Natural Deduction System for the Logic of Binary Relations Based on the Algebraic Tradition
Resumo
We present a proper natural deduction system (ND-system) for a logic of binary relations based on the algebraic tradition. Our system is an evolution from [W. W. Wadge. TCR 5, The University of Warwick, 1975]. We point out some aspects where Wadge's formalism fails as an ND-system and fix them all.
Palavras-chave:
Calculus of Binary Relations Relational algebra, Natural deduction
Referências
Andréka, H. (1997). Complexity of equations valid in algebras of relations, part i: Strong non-finitizability. Ann. Pure. Appl. Logic, 89:49–209.
De morgan, A. (1860). On the syllogism, No. IV., and on the logic of relations. Transactions of the Cambridge Philosophical Society, 10:331--358
Indrzejczak, A. (2010). Natural Deduction, Hybrid Systems and Modal Logics. Springer.
Jónsson, B. (1991). The theory of binary relations. In H. Andréka, D, Monk, I. Németi, editors, Algebraic Logic. North-Holland.
Jónsson, B. and Tarski, A. (1948). Representation problems for relation algebras. Bulletin of the Americam Mathematical Society, 54:80.
Lyndon, R. C. (1950). The representaion of relational algebras. Annals of Mathematics, 51:707–729.
Lyndon, R. C. (1956). The representation of relational algebras, ii. Annals of Mathematics, 63:294–307.
Maddux, R. D. (1983). A sequent calculus for relation algebras. Ann. Pure Appl. Logic, 25:73–101.
Monk, J. D. (1964). On representable relation algebras. Michigan Mathematical Journal, 11:207–210.
Orlowska, E. (1991). Relational interpretation of modal logics. In H. Andréka, D, Monk, I. Németi, editors, Algebraic Logic. North-Holland.
Peirce, C. S. (1883) Note B: the logic of relatives. Studires in Logic by Members or the John Hopkins University, pages 187–203.
Schröder, F. W. K. E. (1895). Vorlesungen über die algebra der logik, volume 3, “Algebra und logik de relative”, part I. B. G. Teubner, Leipzig.
Suguitan, L. (2013). Sobre a lógica e a aritmética das relações. PhD thesis, UNICAMP.
Sundholm, B. G. (1991). Systems of deduction. In D. Gabbay, F. G., editors, Handbook of Philosophical Logic, Vol.I: Elements of Classical Logic, pages 133–188. D. Reidel.
Tarski, A. (1941). On the calculus of relations. Journal of Symbolic Logic, 6:73–89.
Tarski, A. (1955). Contributions to the theory of models iii. Koninkl. Nederl. Akad. Wetensch, pages 56–64.
Wadge. W. W. (1975). A complete natural deduction system for the relational calculus. Technical report, Department of Computer Science, University of Warwick, Coventry, UK.
De morgan, A. (1860). On the syllogism, No. IV., and on the logic of relations. Transactions of the Cambridge Philosophical Society, 10:331--358
Indrzejczak, A. (2010). Natural Deduction, Hybrid Systems and Modal Logics. Springer.
Jónsson, B. (1991). The theory of binary relations. In H. Andréka, D, Monk, I. Németi, editors, Algebraic Logic. North-Holland.
Jónsson, B. and Tarski, A. (1948). Representation problems for relation algebras. Bulletin of the Americam Mathematical Society, 54:80.
Lyndon, R. C. (1950). The representaion of relational algebras. Annals of Mathematics, 51:707–729.
Lyndon, R. C. (1956). The representation of relational algebras, ii. Annals of Mathematics, 63:294–307.
Maddux, R. D. (1983). A sequent calculus for relation algebras. Ann. Pure Appl. Logic, 25:73–101.
Monk, J. D. (1964). On representable relation algebras. Michigan Mathematical Journal, 11:207–210.
Orlowska, E. (1991). Relational interpretation of modal logics. In H. Andréka, D, Monk, I. Németi, editors, Algebraic Logic. North-Holland.
Peirce, C. S. (1883) Note B: the logic of relatives. Studires in Logic by Members or the John Hopkins University, pages 187–203.
Schröder, F. W. K. E. (1895). Vorlesungen über die algebra der logik, volume 3, “Algebra und logik de relative”, part I. B. G. Teubner, Leipzig.
Suguitan, L. (2013). Sobre a lógica e a aritmética das relações. PhD thesis, UNICAMP.
Sundholm, B. G. (1991). Systems of deduction. In D. Gabbay, F. G., editors, Handbook of Philosophical Logic, Vol.I: Elements of Classical Logic, pages 133–188. D. Reidel.
Tarski, A. (1941). On the calculus of relations. Journal of Symbolic Logic, 6:73–89.
Tarski, A. (1955). Contributions to the theory of models iii. Koninkl. Nederl. Akad. Wetensch, pages 56–64.
Wadge. W. W. (1975). A complete natural deduction system for the relational calculus. Technical report, Department of Computer Science, University of Warwick, Coventry, UK.
Publicado
26/08/2020
Como Citar
CERIOLI, Márcia Rosana; SUGUITANI, Leandro Oliva; VIANA, Jorge Petrúcio.
Natural Deduction System for the Logic of Binary Relations Based on the Algebraic Tradition. In: WORKSHOP BRASILEIRO DE LÓGICA (WBL), 1. , 2020, Evento Online.
Anais [...].
Porto Alegre: Sociedade Brasileira de Computação,
2020
.
p. 49-56.
ISSN 2763-8731.
DOI: https://doi.org/10.5753/wbl.2020.11458.
