Equality Logic

Shokoofeh Ghorbani

Bulletin of the Section of Logic (2020)

  • Volume: 49, Issue: 3, page 291-324
  • ISSN: 0138-0680

Abstract

top
In this paper, we introduce and study a corresponding logic to equality-algebras and obtain some basic properties of this logic. We prove the soundness and completeness of this logic based on equality-algebras and local deduction theorem. We show that this logic is regularly algebraizable with respect to the variety of equality∆-algebras but it is not Fregean. Then we introduce the concept of (prelinear) equality∆-algebras and investigate some related properties. Also, we study ∆-deductive systems of equality∆-algebras. In particular, we prove that every prelinear equality ∆-algebra is a subdirect product of linearly ordered equality∆-algebras. Finally, we construct prelinear equality ∆ logic and prove the soundness and strong completeness of this logic respect to prelinear equality∆-algebras.

How to cite

top

Shokoofeh Ghorbani. "Equality Logic." Bulletin of the Section of Logic 49.3 (2020): 291-324. <http://eudml.org/doc/296797>.

@article{ShokoofehGhorbani2020,
abstract = {In this paper, we introduce and study a corresponding logic to equality-algebras and obtain some basic properties of this logic. We prove the soundness and completeness of this logic based on equality-algebras and local deduction theorem. We show that this logic is regularly algebraizable with respect to the variety of equality∆-algebras but it is not Fregean. Then we introduce the concept of (prelinear) equality∆-algebras and investigate some related properties. Also, we study ∆-deductive systems of equality∆-algebras. In particular, we prove that every prelinear equality ∆-algebra is a subdirect product of linearly ordered equality∆-algebras. Finally, we construct prelinear equality ∆ logic and prove the soundness and strong completeness of this logic respect to prelinear equality∆-algebras.},
author = {Shokoofeh Ghorbani},
journal = {Bulletin of the Section of Logic},
keywords = {many-valued logic; equality logic; completness; prelinear equality∆-algebra; prelinear equality∆ logic},
language = {eng},
number = {3},
pages = {291-324},
title = {Equality Logic},
url = {http://eudml.org/doc/296797},
volume = {49},
year = {2020},
}

TY - JOUR
AU - Shokoofeh Ghorbani
TI - Equality Logic
JO - Bulletin of the Section of Logic
PY - 2020
VL - 49
IS - 3
SP - 291
EP - 324
AB - In this paper, we introduce and study a corresponding logic to equality-algebras and obtain some basic properties of this logic. We prove the soundness and completeness of this logic based on equality-algebras and local deduction theorem. We show that this logic is regularly algebraizable with respect to the variety of equality∆-algebras but it is not Fregean. Then we introduce the concept of (prelinear) equality∆-algebras and investigate some related properties. Also, we study ∆-deductive systems of equality∆-algebras. In particular, we prove that every prelinear equality ∆-algebra is a subdirect product of linearly ordered equality∆-algebras. Finally, we construct prelinear equality ∆ logic and prove the soundness and strong completeness of this logic respect to prelinear equality∆-algebras.
LA - eng
KW - many-valued logic; equality logic; completness; prelinear equality∆-algebra; prelinear equality∆ logic
UR - http://eudml.org/doc/296797
ER -

References

top
  1. [1] W. J. Blok, D. Pigozzi, Algebraizable logics, vol. 77, American Mathematical Society (1989), DOI: http://dx.doi.org/10.1090/memo/0396 
  2. [2] W. J. Blok, D. Pigozzi, Abstract algebraic logic and the deduction theorem (2001), URL: https://orion.math.iastate.edu/dpigozzi/papers/aaldedth.pdf 
  3. [3] R. Borzooei, F. Zebardast, M. Aaly Kologani, Some types of filters in equality algebras, Categories and General Algebraic Structures with Applications, vol. 7 (Special Issue on the Occasion of Banaschewski's 90th Birthday (II)) (2017), pp. 33–55, DOI: http://dx.doi.org/10.1007/s00500-005-0534-4 
  4. [4] R. A. Borzooei, M. Zarean, O. Zahiri, Involutive equality algebras, Soft Computing, vol. 22(22) (2018), pp. 7505–7517, DOI: http://dx.doi.org/10.1007/s00500-018-3032-1 
  5. [5] J. R. Büchi, T. M. Owens, Skolem rings and their varieties, [in:] The Collected Works of J. Richard Büchi, Springer (1990), pp. 161–221, DOI: http://dx.doi.org/10.1007/978-1-4613-8928-6-11 
  6. [6] L. C. Ciungu, Internal states on equality algebras, Soft computing, vol. 19(4) (2015), pp. 939–953, DOI: http://dx.doi.org/10.1007/s00500-014-1494-3 
  7. [7] J. Czelakowski, Protoalgebraic logics, [in:] Protoalgebraic Logics, Springer (2001), pp. 69–122, DOI: http://dx.doi.org/10.1007/978-94-017-2807-2-3 
  8. [8] M. Dyba, M. El-Zekey, V. Novák, Non-commutative first-order EQ-logics, Fuzzy Sets and Systems, vol. 292 (2016), pp. 215–241, DOI: http://dx.doi.org/10.1016/j.fss.2014.11.019 
  9. [9] M. Dyba, V. Novák, EQ-logics: Non-commutative fuzzy logics based on fuzzy equality, Fuzzy Sets and Systems, vol. 172(1) (2011), pp. 13–32, DOI: http://dx.doi.org/10.1016/j.fss.2010.11.011 
  10. [10] M. El-Zekey, Representable good EQ-algebras, Soft Computing, vol. 14(9) (2010), pp. 1011–1023, DOI: http://dx.doi.org/10.1007/s00500-009-0491-4 
  11. [11] M. El-Zekey, V. Novák, R. Mesiar, On good EQ-algebras, Fuzzy Sets and Systems, vol. 178(1) (2011), pp. 1–23, DOI: http://dx.doi.org/10.1016/j.fss.2011.05.011 
  12. [12] S. Ghorbani, Monadic pseudo-equality algebras, Soft Computing, vol. 23(24) (2019), pp. 12937–12950, DOI: http://dx.doi.org/10.1007/s00500-019-04243-5 
  13. [13] S. Jenei, Equality algebras, Studia Logica, vol. 100(6) (2012), pp. 1201–1209, DOI: http://dx.doi.org/10.1007/s11225-012-9457-0 
  14. [14] S. Jenei, L. Kóródi, On the variety of equality algebras, [in:] Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology, Atlantis Press (2011), pp. 153–155, DOI: http://dx.doi.org/10.2991/eusat.2011.1 
  15. [15] J. Kühr, Pseudo BCK-semilattices, Demonstratio Mathematica, vol. 40(3) (2007), pp. 495–516, DOI: http://dx.doi.org/10.1515/dema-2007-0302 
  16. [16] V. Novák, On fuzzy type theory, Fuzzy Sets and Systems, vol. 149(2) (2005), pp. 235–273, DOI: http://dx.doi.org/10.1016/j.fss.2004.03.027 
  17. [17] V. Novák, EQ-algebras: primary concepts and properties, [in:] Proceedings of International Joint Czech Republic-Japan & Taiwan-Japan Symposium, Kitakyushu, Japan, August 2006 (2006), pp. 219–223. 
  18. [18] V. Novák, EQ-algebra-based fuzzy type theory and its extensions, Logic J2ournal of the IGPL, vol. 19(3) (2011), pp. 512–542, DOI: http://dx.doi.org/10.1093/jigpal/jzp087 
  19. [19] V. Novák, B. De Baets, EQ-algebras, Fuzzy Sets and Systems, vol.160(20) (2009), pp. 2956–2978, DOI: http://dx.doi.org/10.1016/j.fss.2009.04.010 
  20. [20] R. Suszko, Non-Fregean logic and theories, Analele Universitatii Bucuresti, Acta Logica, vol. 11 (1968), pp. 105–125. 
  21. [21] J. T. Wang, X. L. Xin, Y. B. Jun, Very true operators on equality algebras, Journal of Computational Analysis and Applications, vol. 24(3) (2018), DOI: http://dx.doi.org/10.1515/math-2016-0086 
  22. [22] M. Zarean, R. A. Borzooei, O. Zahiri, On state equality algebras, Quasi-groups and Related Systems, vol. 25(2) (2017), pp. 307–326. 
  23. [23] F. Zebardast, R. A. Borzooei, M. A. Kologani, Results on equality algebras, Information Sciences, vol. 381 (2017), pp. 270–282, DOI: http://dx.doi.org/10.1016/j.ins.2016.11.027 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.