathematics of Metamathematics, Monogr. Mat., 41, PWN, Warszawa, 1963 Zbl0122.24311
- [13] Turunen E., Mathematics Behind Fuzzy Logic, Adv. Soft Comput., Physica-Verlag, Heidelberg, 1999 Zbl0940.03029
- [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