Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T19:56:12.516Z Has data issue: false hasContentIssue false
  • English
  • Français

PROOF ANALYSIS FOR LEWIS COUNTERFACTUALS

Published online by Cambridge University Press:  01 December 2015

SARA NEGRI  and
GIORGIO SBARDOLINI
SARA NEGRI*
Affiliation:
Department of Philosophy University of Helsinki
GIORGIO SBARDOLINI*
Affiliation:
Department of Philosophy The Ohio State University
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF HELSINKI E-mail: [email protected]
†DEPARTMENT OF PHILOSOPHY THE OHIO STATE UNIVERSITY E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A deductive system for Lewis counterfactuals is presented, based directly on the influential generalisation of relational semantics through ternary similarity relations introduced by Lewis. This deductive system builds on a method of enriching the syntax of sequent calculus by labels for possible worlds. The resulting labelled sequent calculus is shown to be equivalent to the axiomatic system VC of Lewis. It is further shown to have the structural properties that are needed for an analytic proof system that supports root-first proof search. Completeness of the calculus is proved in a direct way, such that for any given sequent either a formal derivation or a countermodel is provided; it is also shown how finite countermodels for unprovable sequents can be extracted from failed proof search, by which the completeness proof turns into a proof of decidability.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 1 , March 2016 , pp. 44 - 75
Copyright
Copyright © Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Blackburn, P. (2000). Internalizing labelled deduction. Journal of Logic and Computation, 10, 137–168.Google Scholar
Blackburn, P., Bolander, T., Braüner, T., & Frovin Jørgensen, K. (2015). Completeness and termination for a Seligman-style tableau system. Journal of Logic and Computation. To appear.Google Scholar
Chellas, B. F. (1975). Basic conditional logic. Journal of Philosophical Logic, 4, 133–153.CrossRefGoogle Scholar
Chisholm, R. (1946). The contrary-to-fact conditional. Mind, 55, 289–307.Google Scholar
Dyckhoff, R. & Negri, S. (2012). Proof analysis in intermediate logics. Archive for Mathematical Logic, 51, 71–92, 2012.Google Scholar
Dyckhoff, R. & Negri, S. (2013). A cut-free sequent system for Grzegorczyk logic with an application to the Gödel-McKinsey-Tarski embedding. Journal of Logic and Computation, doi 10.1093/logcom/ext036.Google Scholar
Dyckhoff, R. & Negri, S. (2015). Geometrization of first-order logic. The Bulletin of Symbolic Logic, 21, 123–163.Google Scholar
Fitting, M. (2007). Modal proof theory. In Blackburn, P., Van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic, pp. 85–138, New York: Elsevier.Google Scholar
Gabbay, D. M. & Governatori, G. (1998). Dealing with label dependent deontic modalities. In McNamara, P. and Prakken, H., editors. Norms, Logics and Information Systems, Amsterdam: IOS Press.Google Scholar
Garg, D., Genovese, V. & Negri, S. (2012). Countermodels from sequent calculi in multi-modal logics. LICS 2012, IEEE Computer Society, pp. 315–324.Google Scholar
Gasquet, O. & Herzig, A. (1996). From classical to normal modal logics. In Wansing, H., editor. Proof Theory of Modal Logic Applied Logic Series, vol. 2, pp. 293–311, New York: Springer.CrossRefGoogle Scholar
Gent, I. P. (1992). A sequent- or tableau-style system for Lewis’s counterfactual logic VC. Notre Dame Journal of Formal Logic, 33, 369–382.Google Scholar
Giordano, L., Gliozzi, V., Olivetti, N., & Schwind, C. (2008). Tableau calculi for preference-based conditional logics: PCL and its extensions. ACM Transactions on Computational Logic, 10, no. 3/21, 1–45.Google Scholar
Goodman, N. (1947). The problem of counterfactual conditionals. Journal of Philosophy, 44, 113–128.CrossRefGoogle Scholar
Hakli, R. & Negri, S. (2012). Does the deduction theorem fail for modal logic? Synthese, 187, 849–867.Google Scholar
Kripke, S. (1963). Semantical analysis of modal logic I. Normal modal propositional calculi. Zetschrift für mathematische Logik und Grundlagen der Math., 9, 67–96.Google Scholar
Lellman, B. & Pattinson, D. (2012). Sequent systems for Lewis’ conditional logics. In Farinas del Cerro, L., Herzig, A. and Mengin, J., editors. Logics in Artificial Intelligence, pp. 320–332.Google Scholar
Lewis, D. (1971). Completeness and decidability of three logics of counterfactual conditionals. Theoria, 37, 74–85.CrossRefGoogle Scholar
Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.Google Scholar
Lewis, D. (1973a). Counterfactuals and comparative possibility. Journal of Philosophical Logic, 2, pp. 418–446, 1973.Google Scholar
Negri, S. (2003). Contraction-free sequent calculi for geometric theories, with an application to Barr’s theorem. Archive for Mathematical Logic, 42, 389–401.CrossRefGoogle Scholar
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34, 507–544.Google Scholar
Negri, S. (2009). Kripke completeness revisited. In Primiero, G. and Rahman, S.. editors. Acts of Knowledge - History, Philosophy and Logic, pp. 247–282, London: College Publications.Google Scholar
Negri, S. (2014). Proof analysis beyond geometric theories: from rule systems to systems of rules. Journal of Logic and Computation, doi: 10.1093/logcom/exu037.Google Scholar
Negri, S. (2014a). Proofs and countermodels in non-classical logics. Logica Universalis, 8, 25–60.Google Scholar
Negri, S. & von Plato, J. (1998). Cut elimination in the presence of axioms. The Bulletin of Symbolic Logic, 4, 418–435.CrossRefGoogle Scholar
Negri, S. & von Plato, J. (2001). Structural Proof Theory. Cambridge: Cambridge University Press.Google Scholar
Negri, S. & von Plato, J. (2011). Proof Analysis. Cambridge: Cambridge University Press.Google Scholar
Negri, S. & von Plato, J. (2015). Meaning in use. In Wansing, H., editor. Dag Prawitz on Proofs and Meaning, Trends in Logic, Berlin: Springer, 239–257.Google Scholar
Olivetti, N., Pozzato, G. L. & Schwind, C. (2007). A Sequent Calculus and a Theorem Prover for Standard Conditional Logics. ACTM Transactions on Computational Logics (TOCL), 8/4, 22/pp. 1–51.Google Scholar
Priest, G. (2008). An Introduction to Non-Classical Logic, Cambridge: Cambridge University Press.Google Scholar
Read, S. (2015). Semantic pollution and syntactic purity. The Review of Symbolic Logic, doi: http://dx.doi.org/10.1017/S1755020315000210.Google Scholar
Sano, K. (2009). Hybrid counterfactual logics: David Lewis meets Arthur Prior again. Journal of Logic, Language and Information, 18, 515–539.CrossRefGoogle Scholar
de Swart, H. C. M. (1983). A Gentzen- or Beth-type system, a practical decision procedure and a constructive completeness proof for the counterfactual logics VC and VCS. The Journal of Symbolic Logic, 48, 1–20.Google Scholar
Stalnaker, R. (1968). A theory of conditionals. In Rescher, N., editor. Studies in Logical Theory, Oxford, pp. 98–112.Google Scholar