The Rule of Existential Generalisation and Explicit Substitution

Jiří Raclavský

doi:10.12775/LLP.2021.011

Authors

Jiří Raclavský Department of Philosophy, Masaryk University, Brno https://orcid.org/0000-0002-5123-9965

DOI:

https://doi.org/10.12775/LLP.2021.011

Keywords

existential generalisation, quantifying in, explicit substitution, hyperintensional logic, natural deduction in sequent style

Abstract

The present paper offers the rule of existential generalization (EG) that is uniformly applicable within extensional, intensional and hyperintensional contexts. In contradistinction to Quine and his followers, quantification into various modal contexts and some belief attitudes is possible without obstacles. The hyperintensional logic deployed in this paper incorporates explicit substitution and so the rule (EG) is fully specified inside the logic. The logic is equipped with a natural deduction system within which (EG) is derived from its rules for the existential quantifier, substitution and functional application. This shows that (EG) is not primitive, as often assumed even in advanced writings on natural deduction. Arguments involving existential generalisation are shown to be valid if the sequents containing their premises and conclusions are derivable using the rule (EG). The invalidity of arguments seemingly employing (EG) is explained with recourse to the definition of substitution.

References

Abadi, M., L. Cardelli, P. Curien, and J. Levy, 1991, “Explicit substitutions”, Journal of Functional Programming 1 (4): 375–416. DOI: https://doi.org/10.1017/S0956796800000186

Berto, F., and D. Nolan, 2021, “Hyperintensionality”, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Spring 2021 edition. https://plato.stanford.edu/archives/spr2021/entries/hyperintensionality/

Blackburn, P., M. de Rijke, and Y. Venema, 2001, Modal Logic, Cambridge University Press.

Blamey, S., 2002, “Partial logic”, pages 261–353 in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic 5, Springer. DOI: https://doi.org/10.1007/978-94-017-0458-8_5

Cresswell, M. J., 1975, “Hyperintensional logic”, Studia Logica 34 (1): 26–38. DOI: https://doi.org/10.1007/BF02314421

Duží, M., and B. Jespersen, 2015, “Transparent quantification into hyperintensional objectual attitudes”, Synthese 192 (3): 635–677. DOI: https://doi.org/10.1007/s11229-014-0578-z

Farmer, W. M., 1990, “A partial functions version of Church’s simple theory of types”, Journal of Symbolic Logic 55 (3): 1269–1291. DOI: https://doi.org/10.2307/2274487

Feferman, S., 1995, “Definedness”, Erkenntnis 43 (3): 295–320. DOI: https://doi.org/10.1007/BF01135376

Fitting, M., 2015, “Intensional logic”, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Summer 2015 edition. http://plato.stanford.edu/archives/sum2015/entries/logic-intensional

Girard, J., P. Taylor, and Y. Lafont, 1989, Proofs and Types, Cambridge University Press.

Hintikka, J., 1975, “Impossible possible worlds vindicated”, Journal of Philosophical Logic 4 (4): 475–484. DOI: https://doi.org/10.1007/BF00558761

Indrzejczak, A., 2010, Natural Deduction, Hybrid Systems and Modal Logics, Springer.

Kaplan, D., “Quantifying in”, 1968, Synthese 19 (1–2): 178–214. DOI: https://doi.org/10.1007/BF00568057

Kuchyňka, P., and J. Raclavský, 2021 ,“β-reduction-by-name, by-value and η-reduction in partial type λ-calculus/partial type theory”, Logic Journal of IGPL, cond. accepted.

Lewis, D. K., 1970, “General semantics”, Synthese 22 (1–2): 18–67. DOI: https://doi.org/10.1007/BF00413598

Montague, R., 1974, Formal Philosophy. Selected Papers of Richard Montague edited by R. Thomason, Yale University Press.

Moschovakis, Y. N., 2005, “A logical calculus of meaning and synonymy”, Linguistics and Philosophy 29 (1): 27–89. DOI: https://doi.org/10.1007/s10988-005-6920-7

Muskens, R., 1995, Meaning and Partiality, CSLI.

Muskens, R., 2005, “Sense and the computation of reference”, Linguistics and Philosophy 28 (4): 473–504. DOI: https://doi.org/10.1007/s10988-004-7684-1

Muskens, R., 2007, “Higher order modal logic”, pages 621–653 in P. Blackburn, J. F. A. K. van Benthem, and F. Wolte (eds.), The Handbook of Modal Logic, Elsevier. DOI: https://doi.org/10.1016/S1570-2464(07)80013-9

Negri, S., J. von Plato, and A. Ranta, 2001, Structural Proof Theory, Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511527340

Pezlar, I., 2016, “Towards a more general concept of inference”, Logica Universalis 8 (1): 61–81. DOI: https://doi.org/10.1007/s11787-014-0095-3

Prawitz, D., 2006, Natural Deduction: A Proof-Theoretical Study, Dover Publications.

Queiroz, R. J G B de, A. G de Oliveira, D. M. Gabbay, The Functional Interpretation of Logical Deduction, 2011, World Scientific.

Quine, W. V. O., 1943, “Notes on existence and necessity”, The Journal of Philosophy 40 (5): 113–127. DOI: https://doi.org/10.2307/2017458

Quine, W. V. O., 1956, “Quantifiers and propositional attitudes”, The Journal of Philosophy 53 (5): 177–187. DOI: https://doi.org/10.2307/2022451

Raclavský, J., 2009, Jména a deskripce: logicko-sémantická zkoumání (in Czech, Names and Descriptions: Logico-Semantical Considerations), Nakladatelství Olomouc.

Raclavský, J., 2010, “On partiality and Tichý’s Transparent Intensional Logic”, Hungarian Philosophical Review 54 (4): 120–128.

Raclavský, J., 2011, “Semantic concept of existential presupposition”, Human Affairs 21 (3): 249–261. DOI: https://doi.org/10.2478/s13374-011-0026-4

Raclavský, J., 2014, “On interaction of semantics and deduction in Transparent Intensional Logic (Is Tichý’s logic a logic?)”, Logic and Logical Philosophy 23 (1): 57–68. DOI: https://doi.org/10.12775/LLP.2013.035

Raclavský, J., 2014, “Explicating truth in Transparent Intensional Logic”, pages 167–177 in R. Ciuni, H. Wansing, and C. Willkommen (eds.), Recent Trends in Philosophical Logic, Springer Verlag. DOI: https://doi.org/10.1007/978-3-319-06080-4_12

Raclavský, J., 2018, “Existential import and relations of categorical and modal categorical statements”, Logic and Logical Philosophy 27 (3): 271–300. DOI: https://doi.org/10.12775/LLP.2017.026

Raclavský, J., 2020, Belief Attitudes, Fine-Grained Hyperintensionality and Type-Theoretic Logic, College Publications (Studies in Logic 88).

Raclavský, J., P. Kuchyňka, and I. Pezlar, 2015, Transparent Intensional Logic as Characteristica Universalis and Calculus Ratiocinator (in Czech), Munipress.

Raclavský, J. and I. Pezlar, 2019, “Explicitní/implicitní přesvědčení a derivační systémy” (in Czech,

“Explicit/implicit belief and derivation systems”, Filosofický časopis 67 (1): 89–120.

Russell, B., 1905, “On denoting”, Mind 14 (56): 479–493.

Schroeder-Heister, P., 2006, “Validity concepts in proof-theoretic semantics”, Synthese 148 (3): 525–571.

Tichý, P., “Two kinds of intensional logic”, 1978, Epistemologia 1 (1): 143–164.

Tichý, P., “The logic of temporal discourse”, 1980, Linguistics and Philosophy 3 (3): 343–369. DOI: https://doi.org/10.1007/BF00401690

Tichý, P., “Foundations of partial type theory”, 1982, Reports on Mathematical Logic 14: 57–72.

Tichý, P., 1986, “Constructions”, Philosophy of Science 53 (4): 514–534. DOI: https://doi.org/10.1086/289338

Tichý, P., 1986, “Indiscernibility of identicals”, Studia Logica 45 (3): 251–273. DOI: https://doi.org/10.1007/BF00375897

Tichý, P., 1988, The Foundations of Frege’s Logic, Walter de Gruyter.

Tichý, P., 2004, Pavel Tichý’s Collected Papers in Logic and Philosophy, V. Svoboda, B. Jespersen, C. Cheyne (eds.), The University of Otago Press and Filosofia.

Williamson, T., 2013, Modal Logic as Metaphysics, Oxford University Press.