Intuitionistic-like unsharp implication and negation defined on a poset
Ivan Chajda, Helmut Länger
Received December 27, 2023. Published online December 17, 2024.
Abstract: The aim of the present paper is to show that the concepts of the intuitionistic implication and negation formalized by means of a Heyting algebra can be generalized in such a way that these concepts are formalized by means of a bounded poset. In this case it is not assumed that the poset is relatively pseudocomplemented. The considered logical connectives negation, implication or even conjunction are not operations in this poset but so-called operators since they assign to given entries not necessarily an element of the poset as a result but a subset of mutually incomparable elements. We show that these operators for negation and implication can be characterized by several simple conditions formulated in the language of posets together with the operator of taking the lower cone. Moreover, our implication and conjunction form an adjoint pair. We call these connectives "unsharp" or "inexact" in accordance with the existing literature. We also introduce the concept of a deductive system of a bounded poset with implication and prove that it induces an equivalence relation satisfying a certain substitution property with respect to implication. Moreover, the restriction of this equivalence to the base set is uniquely determined by its kernel, i.e., the class containing the top element.
Keywords: bounded poset; logical connectives defined on a poset; unsharp negation; unsharp implication; adjoint operator; Modus Ponens; deductive system; equivalence relation induced by a deductive system
References: [1] L. E. J. Brouwer: De onbetrouwbaarheid der logische principes. Tijdschrift Wijsbegeerte 2 (1908), 152-158. (In Dutch.)
[2] L. E. J. Brouwer: Intuitionism and formalism. Bull. Am. Math. Soc. 20 (1913), 81-96. DOI 10.1090/S0002-9904-1913-02440-6 | MR 1559427 | JFM 44.0085.06
[3] I. Chajda: An extension of relative pseudocomplementation to non-distributive lattices. Acta Sci. Math. 69 (2003), 491-496. MR 2034188 | Zbl 1048.06005
[4] I. Chajda: Pseudocomplemented and Stone posets. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 51 (2012), 29-34. MR 3060006 | Zbl 1302.06001
[5] I. Chajda, G. Eigenthaler, H. Länger: Congruence Classes in Universal Algebra. Research and Exposition in Mathematics 26. Heldermann, Lemgo (2012). MR 1985832 | Zbl 1014.08001
[6] I. Chajda, H. Länger: Implication in finite posets with pseudocomplemented sections. Soft Comput. 26 (2022), 5945-5953. DOI 10.1007/s00500-022-07052-5
[7] I. Chajda, H. Länger: The logic of orthomodular posets of finite height. Log. J. IGPL 30 (2022), 143-154. DOI 10.1093/jigpal/jzaa067 | MR 4367693 | Zbl 1507.03105
[8] I. Chajda, H. Länger: Operator residuation in orthomodular posets of finite height. Fuzzy Sets Syst. 467 (2023), Article ID 108589, 11 pages. DOI 10.1016/j.fss.2023.108589 | MR 4598488 | Zbl 1543.06001
[9] I. Chajda, H. Länger: Algebraic structures formalizing the logic with unsharp implication and negation. To appear in Log. J. IGPL. (2023), 13 pages. DOI 10.1093/jigpal/jzad023
[10] I. Chajda, H. Länger, J. Paseka: Sectionally pseudocomplemented posets. Order 38 (2021), 527-546. DOI 10.1007/s11083-021-09555-6 | MR 4325977 | Zbl 1490.06001
[11] P. D. Finch: On orthomodular posets. J. Aust. Math. Soc. 11 (1970), 57-62. DOI 10.1017/S1446788700005978 | MR 0255449 | Zbl 0211.02701
[12] O. Frink: Pseudo-complements in semi-lattices. Duke Math. J. 29 (1962), 505-514. DOI 10.1215/S0012-7094-62-02951-4 | MR 0140449 | Zbl 0114.01602
[13] R. Giuntini, H. Greuling: Toward a formal language for unsharp properties. Found. Phys. 19 (1989), 931-945. DOI 10.1007/BF01889307 | MR 1013913
[14] A. Heyting: Die formalen Regeln der intuitionistischen Logik. I. Sitzungsberichte Akad. Berlin 1930 (1930), 42-56. (In German.) JFM 56.0823.01
[15] P. Köhler: Brouwerian semilattices: the lattice of total subalgebras. Banach Center Publ. 9 (1982), 47-56. DOI 10.4064/-9-1-47-56 | MR 0738800 | Zbl 0514.06009
[16] A. Monteiro: Axiomes indépendants pour les algèbres de Brouwer. Rev. Un. Mat. Argentina 17 (1956), 149-160. (In French.) MR 0084483 | Zbl 0072.25004
[17] L. Monteiro: Les algèbres de Heyting et de Lukasiewicz trivalentes. Notre Dame J. Formal Logic 11 (1970), 453-466. (In French.) DOI 10.1305/ndjfl/1093894076 | MR 0286633 | Zbl 0177.01001
[18] P. Pták, S. Pulmannová: Orthomodular Structures as Quantum Logics. Fundamental Theories of Physics 44. Kluwer, Dordrecht (1991). MR 1176314 | Zbl 0743.03039
Affiliations: Ivan Chajda, Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: ivan.chajda@upol.cz; Helmut Länger (corresponding author), TU Wien, Fakultät für Mathematik und Geoinformation, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria, and Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: helmut.laenger@tuwien.ac.at
