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In this paper, I motivate a cut free sequent calculus for classical logic with first order quantification, allowing for singular terms free of existential import. Along the way, I motivate a criterion for rules designed to answer Prior’s question about what distinguishes rules for logical concepts, like conjunction from apparently similar rules for putative concepts like Prior’s tonk, and I show that the rules for the quantifiers—and the existence predicate—satisfy that condition.

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Anellis, I. H. (1990). From semantic tableaux to Smullyan trees: A history of the development of the falsifiability tree method. Modern Logic, 1(1), 3669.
Avron, A. (1993). Gentzen-type systems, resolution and tableaux. Journal of Automated Reasoning, 10(2), 265281.
Battilotti, G. & Sambin, G. (1999). Basic logic and the cube of its extensions. In Cantini, A., Casari, E., and Minari, P., editors. Logic and Foundations of Mathematics. Dordrecht: Kluwer, pp. 165185.
Belnap, N. D. (1962). Tonk, plonk and plink. Analysis, 22, 130134.
Berto, F. (2015). “There is an ‘is’ in ‘there is”’: Meinongian quantification and existence. In Torza, A., editor. Quantifiers, Quantifiers, and Quantifiers. Synthese Library. Heidelberg: Springer.
Curry, H. B. & Feys, R. (1958). Combinatory Logic, Vol. 1. Amsterdam: North-Holland.
Došen, K. (1989). Logical constants as punctuation marks. Notre Dame Journal of Formal Logic, 30(3), 362381.
Faggian, C. & Sambin, G. (1998). From basic logic to quantum logics with cut-elimination. International Journal of Theoretical Physics, 37(1), 3137.
Feferman, S. (1995). Definedness. Erkenntnis, 43(3), 295320.
Gratzl, N. (2010). A sequent calculus for a negative free logic. Studia Logica, 96(3), 331348.
Lambert, K. (1997). Free Logics: Their Foundations, Character, and Some Applications Thereof. Germany: Academia Verlag, Sankt Augustin.
Meinong, A. (1983). On Aassumptions. Berkeley and Los Angeles, California: University of California Press. Translated and edited by Heanue, James.
Priest, G. (2006). Doubt Truth to be a Liar. Oxford: Oxford University Press.
Prior, A. N. (1960). The runabout inference-ticket. Analysis, 21(2), 3839.
Quine, W. V. (1954). Quantification and the empty domain. Journal of Symbolic Logic, 19(3), 177179.
Restall, G. (2005). Multiple conclusions. In Hájek, P., Valdés-Villanueva, L., and Westerståhl, D., editors. Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress. KCL Publications, pp. 189205. Available at:
Restall, G. (2009). Truth values and proof theory. Studia Logica, 92(2), 241264. Available at
Restall, G. (2012). A cut-free sequent system for two-dimensional modal logic, and why it matters. Annals of Pure and Applied Logic, 163(11), 16111623. Available at:
Routley, R. (1979). Exploring Meinong’s Jungle. Canberra: Australian National University.
Sambin, G., Battilotti, G., & Faggian, C. (2014). Basic logic: Reflection, symmetry, visibility. The Journal of Symbolic Logic, 65(3), 9791013.
Scales, R. D. (1969). Attribution and Existence. Ph.D. Thesis, Irvine: University of California.
Schock, R. (1968). Logics Without Existence Assumptions. Stockholm: Almqvist & Wiskell.
Schütte, K. (1956). Ein system des verknüpfenden schliessens. Archiv für mathematische Logik und Grundlagenforschung, 2(2–4), 5567. (German).
Smullyan, R. M. (1995). First-Order Logic. Berlin: Springer-Verlag. Reprinted by Dover Press.
Takeuti, G. (1987). Proof Theory (second edition). Studies in Logic and the Foundations of Mathematics, Vol. 81. Amsterdam: North-Holland.
Textor, M. (2017). Towards a Neo-Brentanian theory of existence. Philosophers’ Imprint, 17(6), 120.
Wadler, P. (2005). Call-by-value is dual to call-by-name, reloaded. In Giesl, J., editor. Rewriting Techniques and Application, RTA’05. Lecture Notes in Computer Science, Vol. 3467. Berlin: Springer, pp. 185203.
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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
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