A meta-logic of inference rules: Syntax

Alex Citkin

DOI: http://dx.doi.org/10.12775/LLP.2015.007

Abstract


This work was intended to be an attempt to introduce the meta-language for working with multiple-conclusion inference rules that admit asserted propositions along with the rejected propositions. The presence of rejected propositions, and especially the presence of the rule of reverse substitution, requires certain change the definition of structurality.

Keywords


propositional logic; multiple-conclusion rule; rejected proposition; Ł-system; admissible rule; deductive system

Full Text:

PDF

References


Bonatti, P., and A. C. Varzi, “On the meaning of complementary systems”, in 10th International Congress of Logic, Methodology and Philosophy of Science. Volume of Abstracts, 1995.

Caferra, R., and N. Peltier, “Accepting/rejecting propositions from accepted/rejected propositions: A unifying overview”, International Journal of Intelligent Systems, 23 (2008), 999–1020. DOI: 10.1002/int.20304

Caferra, R., and N. Zabel, “A method for simultaneous search for refutations and models by equational constraint solving”, J. Symbolic Comput., 13, 6 (1992): 613–641. DOI: 10.1016/S0747-7171(10)80014-8

Carnap, R., Introduction to Semantics, Harvard University Press, Cambridge,

Mass., 1942.

Carnap, R., Formalization of Logic, Harvard University Press, Cambridge, Mass., 1943.

Church, A., “Review of the book Formalization of Logic by R. Carnap”, The Journal of Symbolic Logic, 53, 5 (1953): 493–498.

Dummett, M., The Logical Basis of Metaphysics, Harvard University Press, 1991.

Dummett, M., “‘Yes’, ‘no’ and ‘can’t say’”, Mind, 111, 442 (2002): 289–295.

Dutkiewicz, R., “The method of axiomatic rejection for the intuitionistic propositional logic”, Studia Logica, 48, 4 (1989): 449–459. DOI: 10.1007/BF00370199

Gibbard, P., “Price and Rumfitt on rejective negation and classical logic”, Mind, 111, 442 (2002): 297–303.

Goranko, V.. “Refutation systems in modal logic”, Studia Logica, 53, 2 (1994): 299–324. DOI: 10.1007/BF01054714

Hähnle, R., “Tableaux and related methods”, A. Robinson et al. (eds.), Handbook of automated reasoning, in two vols., Amsterdam: North-Hollandm Elsevier, 2001. DOI: 10.1016/B978-044450813-3/50005-9

Humberstone, L., “The revival of rejective negation”, J. Philos. Logic 29, 4 (2000): 331–381. DOI: 10.1023/A:1004747920321

Iemhoff, R., and G. Metcalfe, “Hypersequent systems for the admissible rules of modal and intermediate logics”, pp. 230–245 in Logical foundations of computer science, vol. 5407 of “Lecture Notes in Comput. Sci.”, Springer, Berlin, 2009. DOI: 10.1007/978-3-540-92687-0_16

Iemhoff, R., and G. Metcalfe, “Proof theory for admissible rules”, Ann. Pure Appl. Logic 159, 1–2 (2009): 171–186. DOI: 10.1016/j.apal.2008.10.011

Incurvati, L., and P. Smith, “Rejection and valuations”, Analysis 70, 1 (2010): 3–10. DOI: 10.1093/analys/anp134

Ishimoto, A., “Axiomatic rejection for classical propositional logic”, pp. 257–270, Chapter 18 in Philosophical logic and Logical Philosophy, vol. 257 of “Synthese Library”, Kluwer Acad. Publ., Dordrecht, 1996. DOI: 10.1007/978-94-015-8678-8_18

Jeřabek, E., “Admissible rules of modal logics”, J. Logic Comput. 15, 4 (2005): 411–431. DOI: 10.1093/logcom/exi029

Jeřabek, E., “Canonical rules”, J. Symbolic Logic 74, 4 (2009): 1171–1205.

Johnson, F., “Rejection and truth-value gaps”, Notre Dame J. Formal Logic 40, 4 (1999): 574–577. DOI: 10.1305/ndjfl/1012429721

Kleene, S.C., Introduction to Metamathematics, D. Van Nostrand Co., Inc., New York, N. Y., 1952.

Kneale, W., “The province of logic”, pp. 235–261 in Contemporary British Philosophy, 3rd series, H. Lewis (ed.), G. Allen & Unwin, London, 1956.

Kneale, W., “The province of logic”, Mind 66, 262 (1957): 258.

Kracht, M., “Book review of [36]”, Notre Dame J. Form. Log. 40, 4 (1999): 578–587.

Kracht, M., “Judgment and consequence relations”, J. Appl. Non-Classical Logics 20, 4 (2010): 423–435. DOI: 10.3166/jancl.20.423-435

Kulicki, P., “Remarks on axiomatic rejection in Aristotle’s syllogistic”, Studies in Logic and Theory of Knowledge 5 (2002): 231–236.

Łukasiewicz, J., Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, Oxford, at the Clarendon Press, 1951.

Łukasiewicz, J., “On the intuitionistic theory of deduction”, Nederl. Akad.

Wetensch. Proc., Ser. A., 55 = Indagationes Math., 14 (1952): 202–212.

Malinowski, G., “Q-consequence operation”, Rep. Math. Logic, 24 (1990): 49–59.

Murzi, J., and O. T. Hjortland, “Inferentialism and the categoricity problem: Reply to Raatikainen”, Analysis, 69, 3 (2009): 480–488. DOI: 10.1093/analys/anp071

Prawitz, D., Natural Deduction. A Proof-Theoretical Study, Acta Universitatis Stockholmiensis, Stockholm Studies in Philosophy, no. 3., Almqvist & Wiksell, Stockholm, 1965.

Restall, G., “Multiple conclusions”, pp. 189–205 in Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, P. Hajek, L. Valdes-Villanueva, and D. Westerstahl (eds.), Kings College Publications, 2005.

Rumfitt, I., “‘Yes’ and ‘No’”, Mind 109, 436 (2000): 781–823.

Rumfitt, I., “Unilateralism disarmed: A reply to: “‘Yes’, ‘no’ and ‘can’t say’” Mind, 111 (2002), 442: 289–295, by M. Dummett and “Price and Rumfitt on rejective negation and classical logic”, Mind, 111 (2002), 442: 297–303, by P. Gibbard”, Mind 111, 442 (2002): 305–321.

Rumfitt, I., “Knowledge by deduction”, pp. 61–84 in Knowledge and Questions, L. Franck (ed.), vol. 77 of “Grazer Philosophische Studien”, Rodopi, 2008.

Rybakov, V. V., Admissibility of Logical Inference Rules, vol. 136 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1997.

Scott, D., “Completeness proofs for the intuitionistic sentential calculus”, pp. 231–241 in Summer Institute for Symbolic Logic, Cornell University, Amer. Math. Soc., 1957.

Scott, D., “On engendering an illusion of understanding”, The Journal of Philosophy 68, 21 (1971): 787–807. DOI: 10.2307/2024952

Scott, D.S., “Background to formalization”, pp. 244–273 in Truth, Syntax and Modality. Proc. Conf. Alternative Semantics, Temple, Univ. Philadelphia, Pa., vol. 68 of Studies in Logic and the Foundations of Math., North-Holland, Amsterdam, 1973. DOI: 10.1016/S0049-237X(08)71542-8

Scott, D.S., “Completeness and axiomatizability in many-valued logic”, pp. 411–435 in Proceedings of the Tarski Symposium. Proc. Sympos. Pure Math., vol. XXV, Univ. California, Berkeley, Calif., 1971.

Scott, D.S. “Rules and derived rules”, pp. 147–161 in Logical Theory and Semantic Analysis. Essays dedicated to Stig Kanger, S. Stenlund (ed.), D. Reidel Publishing Company, 1974. DOI: 10.1007/978-94-010-2191-3_13

Shoesmith, D.J., and T.J. Smiley, Multiple-Conclusion Logic, Cambridge University Press, Cambridge, 2008. Reprint of the 1978 original [MR0500331]. DOI: 10.1017/CBO9780511565687

Skura, T., “A complete syntactical characterization of the intuitionistic logic”, Reports on Mathematical Logic 23 (1989): 75–80.

Skura, T., “Aspects of refutation procedures in the intuitionistic logic and related modal systems, Acta Universitatis Wratislaviensis 2190, Wrocław, 1998.

Skura, T., “On refutation rules”, Log. Univers., 5, 2 (2011): 249–254. DOI: 10.1007/s11787-011-0035-4

Skura, T., “Refutation systems in propositional logic”, pp. 115–157 in vol. 16 of Handbook of Philosophical Logic, D.M. Gabbay and F. Guenthner (eds.), Springer, 2011. DOI:˙10.1007/978-94-007-0479-4_2

Skura, T., Refutation Methods in Modal Propositional Logic, Semper, 2013.

Słupecki, J., Z badań nad sylogistyka Arystotelesa, Wrocławskie Towarzystwo Naukowe, 1948.

Słupecki, J., G. Bryll, and U. Wybraniec-Skardowska, “Theory of rejected propositions. I”, Studia Logica, 29 (1971): 75–123. DOI: 10.1007/BF02121863

Słupecki, J., G. Bryll, and U. Wybraniec-Skardowska, “The theory of rejected propositions. II”, Studia Logica 30 (1972): 97–145. DOI: 10.1007/BF02120839

Smiley, T., “Rejection”, Analysis, 56, 1 (1996), 1–9. DOI: 10.1093/analys/56.1.1

Sochacki, R., “Axiomatic rejection in the implicational-negational invariant sentential calculi of Łukasiewicz”, Bull. Sect. Logic Univ. Łódź, 36, 1–2 (2007), 1–6.

Sochacki, R., Metody refutacyjne w badaniach nad systemami logicznymi (in Polish), Universytet Opolski, 2010.

Staszek, W., “On proofs of rejection”, Studia Logica, 29 (1971): 17–25. DOI: 10.1007/BF02121854

Tanaka, K., F. Berto, E. Mares, and F. Paoli (eds.), Paraconsistency: Logic and Applications, vol. 26 of “Logic, Epistemology, and the Unity of Sscience”, Springer, 2013. DOI: 10.1007/978-94-007-4438-7

Tiomkin, M., “Proving unprovability”, pp. 22–26 in Proceedings. Third Annual Information Symposium on Logic in Computer Science, 1988. DOI: 10.1109/LICS.1988.5097

Varzi, A.C., “Complementary logics for classical propositional languages”, Kriterion. Zeitschrift fur Philosophie, 4 (1992): 20–24.

Wójcicki, R., “Dual counterparts of consequence operations”, Polish Acad. Sci. Inst. Philos. Sociology Bull. Sect. Logic 2, 1 (1973): 54–57.

Wybraniec-Skardowska, U., and J. Waldmajer, “On pairs of dual consequence operations”, Log. Univers., 5, 2 (2011): 177–203. DOI: 10.1007/s11787-011-0030-9




Bookmark and Share




Print ISSN: 1425-3305
Online ISSN: 2300-9802

Partnerzy platformy czasopism