An Approach to QST-based Nmatrices Semantics
DOI:
https://doi.org/10.5007/1808-1711.2023.e91732Keywords:
Nmatrices, Quasets, Rough Sets, Quantum logic, ZFAAbstract
This paper introduces the theory QST of quasets as a formal basis for the Nmatrices. The main aim is to construct a system of Nmatrices by substituting standard sets by quasets. Since QST is a conservative extension of ZFA (the Zermelo-Fraenkel set theory with Atoms), it is possible to obtain generalized Nmatrices (Q-Nmatrices). Since the original formulation of QST is not completely adequate for the developments we advance here, some possible amendments to the theory are also considered. One of the most interesting traits of such an extension is the existence of complementary quasets which admit elements with undetermined membership. Such elements can be interpreted as quantum systems in superposed states. We also present a relationship of QST with the theory of Rough Sets RST, which grants the existence of models for SQT formed by rough sets. Some consequences of the given formalism for the relation of logical consequence are also analysed.
References
Avron, A. 2007. Non-deterministic semantics for families of paraconsistent logics. School of Computer Science. Tel-Aviv University.
Avron, A.; Lev, I. 2005. Non-deterministic Multiple-valued Structures. Journal of Logic and Computation 15(3): 241–61.
Avron, A.; Konikowska, B. 2004. Proof systems for logics based on non-deterministic multiple-valued structures. Prace Instytutu Podstaw Informatyki Polskiej Akademii Nauk: 1–26.
Avron, A.; Konikowska, B. 2008. Rough sets and 3-valued logics. Studia Logica 90: 69–92.
Avron, A.; Lev, I. 2005. Canonical propositional Gentzen-type systems. Proceedings of the 1st International Joint Conference on Automated Reasoning 13: 365–87.
Avron, A.; Zamansky, A. 2005. Non-deterministic semantics for logical systems. Handbook of Philosophical Logic 16: 227–304.
Barrio, E. A.; Pailos, F. 2021. Why a logic is not only its set of valid inferences. Análisis Filosófico 41(2): 261–72.
Chemla, E.; Egré, P.; Spector, B. 2017. Characterizing logical consequence in many-valued logic. Journal of Logic and Computation 27(7): 2193–226.
Cobreros, P.; Egré, P.; Ripley, D.; van Rooij, R. 2012. Tolerant, classical, strict. Journal of Philosophical Logic 41(2): 347–85.
Dalla Chiara, M. L.; Giuntini, R.; Krause, D. 1998. Quasiset theories for microobjects: a comparison. In: E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, pp.142–52. Princeton: Princeton University Press.
Dalla Chiara, M. L.; Toraldo di Francia, G. 1993. Individuals, kinds and names in physics. In: G. Corsi; M. L. Dalla Chiara; G. C. Ghirardi (eds.) Bridging the Gap: Philosophy, Mathematics, and Physics, pp.261–84. Kluwer Ac. Pu.
de Barros, J. A.; Holik, F.; Krause, D. 2022. Distinguishing indistinguishabilities: Differences Between Classical and Quantum Regimes. Springer, forthcoming.
do Nascimento, M. C.; Krause, D.; de Aráujo Feitosa, H. 2011. The quasilattice of indiscernible elements. Studia Logica 97(1): 101–26.
Gleason, A. M. 1957. Measures on the Closed Subspaces of a Hilbert Space. Journal of Mathematics and Mechanics 6(6): 885–93.
Holik, F.; Jorge, J.; Krause, D.; Lombardi, O. 2022. Quasi-set theory: a formal approach to a quantum ontology of properties. Synthese: 401.
Holik, F.; Jorge, J. P.; Massri, C. 2020. Indistinguishability right from the start in standard quantum mechanics. Arxiv arxiv:2011.10903v1.
Jorge, J. P.; Holik, F. 2020. Non-deterministic semantics for quantum states. Entropy 22(2): 156.
Jorge, J. P.; Holik, F. 2022. Lógica cuántica, Nmatrices y adecuación, I. Teorema 41(3): 65–88.
Jorge, J. P.; Holik, F. 2023. Lógica cuántica, Nmatrices y adecuación, II. Teorema 42(1): 149–169.
Krause, D. 1992. On a quasi-set theory. Notre Dame Journal of Formal Logic 33(3): 402–11.
Krause, D. 2012. On a calculus of non-individuals: ideas for a quantum mereology. In: L. H. d. A. Dutra; A. M. Luz (eds.) Linguagem, Ontologia e Ação, volume 10 of Coleção Rumos da Epistemologia, pp.92–106. NEL/UFSC.
Krause, D. 2017. Quantum mereology. In: H. Burkhard; J. Seibt; G. Imaguire; S. Georgiogakis (eds.) Handbook of Mereology, pp.469–72. München: Springer-Verlag.
Krause, D. 2020. Quantum mechanics, ontology, and non-reflexive logics. In: J. A. de Barros; D. Krause (eds.) A True Polimath: A Tribute to Francisco Antonio Doria, pp.75–130. London: College Publications.
Mahajan, P.; Kandwal, R.; Vijay, R. 2012. Article: Rough set approach in machine learning: A review. International Journal of Computer Applications 56(10): 1–13.
Manero Orozco, J. A. 2018. Sobre la estructura que subyace a la mecánica cuántica Bohmiana. PhD thesis. Universidad Nacional autônoma de México, Instituto de Investigaciones Filosóficas.
Manin, Y. I. 1976. Foundations. In: F. Browder, F.; A. M. Society (eds.) Mathematical Developments Arising from Hilbert Problems, p.36. American Mathematical Society.
Orlowska, E. 1982. Logics of vague concepts. Bulletin of the Section of Logic: 115–26.
Orlowska, E. 1985. Semantics of Vague Concepts. Boston: Springer US, p.465–482.
Pailos, F. M. 2020. A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic 13(2): 249–68.
Pawlak, Z. 1982. Rough sets. International Journal of Computer & Information Sciences 11(1): 341–56.
Pawlak, Z. 1986. On rough relations. Bulletin of the Polish Academy of Sciences 34(9): 587–90.
Pawlak, Z. 1987. On rough functions. Bulletin of the Polish Academy of Sciences 35(5–6): 250–2.
Pawlak, Z.; Skowron, A. 2007a. Rough sets and boolean reasoning. Information Sciences 177(1): 41–73.
Pawlak, Z.; Skowron, A. 2007b. Rough sets: Some extensions. Information Sciences 177(1): 28–40.
Pawlak, Z.; Skowron, A. 2007c. Rudiments of rough sets. Inf. Sci. 177: 3–27.
Pedrycz, W.; Han, L.; Peters, J.; Ramanna, S.; Zhai, R. 2001. Calibration of software quality: Fuzzy neural and rough neural computing approaches. Neurocomputing 36(1): 149–70.
Peters, J. F.; Skowron, A.; Synak, P.; Ramanna, S. 2003. Rough sets and information granulation. In: T. Bilgiç; B. De Baets; O. Kaynak (eds.) Fuzzy Sets and Systems | IFSA 2003, pp.370–7. Berlin, Heidelberg: Springer.
Ripley, D. 2012. Conservatively extending classical logic with transparent truth. The Review of Symbolic Logic 5(2): 354–78.
Skowron, A. 2005. Rough sets in perception-based computing. In: S. K. Pal; S. Bandyopadhyay; S. Biswas (eds.) Pattern Recognition and Machine Intelligence, pp.21–9. Berlin, Heidelberg: Springer.
Solé Bellet, A. 2010. Realismo e interpretación en mecânica bohmiana. Madrid: Universidad Complutense de Madrid.
Suraj, Z. 2004. An introduction to rough set theory and its applications a tutorial. Cairo: ICENCO.
Takeuti, G. 1981. Quantum set theory. In: E. G. Beltrametti; B. C. van Fraassen (eds.) Current Issues in Quantum Logic, pp.303–22. New York and London: Plenum Press.
Weber, Z.; Badia, G.; Girard, P. 2016. What is an inconsistent truth table? Australasian Journal of Philosophy 94(3): 533–48.
Weidner, A. J. 1981. Fuzzy sets and boolean-valued universes. Fuzzy Sets and Systems 6: 61–72.
Wintein, S. 2014. On the strict/tolerant conception of truth. Australasian Journal of Philosophy 92(1): 71–90.
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