Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction
Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert’s style of proof and Gentzen’s deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equiva...
- Autores:
-
Bohórquez, Jaime
- Tipo de recurso:
- Article of journal
- Fecha de publicación:
- 2008
- Institución:
- Escuela Colombiana de Ingeniería Julio Garavito
- Repositorio:
- Repositorio Institucional ECI
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.escuelaing.edu.co:001/1910
- Acceso en línea:
- https://repositorio.escuelaing.edu.co/handle/001/1910
- Palabra clave:
- Estilo de cálculo
Deducción ecuacional
Lógica intuicionista
calculational style
equational deduction
Intuitionistic logic
- Rights
- openAccess
- License
- http://purl.org/coar/access_right/c_abf2
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Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction |
| title |
Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction |
| spellingShingle |
Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction Estilo de cálculo Deducción ecuacional Lógica intuicionista calculational style equational deduction Intuitionistic logic |
| title_short |
Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction |
| title_full |
Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction |
| title_fullStr |
Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction |
| title_full_unstemmed |
Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction |
| title_sort |
Intuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction |
| dc.creator.fl_str_mv |
Bohórquez, Jaime |
| dc.contributor.author.none.fl_str_mv |
Bohórquez, Jaime |
| dc.contributor.researchgroup.spa.fl_str_mv |
Informática |
| dc.subject.armarc.spa.fl_str_mv |
Estilo de cálculo Deducción ecuacional Lógica intuicionista |
| topic |
Estilo de cálculo Deducción ecuacional Lógica intuicionista calculational style equational deduction Intuitionistic logic |
| dc.subject.proposal.eng.fl_str_mv |
calculational style equational deduction Intuitionistic logic |
| description |
Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert’s style of proof and Gentzen’s deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations (in Dijkstra’s words, “letting the symbols do the work”) have led to the “calculational style,” an impressive array of techniques for elegant proof constructions. In this paper, we formalize intuitionistic predicate logic according to CED with similar success. In this system (I-CED), we prove Leibniz’s principle for intuitionistic logic and also prove that any (intuitionistic) valid formula of predicate logic can be proved in I-CED. |
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2008 |
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2008 |
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2021-12-07T16:42:59Z |
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2021-12-07T16:42:59Z |
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Artículo de revista |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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http://purl.org/coar/resource_type/c_6501 |
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Text |
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00294527 19390726 |
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https://repositorio.escuelaing.edu.co/handle/001/1910 |
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eng |
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eng |
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384 |
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4 |
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361 |
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49 |
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N/A |
| dc.relation.ispartofjournal.eng.fl_str_mv |
Notre Dame Journal Of Formal Logic |
| dc.relation.references.spa.fl_str_mv |
Backhouse, R., “Mathematics and Programming. A Revolution in the Art of Effective Reasoning,” 2001. http://www.cs.nott.ac.uk/~rcb/inaugural.pdf Backhouse, R., Program Construction: Calculating Implementations from Specifications, John Wiley and Sons, Inc., Hoboken, 2003. Bell, J. L., and M. Machover, A Course in Mathematical Logic, North-Holland Publishing Co., Amsterdam, 1977. Bijlsma, L., and R. Nederpelt, “Dijkstra-Scholten predicate calculus: Concepts and misconceptions,” Acta Informatica, vol. 35 (1998), pp. 1007–1036. Birkhoff, G., Lattice Theory, 3d edition, vol. 25 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1979. Bohorquez, J., and C. Rocha, “Towards the effective use of formal logic in the teaching of discrete math,” 6th International Conference on Information Technology Based Higher Education and Training (ITHET), 2005. Dijkstra, E. W., “Boolean connectives yield punctual expressions,” EWD 1187 in The Writings of Edsger W. Dijkstra, 2000. Dijkstra, E. W., and C. S. Scholten, Predicate Calculus and Program Semantics, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1990. Gries, D., and F. B. Schneider, A Logical Approach to Discrete Math, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1993. Gries, D., and F. B. Schneider, “Equational propositional logic,” Information Processing Letters, vol. 53 (1995), pp. 145–52. Gries, D., and F. B. Schneider, “Formalizations of Substitution of Equals for Equals,” Technical Report TR98-1686, Cornell University-Computer Science, Ithaca, 1998. Lifschitz, V., “On calculational proofs (First St. Petersburg Conference on Days of Logic and Computability, 1999),” Annals of Pure and Applied Logic, vol. 113 (2002), pp. 207–24. Nerode, A., “Some lectures on intuitionistic logic,” pp. 12–59 in Logic and Computer Science (Montecatini Terme, 1988), edited by A. Dold, B. Eckmann, and F. Takens, vol. 1429 of Lecture Notes in Mathematics, Springer, Berlin, 1990. Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, 3d edition, Monografie Matematyczne, Tom 41. PWN-Polish Scientific Publishers, Warsaw, 1970. Simmons, G. F., Introduction to Topology and Modern Analysis, Robert E. Krieger Publishing Co. Inc., Melbourne, 1983. Reprint of the 1963 original. Tourlakis, G., “A basic formal equational predicate logic,” Technical Report CS1998-09, York University-Computer Science, Toronto,1998. Tourlakis, G., “On the soundness and completeness of equational predicate logics,” Journal of Logic and Computation, vol. 11 (2001), pp. 623–53. |
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24 páginas. |
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University of Notre Dame |
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Bohórquez, Jaime34ca64f10c7c3bbadec92bdb453a4170600Informática2021-12-07T16:42:59Z2021-12-07T16:42:59Z20080029452719390726https://repositorio.escuelaing.edu.co/handle/001/1910Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert’s style of proof and Gentzen’s deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations (in Dijkstra’s words, “letting the symbols do the work”) have led to the “calculational style,” an impressive array of techniques for elegant proof constructions. In this paper, we formalize intuitionistic predicate logic according to CED with similar success. In this system (I-CED), we prove Leibniz’s principle for intuitionistic logic and also prove that any (intuitionistic) valid formula of predicate logic can be proved in I-CED.Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert’s style of proof and Gentzen’s deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations (in Dijkstra’s words, “letting the symbols do the work”) have led to the “calculational style,” an impressive array of techniques for elegant proof constructions. In this paper, we formalize intuitionistic predicate logic according to CED with similar success. In this system (I-CED), we prove Leibniz’s principle for intuitionistic logic and also prove that any (intuitionistic) valid formula of predicate logic can be proved in I-CED.24 páginas.application/pdfengUniversity of Notre Damehttps://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-49/issue-4/Intuitionistic-Logic-according-to-Dijkstras-Calculus-of-Equational-Deduction/10.1215/00294527-2008-017.full?tab=ArticleLinkIntuitionistic Logic according to Dijkstra’s Calculus of Equational DeductionArtículo de revistainfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARThttp://purl.org/coar/version/c_970fb48d4fbd8a85384436149N/ANotre Dame Journal Of Formal LogicBackhouse, R., “Mathematics and Programming. A Revolution in the Art of Effective Reasoning,” 2001. http://www.cs.nott.ac.uk/~rcb/inaugural.pdfBackhouse, R., Program Construction: Calculating Implementations from Specifications, John Wiley and Sons, Inc., Hoboken, 2003.Bell, J. L., and M. Machover, A Course in Mathematical Logic, North-Holland Publishing Co., Amsterdam, 1977.Bijlsma, L., and R. Nederpelt, “Dijkstra-Scholten predicate calculus: Concepts and misconceptions,” Acta Informatica, vol. 35 (1998), pp. 1007–1036.Birkhoff, G., Lattice Theory, 3d edition, vol. 25 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1979.Bohorquez, J., and C. Rocha, “Towards the effective use of formal logic in the teaching of discrete math,” 6th International Conference on Information Technology Based Higher Education and Training (ITHET), 2005.Dijkstra, E. W., “Boolean connectives yield punctual expressions,” EWD 1187 in The Writings of Edsger W. Dijkstra, 2000.Dijkstra, E. W., and C. S. Scholten, Predicate Calculus and Program Semantics, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1990.Gries, D., and F. B. Schneider, A Logical Approach to Discrete Math, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1993.Gries, D., and F. B. Schneider, “Equational propositional logic,” Information Processing Letters, vol. 53 (1995), pp. 145–52.Gries, D., and F. B. Schneider, “Formalizations of Substitution of Equals for Equals,” Technical Report TR98-1686, Cornell University-Computer Science, Ithaca, 1998.Lifschitz, V., “On calculational proofs (First St. Petersburg Conference on Days of Logic and Computability, 1999),” Annals of Pure and Applied Logic, vol. 113 (2002), pp. 207–24.Nerode, A., “Some lectures on intuitionistic logic,” pp. 12–59 in Logic and Computer Science (Montecatini Terme, 1988), edited by A. Dold, B. Eckmann, and F. Takens, vol. 1429 of Lecture Notes in Mathematics, Springer, Berlin, 1990.Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, 3d edition, Monografie Matematyczne, Tom 41. PWN-Polish Scientific Publishers, Warsaw, 1970.Simmons, G. F., Introduction to Topology and Modern Analysis, Robert E. Krieger Publishing Co. Inc., Melbourne, 1983. Reprint of the 1963 original.Tourlakis, G., “A basic formal equational predicate logic,” Technical Report CS1998-09, York University-Computer Science, Toronto,1998.Tourlakis, G., “On the soundness and completeness of equational predicate logics,” Journal of Logic and Computation, vol. 11 (2001), pp. 623–53.info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Estilo de cálculoDeducción ecuacionalLógica intuicionistacalculational styleequational deductionIntuitionistic logicORIGINALIntuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction.pdfIntuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction.pdfArtículo principal.application/pdf325356https://repositorio.escuelaing.edu.co/bitstream/001/1910/1/Intuitionistic%20Logic%20according%20to%20Dijkstra%e2%80%99s%20Calculus%20of%20Equational%20Deduction.pdffc6966a5ebc8a43beba24be5dae806adMD51metadata only accessLICENSElicense.txtlicense.txttext/plain; charset=utf-81881https://repositorio.escuelaing.edu.co/bitstream/001/1910/2/license.txt5a7ca94c2e5326ee169f979d71d0f06eMD52open accessTEXTIntuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction.pdf.txtIntuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction.pdf.txtExtracted texttext/plain42979https://repositorio.escuelaing.edu.co/bitstream/001/1910/3/Intuitionistic%20Logic%20according%20to%20Dijkstra%e2%80%99s%20Calculus%20of%20Equational%20Deduction.pdf.txt83e8559ea9b38e6018ba958a4bf51fd2MD53open accessTHUMBNAILIntuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction.pdf.jpgIntuitionistic Logic according to Dijkstra’s Calculus of Equational Deduction.pdf.jpgGenerated Thumbnailimage/jpeg10002https://repositorio.escuelaing.edu.co/bitstream/001/1910/4/Intuitionistic%20Logic%20according%20to%20Dijkstra%e2%80%99s%20Calculus%20of%20Equational%20Deduction.pdf.jpg7b7b9eb97e8df0135ed90e429421409fMD54open access001/1910oai:repositorio.escuelaing.edu.co:001/19102022-07-01 09:32:55.596metadata only accessRepositorio Escuela Colombiana de Ingeniería Julio Garavitorepositorio.eci@escuelaing.edu.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 |