The categorical and the hypothetical: a critique of some fundamental assumptions of standard semantics
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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.
KeywordsConsequence Inference Proof-theoretic semantics Definitional reflection
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- 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.Google Scholar
- Dummett, M. (1978). The justification of deduction (1973). In Truth and Other Enigmas. London: Duckworth.Google Scholar
- 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.Google Scholar
- Hallnäs, L., & Schroeder-Heister, P. (2012). A survey of definitional reflection (in preparation).Google Scholar
- 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.Google Scholar
- Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik (2nd Edn. 1969). Berlin: Springer.Google Scholar
- 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.Google Scholar
- Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study. Almqvist & Wiksell: Stockholm (Reprinted Mineola NY: Dover Publ., 2006).Google Scholar
- 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-HollandGoogle Scholar
- 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: KluwerGoogle Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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, LondonGoogle Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- Statman R. (1979) Lower bounds on Herbrand’s theorem. Proceedings of the American Mathematical Society 75: 104–107Google Scholar