Abstract
This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.
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03 January 2019
The original version of the article unfortunately contained a mistake. The author missed to mention the support by a EU-funded research network that he is involved in. See below. This work was supported by the Marie-Sklodowska-Curie Innovative Training Network DIAPHORA.
References
Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231–33.
Anderson, A.R., & Belnap, N.D. Jr. (1962). Tautological entailments. Philosophical Studies, 13, 9–24.
Anderson, A.R., & Belnap, N.D. Jr. (1975). Entailment: the logic of relevance and necessity Vol. 1. Princeton: Princeton University Press.
Anderson, A.R., Belnap, N.D. Jr., Dunn, J.M. (1992). Entailment: the logic of relevance and necessity Vol. 2. Princeton: Princeton University Press.
Angell, R.B. (1977). Three systems of first degree entailment. The Journal of Symbolic Logic, 42, 147.
Angell, R.B. (2002). A-Logic. Lanham: University Press of America.
Bacon, A. (2013). A new conditional for naive truth theory. Notre Dame Journal of Formal Logic, 54, 87–104.
Barwise, J., & Etchmendy, J. (1989). The liar: an essay in truth and circularity. Oxford: Oxford University Press.
Barwise, J., & Perry, J. (1983). Situations and attitudes. Cambridge: The MIT Press.
Beall, J. (2007). Revenge of the liar. New essays on the paradox. Oxford: Oxford University Press.
Beall, J. (2009). Spandrels of truth. Oxford: Oxford University Press.
Belnap, N.D. (1977). A useful four-valued logic. In Dunn, J.M., & Epstein, G. (Eds.) Modern uses of multiple-valued logic (pp. 8–37). Dordrecht: Reidel.
Berto, F. (2015). A modality called ‘negation’. Mind, 124, 761–93.
Bialynicki-Birula, A., & Rasiowa, H. (1957). On the representation of quasi-Boolean algebras. Bulletin Academie Polen. Sci. Cl., 3, 259–61.
Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–43.
Briggs, R. (2012). Interventionist counterfactuals. Philosophical Studies, 160, 139–66.
Brunner, A.B., & Carnielli, W.A. (2005). Anti-intuitionism and paraconsistency. Journal of Applied Logic, 3, 161–84.
Carnap, R. (1947). Meaning and necessity. Chicago: University of Chicago Press.
Charlwood, G.W. (1981). An axiomatic version of positive semi-lattice relevance logic. Journal of Symbolic Logic, 46, 233–39.
Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40, 55–94.
Cobreros, P., Egré, P., Ripley, D., van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41, 347–85.
Conforti, M., Cornuéjols, G., Zambelli, G. (2014). Integer programming. Heidelberg: Springer.
Cordovil, R., Fukuda, K., Moreira, M.L. (1991). Clutters and matroids. Discrete Mathematics, 89, 161–71.
Correia, F. (2016). On the logic of factual equivalence. The Review of Symbolic Logic, 9, 103–22.
Cresswell, M.J. (1975). Hyperintensional logic. Studia Logica, 34, 25–38.
Dunn, J.M. (1966). The algebra of intensional logics. PhD thesis, University of Pittsburgh.
Dunn, J.M., & Belnap, N.D. Jr. (1975). Intensional algebras. In Anderson, A.R. (Ed.) Entailment (pp. 180–206). Princeton: Princeton University Press.
Dunn, J.M. (1976). Intuitive semantics for first-degree entailments and coupled trees. Philosophical Studies, 29, 149–68.
Dunn, J.M. (1993). Star and perp: two treatments of negation. Philosophical Perspectives, 7, 331–357.
Dunn, J.M. (1996). Generalized ortho negation. In Wansing, H. (Ed.) Negation: a concept in focus (pp. 3–26): de Gruyter.
Dunn, J.M. (1999). A comparative study of various semantical treatments of negation: a history of formal negation. In Gabbay, D., & Wansing, H. (Eds.) What is negation? (pp. 23–51). Dordrecht: Kluwer Academic Publishers.
Dunn, J.M. (2000). Partiality and its dual. Studia Logica, 66, 5–40.
Dunn, J.M., & Restall 2002, G. (2002). Relevance logic. In Guenthner, F., & Gabbay, D. (Eds.) Handbook of philosophical logic (Vol. 6, pp. 1–128). Dordrecht: Kluwer.
Duzi, M., Jespersen, B., Materna, P. (2010). Procedural semantics for hyperintensional logic. foundations and applications of transparent intensional logic, logic, epistemology, and the unity of science Vol. 17. Dordrecht: Springer.
Edmonds, J., & Fulkerson, D.R. (1970). Bottleneck extrema. Journal of Combinatorial Theory, 8, 299–306.
Fefermann, S. (1984). Toward useful type-free theories I. Journal of Symbolic Logic, 49, 75–111.
Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.
Field, H. (2014). Naive truth and restricted quantification: saving truth a whole lot better. Review of Symbolic Logic, 7, 1–45.
Field, H. (forthcoming). Indicative conditionals, restricted quantifiers, and naive truth, forthcoming in Review of Symbolic Logic.
Fine, K. (2012). A guide to ground. In Correia, F., & Schnieder, B. (Eds.) Metaphysical grounding (pp. 37–80). Cambridge: Cambridge University Press.
Fine, K. (2012). Counterfactuals without possible worlds. Journal of Philosophy, 109, 221–46.
Fine, K. (2012). The pure logic of ground. Review of Symbolic Logic, 25, 1–25.
Fine, K. (2014). Truth-maker semantics for intuitionistic logic. Journal of Philosophical Logic, 43, 549–577.
Fine, K. (2014). Permission and possible worlds. Dialectica, 68, 317–36.
Fine, K. (2016). Angellic content. Journal of Philosophical Logic, 45, 199–226.
Fine, K. (2017). Truthmaker semantics. In Hale, B., Wright, C., Miller, A. (Eds.) A companion to the philosophy of language. 2nd edn. (pp. 556–77). Chichester: Wiley.
Fine, K. (forthcoming (a)). A theory of truth-conditional content I: conjunction, disjunction and negation, forthcoming in the Journal of Philosophical Logic.
Fine, K. (forthcoming (b)). A theory of truth-conditional content II: subject-matter, common content, remainder and ground, forthcoming in the Journal of Philosophical Logic.
Fox, C., & Lappin, S. (2001). A framework for the hyperintensional semantics of natural language with two implementations, LACL (logical aspects of computational linguistics), lecture notes in computer science, (pp. 175–92). Dordrecht: Springer.
van Fraassen, B. (1969). Facts and tautological entailments. Journal of Philosophy, 66, 477–87.
van Fraassen, B. (1980). The scientific image. Oxford: Oxford University Press.
Gabbay, D., & Maksimova, L. (2005). Interpolation and definability. Oxford: Clarendon Press.
Galatos, N., Jipsen, P., Kowalski, T., Ono, H. (2007). Residuated lattices: an algebraic glimpse at substructural logics, studies in logic and the foundations of mathematics Vol. 151. Amsterdam: Elsevier.
Gärdenfors, P. (2000). Conceptual spaces: the geometry of thought. Cambridge: The MIT Press.
Gärdenfors, P. (2014). The geometry of meaning: semantics based on conceptual spaces. Cambridge: The MIT Press.
Gödel, K. (1933). Zum intuitionistischen Aussagekalkül. Ergebnisse eines mathematischen Kolloquiums, 4, 40.
Goldblatt, R.I. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 19–35.
Görnemann, S. (1971). A logic stronger than intuitionism. The Journal of Symbolic Logic, 36(/2), 249–261.
Groenendijk, J., & Stokhof, M. (1984). Studies on the semantics of questions and the pragmatics of answers. PhD Thesis, University of Amsterdam.
Gupta, A., & Standefer, S. (2017). Conditionals in theories of truth. Journal of Philosophical Logic, 46, 27–63.
Gurevich, Y. (1977). Intuitionistic logic with strong negation. Studia Logica, 36, 49–59.
Halbach, V. (2011). Axiomatic theories of truth. Cambridge: Cambridge University Press.
Halbach, V., & Horsten, L. (2006). Axiomatizing Kripkes theory of truth. Journal of Symbolic Logic, 71, 677–712.
Halpern, J.Y. (2003). Reasoning about uncertainty. Cambridge: The MIT Press.
Hansen, J., Pigozzi, G., van der Torre, L. (2007). Ten philosophical problems in deontic logic. In Boella, G. et al. (Eds.) Normative multi-agent systems. Dagstuhl: Dagstuhl Seminar Proceedings. http://drops.dagstuhl.de/opus/volltexte/2007/941.
Hermes, H. (1967). Einführung in die Verbandstheorie. Berlin: Springer.
Holliday, W., Hoshi, T., Icard, T. (2012). Uniform logic of information dynamics. In Bolander, T. et al. (Eds.) Advances in modal logic (Vol. 9, pp. 348–367). London: College Publications.
Hornischer, L. (2017). Hyperintensionality and synonymy. A logical, philosophical, and cognitive investigation, MSc Thesis, Institute for Logic. Language and Computation, University of Amsterdam.
Horsten, L. (2011). The Tarskian turn. Deflationism and axiomatic truth. Cambridge: MIT Press.
Jago, M. (2014). The impossible. An essay on hyperintensionality. Oxford: Oxford University Press.
Jennings, R.E., & Chen, Y. (2013). FDE: a logic of clutters. In Tanaka, K., Berto, F., Mares, E., Paoli, F. (Eds.) Paraconsistency: logic and applications (pp. 163–72). Dordrecht: Springer.
Kalman, J.A. (1958). Lattices with involution. Transactions of the American Mathematical Society, 87, 485–91.
Kamp, H. (1973). Free choice permission. Proceedings of the Aristotelian Society, 74, 57–74.
Kapsner, A. (2014). Logics and falsifications. A new perspective on constructivist semantics, trends in logic Vol. 40. Heidelberg: Springer.
Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press.
Kraus, S., Lehmann, D., Magidor, M. (1990). Non-monotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207.
Kripke, S.A. (1965). Semantical analysis of intuitionistic logic. In Crossley, J., & Dummett, M.A.E. (Eds.) Formal systems and recursive functions (pp. 92–130). Amsterdam: North-Holland.
Kripke, S.A. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.
von Kutschera, F. (1969). Ein verallgemeinerter Widerlegungsbegriff für Gentzenkalküle. Archiv für Mathematische Logik und Grundlagenforschung, 12, 104–18.
Leitgeb, H. (1999). Truth and the liar in De Morgan-valued models. Notre Dame Journal of Formal Logic, 40, 496–514.
Leitgeb, H. (2005). What truth depends on. Journal of Philosophical Logic, 34, 155–92.
Leitgeb, H. (2007). On the metatheory of field’s ‘solving the paradoxes, escaping revenge’. Beall, 2007, 159–83.
Leitgeb, H. (2010). On the Ramsey test without triviality. Notre Dame Journal of Formal Logic, 51, 21–54.
Leitgeb, H. (2017). The stability of belief. How rational belief coheres with probability. Oxford: Oxford University Press.
Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.
Lewis, D. (1988). Statements partly about observation. Philosophical Papers, 17, 1–31.
Makinson, D. (1993). Five faces of minimality. Studia Logica, 52, 339–79.
Mares, E.D. (1997). Relevant logic and the theory of information. Synthese, 109, 345–360.
Mares, E.D. (2004). Relevant logic: a philosophical interpretation. Cambridge: Cambridge University Press.
Mares, E.D. (2013). Information, negation, and paraconsistency. In Tanaka, K., Berto, F., Mares, E., & Paoli, F. (Eds.) Paraconsistency: logic and applications (pp. 43–55). Dordrecht: Springer.
Martin, R.L., & Woodruff, P.W. (1975). On representing ‘True-in-L’ in L. Philosophia, 5, 213–7.
Moltmann, F. (2007). Events, tropes and truthmaking. Philosophical Studies, 134, 363–403.
Moschovakis, J. (2005). Notes on the foundations of constructive mathematics, unpublished manuscript, available at: http://www.math.ucla.edu/~joan/newnotes.pdf.
Moschovakis, Y.N. (2006). A logical calculus of meaning and synonymy. Linguistics and Philosophy, 29, 27–89.
Muskens, R. (1995). Meaning and partiality. CSLI Publications: Stanford.
Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic, 14, 16–26.
Nolan, D. (2014). Hyperintensional metaphysics. Philosophical Studies, 171, 149–160.
Odintsov, S.P., Speranski, S.O., Shevchenko, I.Yu. (2018). Hintikka’s independence-friendly logic meets Nelson’s realizability. Studia Logica, 106, 637–70.
Perry, J. (1989). Possible worlds and subject matter. In Perry, J. (Ed.) The problem of the essential indexical and other essays (pp. 145–60). CSLI Publications: Stanford.
Pietz, A., & Rivieccio, U. (2013). Nothing but the truth. Journal of Philosophical Logic, 42, 125–35.
Pollard, C. (2008). Hyperintensions. Journal of Logic and Computation, 18, 257–82.
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8, 219–41.
Priest, G. (1987). In contradiction. A study of the transconsistent. Dordrecht: Martinus Nijhoff.
Priest, G. (2002). Paraconsistent logic. In Gabbay, D., & Guenther, F. (Eds.) Handbook of philosophical logic. 2nd Edn. (Vol. 6, pp. 287–393). Dordrecht: Kluwer.
Priest, G. (2008). An introduction to non-classical logic. From if to is. Cambridge: Cambridge University Press.
Punčochář, V. (Unpublished). Substructural inquisitive logics, unpublished manuscript.
Rasiowa, H., & Sikorski, R. (1963). The mathematics of metamathematics. Państwowe Wydawnictwo Naukowe: Warsaw.
Rauszer, C., & Sabalski, B. (1975). Notes on the Rasiowa-Sikorski lemma. Studia Logica, 34, 265–8.
Read, S. (1988). Relevant logic. Oxford: Blackwell.
Restall, G. (1995). Four-valued semantics for relevant logics (and some of their rivals. Journal of Philosophical Logic, 24, 139–60.
Restall, G. (1996). Information flow and relevant logics. In Seligman, J., & Westerstahl, D. (Eds.) Logic, language and computation (pp. 463–77). Stanford: CSLI.
Restall, G. (2000). Modelling truthmaking. Logique et Analyse, 169–170, 211–30.
Rossi, L. (2016). Adding a conditional to Kripke’s theory of truth. Journal of Philosophical Logic, 45, 485–529.
Van Rooij, R. (2017). A fine-grained global analysis of implicatures. In Pistioa-Reda, S., & Domaneschi, F. (Eds.) Linguistic and psycholinguistic approaches on implicatures and presuppositions (pp. 73–110). Cham: Palgrave Macmillan.
Routley, R., & Routley, V. (1972). Semantics of first-degree entailment. Nous, 3, 335–59.
Routley, R., Plumwood, V., Meyer, R.K., Brady, R.T. (1982). Relevant logics and their rivals, Ridgeview Press.
Schindler, T. (2014). Axioms for grounded truth. Review of Symbolic Logic, 7, 73–83.
Schnieder, B. (2011). A logic for ‘because’. Review of Symbolic Logic, 4, 445–65.
Seymour, P.D. (1976). The forbidden minors of binary clutters. Journal of the London Mathematical Society (2), 12, 356–60.
Thomason, R.H. (1969). A semantical study of constructible falsity. Zeitschrift for mathematische Logik und Grundlagen der Mathematik, 15, 247–57.
Urquhart, A. (1972). Semantics for relevant implication. Journal of Symbolic Logic, 37, 15–69.
Veltman, F. (1985). Logics for conditionals. PhD Thesis, University of Amsterdam (UvA).
Visser, A. (1984). Four valued semantics and the liar. Journal of Philosophical Logic, 13, 181–212.
Wansing, H. (1993). The logic of information structures. Berlin: Springer.
Williamson, T. (1994). Never say never. Topoi, 13, 135–145.
Woodruff, P.W. (1984). Paradox, truth and logic I. Journal of Philosophical Logic, 13, 213–232.
Wright, C. (2001). On being in a quandary: relativism, vagueness, logical revisionism. Mind, 110, 45–98.
Yablo, S. (1982). Grounding, dependence, and paradox. Journal of Philosophical Logic, 11, 117–37.
Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.
Yablo, S. (2016). Ifs, ands, and buts: an incremental truthmaker-semantics for indicative conditionals. Analytic Philosophy, 57, 175–213.
Yolov, N. (2016). Unpublished: blocker size via matching minors, unpublished manuscript (arXiv:1606.06263).
Zalta, E.N. (1983). Abstract objects: an introduction to axiomatic metaphysics. Dordrecht: Reidel.
Acknowledgements
I am grateful to Albert Anglberger, JC Beall, Johan van Benthem, Franz Berto, Walter Carnielli, Lucas Champollion, Ivano Ciardelli, J. Michael Dunn, Hartry Field, Martin Fischer, Volker Halbach, Ole Hjortland, Levin Hornischer, Leon Horsten, Johannes Korbmacher, David Makinson, Ed Mares, Joan R. Moschovakis, Yiannis Moschovakis, Julien Murzi, Carlo Nicolai, Lavinia Picollo, Andreas Pietz, Graham Priest, Greg Restall, Floris Roelofsen, Lorenzo Rossi, Benjamin Schnieder, Moritz Schulz, Stanislav Speranski, Andrew Tedder, Frank Veltman, Albert Visser, Philip Welch, Steve Yablo, and an anonymous reviewer for comments on earlier versions of this paper. This work was supported by an Alexander von Humboldt Professorship grant by the Alexander von Humboldt Foundation.
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Leitgeb, H. HYPE: A System of Hyperintensional Logic (with an Application to Semantic Paradoxes). J Philos Logic 48, 305–405 (2019). https://doi.org/10.1007/s10992-018-9467-0
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DOI: https://doi.org/10.1007/s10992-018-9467-0