Element contents
Series: Elements in Philosophy and Logic
Proofs and Models in Philosophical Logic
Published online by Cambridge University Press: 25 March 2022
Summary
This Element is an introduction to recent work proofs and models in philosophical logic, with a focus on the semantic paradoxes the sorites paradox. It introduces and motivates different proof systems and different kinds of models for a range of logics, including classical logic, intuitionistic logic, a range of three-valued and four-valued logics, and substructural logics. It also compares and contrasts the different approaches to substructural treatments of the paradox, showing how the structural rules of contraction, cut and identity feature in paradoxical derivations. It then introduces model theoretic treatments of the paradoxes, including a simple fixed-point model construction which generates three-valued models for theories of truth, which can provide models for a range of different non-classical logics. The Element closes with a discussion of the relationship between proofs and models, arguing that both have their place in the philosophers' and logicians' toolkits.
- Type
- Element
- Information
- Series: Elements in Philosophy and LogicOnline ISBN: 9781009040457Publisher: Cambridge University PressPrint publication: 21 April 2022
References
Barrio, E. , Rosenblatt, L. , & Tajer, D. (2014). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.CrossRefGoogle Scholar
Barrio, E. A. , Pailos, F. , & Szmuc, D. (2019). (Meta)inferential levels of entailment beyond the Tarskian paradigm. Synthese. Online first.Google Scholar
Beall, J. et al. (2012). On the ternary relation and conditionality. Journal of Philosophical Logic, 41(3), 595–612.CrossRefGoogle Scholar
Beall, J. & van Fraassen, B. (2003). Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic. Oxford: Oxford University Press.Google Scholar
Belnap, N. D. (1977a). How a computer should think. In Ryle, G (ed.), Contemporary Aspects of Philosophy (pp. 30–55). Boston: Oriel Press.Google Scholar
Belnap, N. D. (1977b). A useful four-valued logic. In Dunn, J & Epstein, G (eds.), Modern Uses of Multiple-Valued Logics (pp. 8–37). Dordrecht: Reidel.Google Scholar
Berto, F. & Restall, G. (2019). Negation on the Australian plan. Journal of Philosophical Logic, 48(6), 1119–1144.CrossRefGoogle Scholar
Blackburn, P. , de Rijke, M. , & Venema, Y. (2001). Modal Logic. Cambridge University Press.CrossRefGoogle Scholar
Blamey, S. (1986). Partial logic. In Gabbay, D & Guenthner, F (eds.), Handbook of Philosophical Logic, volume III (pp. 261–353). Dordrecht: D. Reidel.Google Scholar
Brady, R. T. (1971). The consistency of the axioms of abstraction and extensionality in a three-valued logic. Notre Dame Journal of Formal Logic, 12, 447–453.CrossRefGoogle Scholar
Brandom, R. B. (2000). Articulating Reasons: An Introduction to Inferentialism. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar
Brouwer, L. E. J. (1913). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20, 91–96. Reprinted as Brouwer (1999).CrossRefGoogle Scholar
Brouwer, L. E. J. (1999). Intuitionism and formalism. Bulletin of the American Mathematical Society, 37(1), 55–64. Reprint of Brouwer (1913).CrossRefGoogle Scholar
Chellas, B. F. (1980). Modal Logic: An Introduction. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Cobreros, P. , Egré, P. , Ripley, D. , & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.CrossRefGoogle Scholar
Cobreros, P. , Egré, P. , Ripley, D. , & van Rooij, R. (2015). Vagueness, truth and permissive consequence. In Achourioti, T, Galinon, H, Martínez Fernández, J, & Fujimoto, K (eds.), Unifying the Philosophy of Truth (pp. 409–430). Dordrecht: Springer Netherlands.CrossRefGoogle Scholar
Coffa, J. A. (1993). The Semantic Tradition from Kant to Carnap. Cambridge, UK: Cambridge University Press. Edited by Linda, Wessels.Google Scholar
Copeland, B. J. (1979). On when a semantics is not a semantics: some reasons for disliking the Routley-Meyer semantics for relevance logic. Journal of Philosophical Logic, 8(1), 399–413.CrossRefGoogle Scholar
Dunn, J. M. (1976). Intuitive semantics for first-degree entailments and ‘coupled trees’. Philosophical Studies, 29(3), 149–168.CrossRefGoogle Scholar
Dunn, J. M. & Restall, G. (2002). Relevance logic. In Gabbay, D. M (ed.), Handbook of Philosophical Logic, volume 6 (pp. 1–136). Dordrecht: Kluwer Academic Publishers, second edition.Google Scholar
Égré, P. (2021). Half-truths and the liar. In Nicolai, C & Stern, J (eds.), Modes of Truth: The Unified Approach to Truth, Modality and Paradox (pp. 18–40). London: Routledge.CrossRefGoogle Scholar
Fjellstand, A. (2015). How a semantics for tonk should be. The Review of Symbolic Logic, 8(3), 488–505.CrossRefGoogle Scholar
French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, an Open Access Journal of Philosophy, 3, 113–31.CrossRefGoogle Scholar
French, R. & Ripley, D. (2018). Valuations: Bi, tri, and tetra. Studia Logica, 107(6), 1313–1346.CrossRefGoogle Scholar
Gamut, L. T. F. (1991). Logic, Language, and Meaning: Volume 2, Intensional Logic and Logical Grammar. Chicago: University of Chicago Press.Google Scholar
Genesereth, M. & Kao, E. J. (2016). Introduction to Logic. Morgan & Claypool Publishers LLC.Google Scholar
Gentzen, G. (1935a). Untersuchungen über das logische schließen. I. Mathematische Zeitschrift, 39(1), 176–210.CrossRefGoogle Scholar
Gentzen, G. (1935b). Untersuchungen über das logische schließen. II. Mathematische Zeitschrift, 39(1), 405–431.CrossRefGoogle Scholar
Gentzen, G. (1969). The Collected Papers of Gerhard Gentzen. Amsterdam: North Holland.Google Scholar
Gilmore, P. C. (1974). The consistency of partial set theory without extensionality. In Scott, Dana S (ed.), Axiomatic Set Theory, volume 13 of Proceedings of Symposia in Pure Mathematics (pp. 147–153). Providence, Rhode Island: American Mathematical Society.Google Scholar
Gupta, A. & Belnap, N. (1993). The Revision Theory of Truth. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Gupta, A. & Standefer, S. (2017). Conditionals in theories of truth. Journal of Philosophical Logic, 46, 27–63.CrossRefGoogle Scholar
Gupta, A. & Standefer, S. (2018). Intersubstitutivity principles and the generalization function of truth. Synthese, 195(3), 1065–1075.CrossRefGoogle Scholar
Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Horsten, L. (2011). The Tarskian Turn: Deflationism and Axiomatic Truth. Cambridge, MA: The MIT Press.CrossRefGoogle Scholar
Hughes, G. & Cresswell, M. (1996). A New Introduction to Modal Logic. London: Routledge.CrossRefGoogle Scholar
Kripke, S. (1975). Outline of a theory of truth. The Journal of Philosophy, 72(19), 690–716.CrossRefGoogle Scholar
Lance, M. & White, W. H. (2007). Stereoscopic vision: Persons, freedom, and two spaces of material inference. Philosophers’ Imprint, 7(4), 1–21.Google Scholar
Martin, R. L. & Woodruff, P. W. (1975). On representing ‘true-in-L’ in L. Philosophia (Israel), 5, 213–217.CrossRefGoogle Scholar
Milne, P. (2002). Harmony, purity, simplicity and a ‘seemingly magical fact’. Monist, 85(4), 498–534.CrossRefGoogle Scholar
Pelletier, F. J. (1999). A brief history of natural deduction. History and Philosophy of Logic, 20(1), 1–31.CrossRefGoogle Scholar
Petersen, U. (2000). Logic without contraction as based on inclusion and unrestricted abstraction. Studia Logica, 64(3), 365–403.CrossRefGoogle Scholar
Petersen, U. (2003). LiDZλ as a basis for PRA. Archive for Mathematical Logic, 42(7), 665–694.CrossRefGoogle Scholar
Poggiolesi, F. (2008). A cut-free simple sequent calculus for modal logic s5. Review of Symbolic Logic, 1, 3–15.CrossRefGoogle Scholar
Poggiolesi, F. (2009). The method of tree-hypersequents for modal propositional logic. In Makinson, D, Malinowski, J, & Wansing, H (eds.), Towards Mathematical Philosophy, volume 28 (pp. 31–51). Dordrecht: Springer Netherlands.CrossRefGoogle Scholar
Poggiolesi, F. (2010). Gentzen Calculi for Modal Propositional Logic. Trends in Logic. Dordrecht: Springer.Google Scholar
Poggiolesi, F. & Restall, G. (2012). Interpreting and applying proof theories for modal logic. In Restall, G & Russell, G (eds.), New Waves in Philosophical Logic (pp. 39–62). Basingstoke, UK: Palgrave Macmillan.CrossRefGoogle Scholar
Prawitz, D. (1965). Natural Deduction: A Proof Theoretical Study. Stockholm: Almqvist and Wiksell.Google Scholar
Prawitz, D. (1973). Towards a foundation of general proof theory. In Suppes, P, Henkin, L, Joja, A, & Moisil, G. C (eds.), Logic, Methodology and Philosophy of Science IV (pp. 225–250). Amsterdam: North Holland.Google Scholar
Prawitz, D. (1974). On the idea of a general proof theory. Synthese, 27, 63–77.CrossRefGoogle Scholar
Prawitz, D. (2019). The fundamental problem of general proof theory. Studia Logica, 107(1), 11–29.CrossRefGoogle Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8(1), 219–241.CrossRefGoogle Scholar
Read, S. (2008). Harmony and modality. In Dégremont, C, Kieff, L, & Rückert, H (eds.), Dialogues, Logics and Other Strange Things: Essays in Honour of Shahid Rahman (pp. 285–303). London: College Publications.Google Scholar
Read, S. (2015). Semantic pollution and syntactic purity. The Review of Symbolic Logic, 8(4), 649–661.CrossRefGoogle Scholar
Restall, G. (2000). An Introduction to Substructural Logics. London: Routledge.CrossRefGoogle Scholar
Restall, G. (2005). Multiple conclusions. In Hájek, P, Valdés-Villanueva, L, & Westerståhl, D (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress (pp. 189–205). London: kcl Publications.Google Scholar
Restall, G. (2009). Truth values and proof theory. Studia Logica, 92(2), 241–264.CrossRefGoogle Scholar
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), 1611–1623.CrossRefGoogle Scholar
Restall, G. (2013). Assertion, denial and non-classical theories. In Tanaka, K, Berto, F, Mares, E, & Paoli, F (eds.), Paraconsistency: Logic and Applications (pp. 81–99). Dordrecht: Springer.CrossRefGoogle Scholar
Restall, G. (2019). Generality and existence 1: Quantification and free logic. Review of Symbolic Logic, 12, 1–29.CrossRefGoogle Scholar
Restall, G. & Standefer, S. (2021). Collection frames for substructural logics. Paper in progress.Google Scholar
Restall, G. & Standefer, S. (2022). Logical Methods. Cambridge, MA: MIT Press. In press.Google Scholar
Ripley, D. (2011). Contradictions at the borders. In Nouwen, R, van Rooij, R, Sauerland, U, & Schmitz, H.-C (eds.), Vagueness in Communication (pp. 169–188). Berlin, Heidelberg: Springer Berlin Heidelberg.CrossRefGoogle Scholar
Ripley, D. (2015b). Comparing substructural theories of truth. Ergo, an Open Access Journal of Philosophy, 2(20190926), 299–328.CrossRefGoogle Scholar
Ripley, D. (2015c). ‘Transitivity’ of consequence relations. In van der Hoek, W, Holliday, W, & Wang, W (eds.), Logic, Rationality, and Interaction (pp. 328–340). Berlin, Heidelberg: Springer Berlin Heidelberg.CrossRefGoogle Scholar
Ripley, D. (2017). Bilateralism, coherence, warrant. In Moltmann, F & Textor, M (eds.), Act-Based Conceptions of Propositional Content (pp. 307–324). Oxford: Oxford University Press.Google Scholar
Routley, R. & Meyer, R. K. (1973). Semantics of entailment. In Leblanc, H (Ed.), Truth, Syntax and Modality (pp. 194–243). North Holland. Proceedings of the Temple University Conference on Alternative Semantics.Google Scholar
Routley, R. & Routley, V. (1972). Semantics of first degree entailment. Noûs, 6(4), 335–359.CrossRefGoogle Scholar
Smullyan, R. M. (1968). First-Order Logic. Berlin: Springer-Verlag Reprinted by Dover Press, 1995.CrossRefGoogle Scholar
Standefer, S. (2015). Solovay-type theorems for circular definitions. The Review of Symbolic Logic, 8(3), 467–487.CrossRefGoogle Scholar
Standefer, S. (2016). Contraction and revision. The Australasian Journal of Logic, 13(3), 58–77.CrossRefGoogle Scholar
Steinberger, F. (2011). Why conclusions should remain single. Journal of Philosophical Logic, 40(3), 333–355.CrossRefGoogle Scholar
Stevenson, J. T. (1961). Roundabout the runabout inference-ticket. Analysis, 21(6), 124–128.CrossRefGoogle Scholar
Van Dalen, D. (1986). Intuitionistic logic. In Gabbay, D & Guenthner, F (eds.), Handbook of Philosophical Logic, volume III (pp. 225–339). Dordrecht: D. Reidel.CrossRefGoogle Scholar
Von Plato, J. (2001). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40, 541–567.CrossRefGoogle Scholar
Williamson, T. (2013). Modal Logic as Metaphysics. Oxford: Oxford University Press.CrossRefGoogle Scholar
Zach, R. (1999). Completeness before Post: Bernays, Hilbert, and the development of propositional logic. Bulletin of Symbolic Logic, 5(3), 331–366.CrossRefGoogle Scholar
Zach, R. (2019). Hilbert’s Program. In Zalta, E. N (ed.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford, CA: Stanford University, Fall 2019 edition.Google Scholar
Zardini, E. (2011). Truth without contra(di)ction. The Review of Symbolic Logic, 4(4), 498–535.CrossRefGoogle Scholar
- 1
- Cited by