Abstract
The hypothetical notion of consequence is normally understood as the transmission of a categorical notion from premisses to conclusion. In model-theoretic semantics this categorical notion is ‘truth’, in standard proof-theoretic semantics it is ‘canonical provability’. Three underlying dogmas, (I) the priority of the categorical over the hypothetical, (II) the transmission view of consequence, and (III) the identification of consequence and correctness of inference are criticized from an alternative view of proof-theoretic semantics. It is argued that consequence is a basic semantical concept which is directly governed by elementary reasoning principles such as definitional closure and definitional reflection, and not reduced to a categorical concept. This understanding of consequence allows in particular to deal with non-wellfounded phenomena as they arise from circular definitions.
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References
Brotherston, J., & Simpson, A. (2007). Complete sequent calculi for induction and infinite descent. In Proceedings of the 22nd annual IEEE symposium on logic in computer science (LICS), (pp. 51–62). Los Alamitos: IEEE Press.
Dummett, M. (1978). The justification of deduction (1973). In Truth and Other Enigmas. London: Duckworth.
Hallnäs L. (1991) Partial inductive definitions. Theoretical Computer Science 87: 115–142
Hallnäs L. (2006) On the proof-theoretic foundation of general definition theory. Synthese 148: 589–602
Hallnäs, L., & Schroeder-Heister, P. (1990/91). A proof-theoretic approach to logic programming: I. Clauses as rules. II. Programs as definitions. Journal of Logic and Computation, 1, 261–283, 635–660.
Hallnäs, L., & Schroeder-Heister, P. (2012). A survey of definitional reflection (in preparation).
Kreuger, P. (1994). Axioms in definitional calculi. In R. Dychhoff (Ed.), Extensions of logic programming. 4th international workshop, ELP’93 (St. Andrews,U.K., March/April 1993). Proceedings (Lecture Notes in Computer Science) (Vol. 798, pp. 196–205). Berlin: Springer.
Lorenzen P. (1950) Konstruktive Begründung der Mathematik. Mathematische Zeitschrift 53: 162–202
Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik (2nd Edn. 1969). Berlin: Springer.
Orevkov, V. P. (1982). Lower bounds for increasing complexity of derivations after cut elimination (Transl., russ. orig. 1979). Journal of Soviet Mathematics, 20, 2337–2350.
Post E. L. (1921) Introduction to a general theory of elementary propositions. American Journal of Mathematics 43: 163–185
Post E. L. (1943) Formal reductions of the general combinatorial decision problem. American Journal of Mathematics 65: 197–215
Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study. Almqvist & Wiksell: Stockholm (Reprinted Mineola NY: Dover Publ., 2006).
Prawitz, D. (1973). Towards a foundation of a general proof theory. In P. Suppes et al. (Eds.), Logic, methodology and philosophy of science IV (pp. 225–250). Amsterdam: North-Holland
Prawitz, D. (1979). Proofs and the meaning and completeness of the logical constants. In J. Hintikka et al. (Eds.), Essays on mathematical and philosophical logic (pp. 25–40). Dordrecht: Kluwer
Schroeder-Heister P. (1984) A natural extension of natural deduction. Journal of Symbolic Logic 49: 1284–1300
Schroeder-Heister P. (1991) Structural frameworks, substructural logics, and the role of elimination inferences. In: Huet G., Plotkin G. (eds) Logical frameworks. Cambridge University Press, New York, pp 385–403
Schroeder-Heister, P. (1992). Cut elimination in logics with definitional reflection. In D. Pearce & H. Wansing (Eds.), Nonclassical logics and information processing: International workshop, Berlin, November 1990, Proceedings (Lecture Notes in Computer Science) (Vol. 619, pp. 146–171). Berlin: Springer.
Schroeder-Heister, P. (2004). On the notion of assumption in logical systems. In R. Bluhm & C. Nimtz (Eds.), Selected papers contributed to the sections of GAP5, fifth international congress of the Society for Analytical Philosophy, Bielefeld, 22–26 September 2003 (pp. 27–48). Mentis: Paderborn http://www.gap5.de/proceedings.
Schroeder-Heister, P. (2008). Proof-theoretic versus model-theoretic consequence. In M. Peliš (Ed.), The Logica Yearbook 2007 (pp. 187–200). Filosofia: Prague.
Schroeder-Heister P. (2009) Sequent calculi and bidirectional natural deduction: On the proper basis of proof-theoretic semantics. In: Peliš M. (eds) The Logica Yearbook 2008. College Publications, London
Schroeder-Heister, P. (2011a). Generalized elimination inferences, higher-level rules, and the implications-as-rules interpretation of the sequent calculus. In E. H. Haeusler, L. C. Pereira, & V. de Paiva (Eds.), Advances in natural deduction.
Schroeder-Heister, P. (2011b). Implications-as-rules vs. implications-as-links: An alternative implication-left schema for the sequent calculus. Journal of Philosophical Logic, 40 95–101.
Schroeder-Heister, P. (2011c). Proof-theoretic semantics. In Ed. Zalta (Ed.), Stanford Encyclopedia of Philosophy. Stanford: Stanford University. http://plato.stanford.edu.
Smullyan, R. (1961). Theory of formal systems. Annals of mathematics studies 47. Princeton: Princeton University Press.
Statman R. (1979) Lower bounds on Herbrand’s theorem. Proceedings of the American Mathematical Society 75: 104–107
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Schroeder-Heister, P. The categorical and the hypothetical: a critique of some fundamental assumptions of standard semantics. Synthese 187, 925–942 (2012). https://doi.org/10.1007/s11229-011-9910-z
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DOI: https://doi.org/10.1007/s11229-011-9910-z