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Series:   Elements in Philosophy and Logic

Logical Consequence

Published online by Cambridge University Press:  20 August 2022

Gila Sher
Gila Sher
Affiliation:
University of California, San Diego

Summary

To understand logic is, first and foremost, to understand logical consequence. This Element provides an in-depth, accessible, up-to-date account of and philosophical insight into the semantic, model-theoretic conception of logical consequence, its Tarskian roots, and its ideas, grounding, and challenges. The topics discussed include: (i) the passage from Tarski's definition of truth (simpliciter) to his definition of logical consequence, (ii) the need for a non-proof-theoretic definition, (iii) the idea of a semantic definition, (iv) the adequacy conditions of preservation of truth, formality, and necessity, (v) the nature, structure, and totality of models, (vi) the logicality problem that threatens the definition of logical consequence (the problem of logical constants), (vii) a general solution to the logicality, formality, and necessity problems/challenges, based on the isomorphism-invariance criterion of logicality, (viii) philosophical background and justification of the isomorphism-invariance criterion, and (ix) major criticisms of the semantic definition and the isomorphism-invariance criterion.
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Keywords

logical consequencelogical semanticsmodel theorycriterion of logicalityinvariance under
Type
Element
Information
Series: Elements in Philosophy and Logic
Online ISBN: 9781108981668
Publisher: Cambridge University Press
Print publication: 08 September 2022

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References

Bagaria, J. 2019. “Set Theory.” Stanford Encyclopedia of Philosophy. Zalta, E. N. (ed.). Stanford, CA: The Metaphysics Research Lab.Google Scholar
Barwise, J. 1972. “Absolute Logics and L∞ ω.” Annals of Mathematical Logic 4: 309–340.CrossRefGoogle Scholar
Barwise, J. 1985. “Model-Theoretic Logics: Background and Aims.” Model-Theoretic Logics. Barwise, J. and Feferman, S. (eds.). New York: Springer-Verlag. pp. 3–23.Google Scholar
Beall, J. C., and Restall, G.. 2006. Logical Pluralism. Oxford: Oxford University Press.Google Scholar
Bonnay, D. 2008. “Logicality and Invariance.” Bulletin of Symbolic Logic 14: 29–68.CrossRefGoogle Scholar
Brandom, R. B. 1994. Making It Explicit. Cambridge, MA: Harvard University Press.Google Scholar
Chihara, C. 1998. “Tarski’s Thesis and the Ontology of Mathematics.” The Philosophy of Mathematics Today. Schirn, M. (ed.). Oxford: Oxford University Press. pp. 157–172.Google Scholar
Dummett, M. 1978. Truth and Other Enigmas. Cambridge, MA: Harvard University Press.Google Scholar
Dutilh Novaes, C. 2014. “The Undergeneration of Permutation Invariance as a Criterion for Logicality.” Erkenntnis 79: 81–97.Google Scholar
Enderton, H. B. 2001. A Mathematical Introduction to Logic. San Diego, CA: Hartcourt.Google Scholar
Etchemendy, J. 1990. The Concept of Logical Consequence. Cambridge, MA: Harvard University Press.Google Scholar
Etchemendy, J. 2008. “Reflections on Consequence.” New Essays on Tarski and Philosophy. Patterson, D. (ed.). Oxford: Oxford University Press. pp. 263–299.CrossRefGoogle Scholar
Feferman, S. 1999. “Logic, Logics, and Logicism.” Notre Dame Journal of Formal Logic 40: 31–54.Google Scholar
Feferman, S. 2000. “Mathematical Intuition vs. Mathematical Monsters.” Synthese 125: 317–322.Google Scholar
Feferman, S. 2010. “Set-theoretical Invariance Criteria for Logicality.” Notre Dame Journal of Formal Logic 51: 3–20.Google Scholar
Field, H. 2009. “What Is the Normative Role of Logic?” Proceedings of the Aristotelian Society Suppl. 83: 251–268.Google Scholar
Field, H. 2015. “What Is Logical Validity.” Foundations of Logical Consequence. Caret, C. R and Hjortland (eds.), O. T. Oxford: Oxford University Press. pp. 33–70.Google Scholar
Fitting, M. 2015. “Intensional Logic.” Stanford Encyclopedia of Philosophy. Zalta, E. N. (ed.). Stanford, CA: The Metaphysics Research Lab.Google Scholar
Frege, G. 1967 (1879). “Begriffsschrift.” From Frege to Gödel. van Heijenoort, J. (ed.). Cambridge, MA: Harvard University Press. pp. 5–82.Google Scholar
Frege, G. 1893. The Basic Laws of Arithmetic. Vol. 1. Berkeley, CA: University of California Press, English translation 1964.Google Scholar
Frege, G. 1918. “Thoughts.” Logical Investigations. Oxford: Basil Blackwell, English translation 1977. pp. 1–30.Google Scholar
Friedman, M. 2001. Dynamics of Reason. Stanford, CA: CSLI.Google Scholar
García-Carpintero, M. 1993. “The Grounds of the Model-theoretic Account of the Logical Properties.” Notre Dame Journal of Formal Logic 34: 107–131.Google Scholar
Gödel, K. 1986 (1929). “On the Completeness of the Calculus of Logic.” Collected Works. Vol. 1. Feferman, S., Dawson, J. W. Jr., Kleene, S. C., et al. (eds.). New York: Oxford University Press. pp. 61–101.Google Scholar
Gödel, K. 1986 (1931). “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I.” Collected Works. Vol. 1. Feferman, S., Dawson, J. D. Jr., Kleene, S. C., et al. (eds.). New York: Oxford University Press. pp. 145–195.Google Scholar
Gómez-Torrente, M. 1996. “Tarski on Logical Consequence.” Notre Dame Journal of Formal Logic 37: 125–151.CrossRefGoogle Scholar
Gómez-Torrente, M. 2002. “The Problem of Logical Constants.” Bulletin of Symbolic Logic 8: 1–37.CrossRefGoogle Scholar
Griffiths, O., and Paseau, A. C.. 2016. “Isomorphism Invariance and Overgeneration.” Bulletin of Symbolic Logic 22: 482–503.Google Scholar
Griffiths, O., and Paseau, A. C. 2022. One True Logic. Oxford: Oxford University Press.Google Scholar
Hanson, W. H. 1997. “The Concept of Logical Consequence.” Philosophical Review 106: 365–409.CrossRefGoogle Scholar
Harman, G. 1986. Change in View. Cambridge, MA: The MIT Press.Google Scholar
Hilbert, D. 1950 (1899). The Foundations of Geometry. La Salle, IL: Open Court.Google Scholar
Hilbert, D., and Ackerman, W.. 1950 (1928). Principles of Mathematical Logic. New York: Chelsea Publishing.Google Scholar
Hodges, W. 1986. “Truth in a Structure.” Proceedings of the Aristotelian Society 86: 135–151.CrossRefGoogle Scholar
Jacquette, D. 1994. “Tarski’s Quantificational Semantics and Meinongian Object Theory Domains.” Pacific Philosophical Quarterly 75: 88–107.Google Scholar
Kant, I. 1929 (1781/1787). Critique of Pure Reason. London: Macmillan.Google Scholar
Keisler, H. J. 1970. “Logic with the Quantifier ‘There Exist Uncountably Many.’” Annals of Mathematical Logic 1: 1–93.CrossRefGoogle Scholar
Klein, F. 1872. “A Comparative Review of Recent Researches in Geometry.” PhD thesis. University of Bonn.Google Scholar
Kreisel, G. 1967. “Informal Rigor and Completeness Proofs.” Problems in the Philosophy of Mathematics. Lakatos, I. (ed.). Amsterdam: North-Holland. pp. 138–186.Google Scholar
Kripke, S. 1970/1980. Naming and Necessity. Cambridge, MA: Harvard University Press.Google Scholar
Levy, A. 1960. “Axiom Schemata of Strong Infinity in Axiomatic Set Theory.” Pacific Journal of Mathematics 10: 223–238.CrossRefGoogle Scholar
Lindström, P. 1966. “First Order Predicate Logic with Generalized Quantifiers.” Theoria 32: 186–195.CrossRefGoogle Scholar
Lindström, P. 1969. “On Extensions of Elementary Logic.” Theoria 35: 1–11.CrossRefGoogle Scholar
Löwenheim, L. 1967 (1915). “On Possibilities in the Calculus of Relatives.” From Frege to Gödel. van Heijenoort, J. (ed.). Cambridge, MA: Harvard University Press. pp. 228–251.Google Scholar
MacFarlane, J. 2000. “What Does It Mean to Say That Logic Is Formal?” PhD thesis. University of Pittsburgh.Google Scholar
MacFarlane, J. 2015. “Logical Constants.” Stanford Encyclopedia of Philosophy. Zalta, E. N. (ed.). Stanford, CA: The Metaphysics Research Lab.Google Scholar
Maddy, P. 2007. Second Philosophy. Oxford: Oxford University Press.Google Scholar
May, R. 1985. Logical Form: Its Structure and Derivation. Cambridge, MA: The MIT Press.Google Scholar
McCarthy, T. 1981. “The Idea of a Logical Constant.” Journal of Philosophy 78: 499–523.Google Scholar
McGee, V. 1992a. “Review of Etchemendy, The Concept of Logical Consequence.” Journal of Symbolic Logic 57: 254–255.Google Scholar
McGee, V. 1992b. “Two Problems with Tarski’s Theory of Consequence.” Proceedings of the Aristotelian Society 92: 273–292.Google Scholar
McGee, V. 1996. “Logical Operations.” Journal of Philosophical Logic 25: 567–580.Google Scholar
McGee, V. 2004. “Tarski’s Staggering Existential Assumptions.” Synthese 142: 371–387.Google Scholar
Montague, R. 1974. “The Proper Treatment of Quantification in Ordinary English.” Formal Philosophy: Selected Papers. Thomason, R. H. (ed.). New Haven, CT: Yale University Press. pp. 247–270.Google Scholar
Mostowski, A. 1957. “On a Generalization of Quantifiers.” Fundamenta Mathematicae 44: 12–36.Google Scholar
Parsons, C. 1974. “Sets and Classes.” Noûs 8: 1–12.CrossRefGoogle Scholar
Peregrin, J. 2014. Inferentialism: Why Rules Matter. London: Palgrave-Macmillan.CrossRefGoogle Scholar
Peters, S., and Westerståhl, D.. 2006. Quantifiers in Language and Logic. Oxford: Oxford University Press.Google Scholar
Posy, C. 2020. Mathematical Intuitionism. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Priest, G. 1995. “Etchemendy and Logical Consequence.” Canadian Journal of Philosophy 25: 283–292.Google Scholar
Quine, W. V. 1970/1986. Philosophy of Logic. Cambridge, MA: Harvard University Press.Google Scholar
Ray, G. 1996. “Logical Consequence: A Defense of Tarski.” Journal of Philosophical Logic 25: 303–313.Google Scholar
Rescher, N. 1962. “Plurality-Quantification.” Abstract. Journal of Symbolic Logic 27: 373–374.Google Scholar
Resnik, M. D. 1981. “Mathematics as a Science of Patterns: Ontology and Reference.” Noûs 15: 529–550.Google Scholar
Russell, B. 1971 (1919). Introduction to Mathematical Philosophy. New York: Simon & Schuster.Google Scholar
Russell, G. 2018. “Logical Nihilism: Could There Be No Logic?” Philosophical Issues 28: 308–324.CrossRefGoogle Scholar
Russell, G. 2020. “Logic Isn’t Normative.” Inquiry 63: 371–388.Google Scholar
Sagi, G. 2015. “The Modal and Epistemic Arguments against the Invariance Criterion for Logical Terms.” Journal of Philosophy 112: 159–167.Google Scholar
Schurz, G. 1994. “Logical Truth: Comments on Etchemendy’s Critique of Tarski.” Sixty Years of Tarski’s Definition of Truth. Twardowski, B and Woleński, J (eds.). Kraków: Philed. pp. 78–95.Google Scholar
Shapiro, L. 2011. “Deflating Logical Consequence.” Philosophical Quarterly 61: 320–342.Google Scholar
Shapiro, S. 1997. Philosophy of Mathematics. Oxford: Oxford University Press.Google Scholar
Shapiro, S. 1998. “Logical Consequence: Models and Reality.” The Philosophy of Mathematics Today. Schirn, M. (ed.). Oxford: Oxford University Press. pp. 131–156.Google Scholar
Shapiro, S. 2014. Varieties of Logic. Oxford: Oxford University Press.Google Scholar
Sher, G. 1991. The Bounds of Logic. Cambridge, MA: The MIT Press.Google Scholar
Sher, G. 1996. “Did Tarski Commit ‘Tarski’s Fallacy’?” Journal of Symbolic Logic 61: 653–686.Google Scholar
Sher, G. 2001. “The Formal-structural View of Logical Consequence.” Philosophical Review 110: 241–261.CrossRefGoogle Scholar
Sher, G. 2003. “A Characterization of Logical Constants Is Possible.” Theoria 18: 189–197.CrossRefGoogle Scholar
Sher, G. 2008. “Tarski’s Thesis.” New Essays on Tarski and Philosophy. Patterson, D (ed.). Oxford: Oxford University Press. pp. 300–339.Google Scholar
Sher, G. 2016. Epistemic Friction: An Essay on Knowledge, Truth, and Logic. Oxford: Oxford University Press.Google Scholar
Sher, G. 2021. “Invariance and Logicality in Perspective.” The Semantic Conception of Logic: Essays on Consequence, Invariance, and Meaning. Sagi, G. and Woods, J. (eds.). Cambridge: Cambridge University Press.Google Scholar
Skolem, T. 1967 (1920). “A Simplified Proof of a Theorem by L. Löwenheim and Generalizations of the Theorem.” From Frege to Gödel. van Heijenoort, J. (ed.). Cambridge, MA: Harvard University Press. pp. 252–263.Google Scholar
Speitel, S. 2020. Logical Constants between Inference and Reference: An Essay in the Philosophy of Logic. PhD thesis. University of California–San Diego.Google Scholar
Steinberger, F. 2019. “Three Ways in Which Logic Might Be Normative.” Journal of Philosophy 116: 5–31.CrossRefGoogle Scholar
Tarski, A. 1966/1986. “What Are Logical Notions?” History and Philosophy of Logic 7: 143–154.Google Scholar
Tarski, A. 1983 (1933). “The Concept of Truth in Formalized Languages.” Logic, Semantics, Metamathematics. J. Corcoran (ed.). Indianapolis, IN: Hackett. pp. 152–278.Google Scholar
Tarski, A. 1983 (1936a). “On the Concept of Logical Consequence.” Logic, Semantics, Metamathematics. J. Corcoran (ed.). Indianapolis, IN: Hackett. pp. 409–420.Google Scholar
Tarski, A. 1983 [1936b]. “The Establishment of Scientific Semantics.” Logic, Semantics, Metamathematics. J. Corcoran (ed.). Indianapolis, IN: Hackett. pp. 401–408.Google Scholar
Tarski, A., and Vaught, R. L.. 1957. “Arithmetical Extensions of Relational Systems.” Compositio Mathematica 13: 81–102.Google Scholar
Väänänen, J. 2019. “Second-order and Higher-order Logic.” Stanford Encyclopedia of Philosophy. Zalta, E. N. (ed.). Stanford, CA: The Metaphysics Research Lab.Google Scholar
Varzi, A. C. 2002. “On Logical Relativity.” Philosophical Issues 12: 197–219.Google Scholar
Vaught, R. L. 1974. “Model Theory before 1945.” Henkin, L., Addison, J., Chang, C.C., et al. (eds.). Proceedings of the Tarski Symposium. Providence, RI:American Mathematical Society. pp. 153–172.Google Scholar
Whitehead, A. N., and Russell, B.. 1910–1913/1925–1927. Principia Mathematica. Vols. I–III. Cambridge: Cambridge University Press.Google Scholar

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Logical Consequence
  • Gila Sher, University of California, San Diego
  • Online ISBN: 9781108981668
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Logical Consequence
  • Gila Sher, University of California, San Diego
  • Online ISBN: 9781108981668
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Logical Consequence
  • Gila Sher, University of California, San Diego
  • Online ISBN: 9781108981668
Available formats
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