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Possibility Semantics (2405.06852v1)

Published 10 May 2024 in math.LO and cs.LO

Abstract: In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness--a world makes a disjunction true only if it makes one of the disjuncts true--which classically implies totality--for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.

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References (175)
  1. A Kripke semantics for the logic of Gelfand quantales. Studia Logica 68 (2001), 173–228.
  2. Ultraproducts of continuous posets. Algebra Universalis 76, 2 (2016), 231–235.
  3. First-order classical modal logic. Studia Logica 84, 2 (2006), 171–210.
  4. Provability logic. In Handbook of Philosophical Logic, D. Gabbay and F. Guenthner, Eds., vol. 13. Springer, Dordrecht, 2005, pp. 229–403.
  5. Beklemishev, L. D. Reflection principles and provability algebras in formal arithmetic. Russian Mathematical Surveys 60, 2 (2005), 197–268.
  6. Bell, J. L. Set Theory: Boolean-Valued Models and Independence Proofs. Oxford Logic Guides. Clarendon Press, Oxford, 2005.
  7. Models and Ultraproducts: An Introduction. North-Holland, Amsterdam, 1974.
  8. van Benthem, J. Syntactic aspects of modal incompleteness theorems. Theoria 45, 2 (1979), 63–77.
  9. van Benthem, J. Points and periods. In Time, Tense, and Quantifiers, C. Rohrer, Ed. Niemeyer Verlag, Tübingen, 1981, pp. 39–58.
  10. van Benthem, J. Possible worlds semantics for classical logic. Tech. Rep. ZW-8018, Department of Mathematics, Rijksuniversiteit, Groningen, December 1981.
  11. van Benthem, J. The Logic of Time: A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse. D. Reidel Publishing Company, Dordrecht, 1983.
  12. van Benthem, J. Modal Logic and Classical Logic. Bibliopolis, Milan, 1983.
  13. van Benthem, J. Partiality and Nonmonotonicity in Classical Logic. Logique et Analyse 29 (1986), 225–247.
  14. van Benthem, J. A Manual of Intensional Logic, 2nd revised and expanded ed. CSLI Publications, Stanford, 1988.
  15. van Benthem, J. Correspondence theory. In Handbook of Philosophical Logic, D. M. Gabbay and F. Guenthner, Eds., 2nd ed., vol. 3. Springer, Dordrecht, 2001, pp. 325–408.
  16. van Benthem, J. Tales from an old manuscript. Tech. Rep. PP-2016-30, ILLC, University of Amsterdam, 2016.
  17. van Benthem, J. Constructive agents. Indagationes Mathematicae 29, 1 (2018), 23–35. Special issue on “L.E.J. Brouwer, fifty years later”.
  18. van Benthem, J. Implicit and explicit stances in logic. Journal of Philosophical Logic 48 (2019), 571–601.
  19. A bimodal perspective on possibility semantics. Journal of Logic and Computation 27, 5 (2017), 1353–1389.
  20. Beth, E. W. Semantic construction of intuitionistic logic. Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen 19, 11 (1956), 357–388.
  21. Leo Esakia on Duality in Modal and Intuitionistic Logic. Springer, Dordrecht, 2017.
  22. Funayama’s theorem revisited. Algebra Universalis 70 (2013), 271–286.
  23. Locales, nuclei, and Dragalin frames. In Advances in Modal Logic, L. Beklemishev, S. Demri, and A. Máté, Eds., vol. 11. College Publications, London, 2016, pp. 177–196.
  24. A semantic hierarchy for intuitionistic logic. Indagationes Mathematicae 30, 3 (2019), 403–469. Special issue on “L.E.J. Brouwer, fifty years later”.
  25. Algebraic and topological semantics for inquisitive logic via choice-free duality. In Logic, Language, Information, and Computation. WoLLIC 2019 (Berlin, 2019), vol. 11541 of LNCS, Springer, pp. 35–52.
  26. Topological possibility frames. In Advances in Modal Logic 2018, Accepted Short Papers. 2019, pp. 16–20.
  27. Choice-free Stone duality. The Journal of Symbolic Logic 85, 1 (2020), 109–148.
  28. Birkhoff, G. Rings of sets. Duke Mathematical Journal 3, 3 (1937), 443–454.
  29. Birkhoff, G. Lattice Theory, 3rd ed. American Mathematical Society, Providence, RI, 1967.
  30. Modal Logic. Cambridge University Press, New York, 2001.
  31. Blok, W. J. Varieties of interior algebras. PhD thesis, University of Amsterdam, 1976.
  32. Boolos, G. The Logic of Provability. Cambridge University Press, Cambridge, 1993.
  33. Bradley, S. Imprecise probabilities. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed. 2019.
  34. Fitch’s Paradox of Knowability. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed., Fall 2019 ed. Metaphysics Research Lab, Stanford University, 2019.
  35. Bull, R. A. On modal logic with propositional quantifiers. The Journal of Symbolic Logic 34, 2 (1969), 257–263.
  36. Cariani, F. Modeling future indeterminacy in possibility semantics. Manuscript, 2021.
  37. Modal Logic. Clarendon Press, Oxford, 1997.
  38. On the semantics and logic of declaratives and interrogatives. Synthese 192 (2015), 1689–1728.
  39. Inquisitive Semantics. Oxford University Press, 2018.
  40. Inquisitive Logic. Journal of Philosophical Logic 40 (2011), 55–94.
  41. Ciardelli, I. A. Inquisitive semantics and intermediate logics. Master’s thesis, University of Amsterdam, 2009. ILLC Master of Logic Thesis Series MoL-2009-11.
  42. Ciardelli, I. A. Questions in Logic. PhD thesis, University of Amsterdam, 2016.
  43. Davey, B. A. On the lattice of subvarieties. Houston Journal of Mathematics 5, 2 (1979), 183–192.
  44. Logic and Probability. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed., Summer 2019 ed. Metaphysics Research Lab, Stanford University, 2019.
  45. Ding, Y. On the logics with propositional quantifiers extending S5ΠΠ\Piroman_Π. In Advances in Modal Logic, G. Bezhanishvili, G. D’Agostino, G. Metcalfe, and T. Studer, Eds., vol. 12. College Publications, London, 2018, pp. 219–235.
  46. Ding, Y. On the logic of belief and propositional quantification. Journal of Philosophical Logic 50 (2021), 1143–1198.
  47. Another problem in possible world semantics. In Advances in Modal Logic, N. Olivetti, R. Verbrugge, S. Negri, and G. Sandu, Eds., vol. 13. College Publications, London, 2020, pp. 149–168.
  48. Doering, A. Topos-based logic for quantum systems and bi-Heyting algebras. arXiv:1202.2750, 2012.
  49. Došen, K. Duality between modal algebras and neighborhood frames. Studia Logica 48, 2 (1989), 219–234.
  50. Dragalin, A. G. Matematicheskii Intuitsionizm: Vvedenie v Teoriyu Dokazatelstv. Matematicheskaya Logika i Osnovaniya Matematiki. “Nauka”, Moscow, 1979.
  51. Dragalin, A. G. Mathematical Intuitionism: Introduction to Proof Theory, vol. 67 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1988.
  52. Modal logics between S4 and S5. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 5 (1959), 250–264.
  53. Mathematical Logic, second ed. Springer, New York, 1994.
  54. Enderton, H. B. A Mathematical Introduction to Logic. Harcourt Academic Press, 2001.
  55. Esakia, L. Heyting Algebras I. Duality Theory. Metsniereba Press, Tbilisi, 1985. (Russian).
  56. Esakia, L. Heyting Algebras. Duality Theory. Springer, Cham, 2019. English translation by A. Evseev, eds. G. Bezhanishvili and W. H. Holliday.
  57. Propositional Lax Logic. Information and Computation 137, 1 (1997), 1–33.
  58. Feferman, S. Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae 49, 1 (1960), 35–92.
  59. Feferman, S. Some applications of the notions of forcing and generic sets. Fundamenta Mathematicae 56, 3 (1964), 325–345.
  60. Fine, K. Propositional quantifiers in modal logic. Theoria 36, 3 (1970), 336–346.
  61. Fine, K. Vagueness, Truth and Logic. Synthese 30, 3-4 (1975), 265–300.
  62. Fine, K. First-order modal theories III: Facts. Synthese 53, 1 (1982), 43–122.
  63. Fine, K. Truthmaker semantics. In A Companion to the Philosophy of Language, B. Hale, C. Wright, and A. Miller, Eds., 2nd ed., Blackwell Companions to Philosophy. John Wiley & Sons Ltd, West Sussex, 2017, pp. 556–577.
  64. Fitch, F. B. A logical analysis of some value concepts. The Journal of Symbolic Logic 28, 2 (1963), 135–142.
  65. Fitting, M. Intuitionistic Logic, Model Theory, and Forcing. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1969.
  66. First-Order Modal Logic. Springer, Dodrecht, 1998.
  67. Term-modal logics. Studia Logica 69, 1 (2001), 133–169.
  68. Flori, C. A Second Course in Topos Quantum Theory. Springer, Cham, 2018.
  69. Sheaves and logic. In Applications of Sheaves, M. P. Fourman, C. J. Mulvey, and D. S. Scott, Eds. Springer, Berlin, 1979, pp. 302–401.
  70. Frege, G. Über Sinn und Bedeutung. Zeitschrift fúr Philosophie und philosophische Kritik 100 (1892), 25–50.
  71. Fritz, P. On Stalnaker’s simple theory of propositions. Journal of Philosophical Logic 50 (2021), pages 1–31.
  72. A negative solution of Kuznetsov’s problem for varieties of bi-Heyting algebras. arXiv:2104.05961 [math.LO], 2021.
  73. Quantification in Nonclassical Logic: Volume 1, vol. 153 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam, 2009.
  74. Garson, J. W. What Logics Mean: From Proof Theory to Model-Theoretic Semantics. Cambridge University Press, Cambridge, 2013.
  75. Bounded lattice expansions. Journal of Algebra 238, 1 (2001), 345–371.
  76. Introduction to Boolean Algebras. Springer, New York, 2009.
  77. Goldblatt, R. Metamathematics of Modal Logic. PhD thesis, Victoria University, Wellington, 1974. Reprinted in [78].
  78. Goldblatt, R. Mathematics of Modality. CSLI Publications, Stanford, CA, 1993.
  79. Goldblatt, R. Maps and monads for modal frames. Studia Logica 83, 1 (2006), 309–331.
  80. Goldblatt, R. I. Semantic analysis of orthologic. Journal of Philosophical Logic 3, 1 (1974), 19–35.
  81. Goldblatt, R. I. The Stone space of an ortholattice. Bulletin of the London Mathematical Society 7, 1 (1975), 45–48.
  82. Temporal logic. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed., summer 2020 ed. Metaphysics Research Lab, Stanford University, 2020.
  83. Naming and identity in epistemic logics part I: The propositional case. Journal of Logic and Computation 3, 4 (1993), 345–378.
  84. Grzegorczyk, A. A philosophically plausible formal interpretation of intuitionistic logic. Indagationes Mathematicae 26 (1964), 596–601.
  85. Hale, B. Necessary Beings. Oxford University Press, Oxford, 2013.
  86. The Boolean prime ideal theorem does not imply the axiom of choice. In Axiomatic Set Theory, D. S. Scott, Ed., vol. 13 of Proceedings of Symposia in Pure Mathematics. American Mathematical Society, Providence, RI, 1971, pp. 83–134.
  87. Knowledge and common knowledge in a distributed environment. Journal of the ACM (JACM) 37, 3 (1990), 549–587.
  88. Reasoning about knowledge of unawareness. Games and Economic Behavior 67, 2 (2009), 503–525.
  89. A propositional modal logic of time intervals. Journal of the ACM 38, 4 (1991), 935–962.
  90. Harrison-Trainor, M. A representation theorem for possibility models. UC Berkeley Working Paper in Logic and the Methodology of Science, http://escholarship.org/uc/item/881757qn, 2017.
  91. Harrison-Trainor, M. First-order possibility models and finitary completeness proofs. The Review of Symbolic Logic 12, 4 (2019), 637–662.
  92. Henkin, L. A. Metamathematical theorems equivalent to the prime ideal theorem for Boolean algebra. Bulletin of the American Mathematical Society 60, 4 (1954), 387–388.
  93. Hochster, M. Prime ideal structure in commutative rings. Transactions of the American Mathematical Society 142 (1969), 43–60.
  94. Holliday, W. H. Partiality and adjointness in modal logic. In Advances in Modal Logic, Vol. 10, R. Goré, B. Kooi, and A. Kurucz, Eds. College Publications, London, 2014, pp. 313–332.
  95. Holliday, W. H. Possibility frames and forcing for modal logic. UC Berkeley Working Paper in Logic and the Methodology of Science, http://escholarship.org/uc/item/5462j5b6, 2015.
  96. Holliday, W. H. On the modal logic of subset and superset: Tense logic over Medvedev frames. Studia Logica 105 (2017), 13–35.
  97. Holliday, W. H. Possibility frames and forcing for modal logic (February 2018). Forthcoming in The Australasian Journal of Logic, UC Berkeley Working Paper in Logic and the Methodology of Science, https://escholarship.org/uc/item/0tm6b30q, 2018.
  98. Holliday, W. H. A note on algebraic semantics for S5 with propositional quantifiers. Notre Dame Journal of Formal Logic 60, 2 (2019), 311–332.
  99. Holliday, W. H. Inquisitive intuitionistic logic. In Advances in Modal Logic, N. Olivetti, R. Verbrugge, S. Negri, and G. Sandu, Eds., vol. 13. College Publications, London, 2020, pp. 329–348.
  100. Holliday, W. H. Three roads to complete lattices: Orders, compatibility, polarity. Algebra Universalis 82, 26 (2021).
  101. One modal logic to rule them all? In Advances in Modal Logic, Vol. 12, G. Bezhanishvili, G. D’Agostino, G. Metcalfe, and T. Studer, Eds. College Publications, London, 2018, pp. 367–386.
  102. Complete additivity and modal incompleteness. The Review of Symbolic Logic 12, 3 (2019), 487–535.
  103. The orthologic of epistemic modals. Manuscript, 2021.
  104. Humberstone, L. Interval semantics for tense logic: Some remarks. Journal of Philosophical Logic 8, 1 (1979), 171–196.
  105. Humberstone, L. From worlds to possibilities. Journal of Philosophical Logic 10, 3 (1981), 313–339.
  106. Humberstone, L. The Connectives. MIT Press, Cambridge, Mass., 2011.
  107. Humberstone, L. Philosophical Applications of Modal Logic. College Publications, London, 2015.
  108. Humberstone, L. Supervenience, dependence, disjunction. Logic and Logical Philosophy 28 (2019), 3–135.
  109. Japaridze, G. K. The polymodal logic of provability. In Intensional Logics and the Logical Structure of Theories: Proceedings of the Fourth Soviet-Finnish Symposium on Logic, Telavi, May 1985, V. A. Smirnov and M. N. Bezhanishvili, Eds. Metsniereba, Tbilisi, 1988, pp. 16–48.
  110. Johnstone, P. T. Stone Spaces, vol. 3 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1982.
  111. Kamp, H. Instants, events and temporal discourse. In Semantics from Different Points of View, R. Bäuerle, U. Egli, and A. von Stechow, Eds. Springer, Berlin, 1979, pp. 376–417.
  112. Kelley, J. L. General Topology. Springer, New York, 1975.
  113. Klement, K. Russell’s logical atomism. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed., spring 2020 ed. Metaphysics Research Lab, Stanford University, 2020.
  114. Normal monomodal logics can simulate all others. Journal of Symbolic Logic 64, 1 (1999), 99–138.
  115. Kratzer, A. Situations in natural language semantics. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed., Summer 2019 ed. Metaphysics Research Lab, Stanford University, 2019.
  116. Reflection principles and their use for establishing the complexity of axiomatic systems. Mathematical Logic Quarterly 14, 7-12 (1968), 97–142.
  117. Kripke, S. A. Semantical analysis of modal logic I. Normal modal propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9 (1963), 67–96.
  118. Kripke, S. A. Semantical considerations on modal logic. Acta Philosophica Fennica 16 (1963), 83–94.
  119. Kripke, S. A. Semantical analysis of intuitionistic logic I. In Formal Systems and Recursive Functions, J. N. Crossley and M. A. E. Dummett, Eds. North-Holland, Amsterdam, 1965, pp. 92–130.
  120. Kuznetsov, A. V. On superintuitionistic logics. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1 (Montreal, Quebec, 1975), Canadian Mathematical Congress, pp. 243–249.
  121. The “Lemmon Notes”: An Introduction to Modal Logic. No. 11 in American Philosophical Quarterly Monograph Series. Basil Blackwell, Oxford, 1977.
  122. Litak, T. An Algebraic Approach to Incompleteness in Modal Logic. PhD thesis, Japan Advanced Institute of Science and Technology, 2005.
  123. Litak, T. On notions of completeness weaker than Kripke completeness. In Advances in Modal Logic, Vol. 5, R. Schmidt, I. Pratt-Hartmann, M. Reynolds, and H. Wansing, Eds. College Publications, London, 2005, pp. 149–169.
  124. Litak, T. A continuum of incomplete intermediate logics. arXiv:1808.06284 [cs.LO], 2018.
  125. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext. Springer-Verlag, New York, 1992.
  126. MacFarlane, J. Future contingents and relative truth. The Philosophical Quarterly 53, 212 (2003), 321–336.
  127. Macnab, D. S. Modal operators on Heyting algebras. Algebra Universalis 12, 1 (1981), 5–29.
  128. Malcev, A. Untersuchungen aus dem Gebiete der mathematischen Logik. Matematicheskii Sbornik 1(43), 3 (1936), 323–336.
  129. Malink, M. Aristotle on one-sided possibility. In Logical Modalities from Aristotle to Carnap: The Story of Necessity, M. Cresswell, E. Mares, and A. Rini, Eds. Cambridge University Press, 2016, pp. 29–49.
  130. Massas, G. Possibility spaces, Q-completions and Rasiowa-Sikorski lemmas for non-classical logics. Master’s thesis, University of Amsterdam, 2016. ILLC Master of Logic Thesis.
  131. Massas, G. B-frame duality for complete lattices. Manuscript, 2020.
  132. Massas, G. Nonstandard analysis via possibility models. Manuscript, 2020.
  133. Choice-free duality for orthocomplemented lattices. arXiv:2010.06763 [math.LO], 2021.
  134. Propositions. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed., Winter 2020 ed. Metaphysics Research Lab, Stanford University, 2020.
  135. The algebra of topology. Annals of Mathematics 45, 1 (1944), 141–191.
  136. On closed elements in closure algebras. Annals of Mathematics 47, 1 (1946), 122–162.
  137. Medvedev, Y. T. Interpretation of logical formulas by means of finite problems. Soviet Mathematics Doklady 7, 4 (1966), 857–860.
  138. Montague, R. Universal Grammar. Theoria 36, 3 (1970), 373–398.
  139. Topological duality and lattice expansions, I: A topological construction of canonical extensions. Algebra Universalis 71, 2 (2014), 109–126.
  140. Naturman, C. A. Interior Algebras and Topology. PhD thesis, University of Capetown, 1976.
  141. Pacuit, E. Neighborhood Semantics for Modal Logic. Short Textbooks in Logic. Springer, Dordrecht, 2017.
  142. The monodic fragment of propositional term modal logic. Studia Logica 107, 3 (2019), 533–557.
  143. Propositional modal logic with implicit modal quantification. In Indian Conference on Logic and Its Applications. ICLA 2019, M. Khan and A. Manuel, Eds. Springer, Berlin, 2019, pp. 6–17.
  144. Two variable fragment of term modal logic. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), P. Rossmanith, P. Heggernes, and J.-P. Katoen, Eds., Leibniz International Proceedings in Informatics. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, 2019, pp. 30:1–30:14.
  145. Frames and Locales: Topology without Points. Frontiers in Mathematics. Birkhäuser, Basel, 2012.
  146. Pincus, D. Adding dependent choice to the prime ideal theorem. In Logic Colloquium 76, R. O. Gandy and J. M. E. Hyland, Eds., vol. 87 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1977, pp. 547–565.
  147. The Mathematics of Metamathematics. Monografie Matematyczne, Tom 41. Państwowe Wydawnictwo Naukowe, Warsaw, 1963.
  148. Röper, P. Intervals and tenses. Journal of Philosophical Logic 9, 4 (1980), 451–469.
  149. Rumfitt, I. The Boundary Stones of Thought: An Essay in the Philosophy of Logic. Oxford University Press, Oxford, 2015.
  150. Topology and duality in modal logic. Annals of Pure and Applied Logic 37 (1988), 249–296.
  151. Santorio, P. Counterfactuals in possibility semantics. Manuscript, 2021.
  152. Schechter, E. Handbook of Analysis and Its Foundations. Academic Press, New York, 1996.
  153. Scott, D. Advice on modal logic. In Philosophical Problems in Logic: Some Recent Developments, K. Lambert, Ed., vol. 29. D. Reidel Publishing Company, Dordrecht, 1970, pp. 143–173.
  154. Scott, D. S. The algebraic interpretation of quantifiers: Intuitionistic and classical. In Andrzej Mostowski and Foundational Studies, A. Ehrenfeucht, V. W. Marek, and M. Srebrny, Eds. IOS Press, Amsterdam, 2008, pp. 289–312.
  155. Uniform density in Lindenbaum algebras. Notre Dame Journal of Formal Logic 55, 4 (2014), 569–582.
  156. Shehtman, V. B. Incomplete propositional logics. Doklady Akademii Nauk SSSR 235, 3 (1977), 542–545. (Russian).
  157. Shehtman, V. B. Topological models of propositional logics. In Semiotics and information science, vol. 15. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1980, pp. 74–98. (Russian).
  158. Shehtman, V. B. On neighbourhood semantics thirty years later. In We Will Show Them! Essays in Honour of Dov Gabbay, S. Artemov, H. Barringer, A. d’Avila Garcez, L. C. Lamb, and J. Woods, Eds., vol. 2. College Publications, London, 2005, pp. 663–692.
  159. Shoenfield, J. R. Mathematical Logic. CRC Press, New York, 2010.
  160. Stone, M. Topological representation of distributive lattices and Brouwerian logics. Časopis pro pěstování matematiky a fysiky 67, 1 (1938), 1–25.
  161. Stone, M. H. The theory of representation for Boolean algebras. Transactions of the American Mathematical Society 40, 1 (1936), 37–111.
  162. Stone, M. H. Topological representation of distributive lattices and Brouwerian logics. Časopis pro pěstování matematiky a fysiky 67, 1 (1937), 1–25.
  163. Axiomatic Set Theory. Springer-Verlag, New York, 1973.
  164. Tarski, A. Zur Grundlegung der Bool’schen Algebra. I. Fundamenta Mathematicae 24 (1935), 177–198.
  165. Tarski, A. Über additive und multiplikative Mengenkörper und Mengenfunktionen. Sprawozdania z Posiedzeń Towarzystwa Naukowego Warszawskiego, Wydział III, Nauk Matematyczno-fizycznych 30 (1937), 151–181.
  166. Tarski, A. Der Aussagenkalkül und die Topologie. Fundamenta Mathematicae 31, 1 (1938), 103–134. English translation in [167], pp. 421-454.
  167. Tarski, A. Logic, Semantics, Metamathematics: Papers from 1923 to 1938. Clarendon Press, Oxford, 1956. Translated by J. H. Woodger.
  168. Thomason, S. K. Categories of Frames for Modal Logic. The Journal of Symbolic Logic 40, 3 (1975), 439–442.
  169. van Dalen, D. Logic and Structure, fifth ed. Springer, Berlin, 2013.
  170. Venema, Y. Algebras and coalgebras. In Handbook of Modal Logic, P. Blackburn, J. van Benthem, and F. Wolter, Eds. Elsevier, Amsterdam, 2007, pp. 331–426.
  171. Vietoris, L. Bereiche zweiter ordnung. Monatshefte für Mathematik und Physik 32, 1 (1922), 258–280.
  172. The relation between intuitionistic and classical modal logics. Algebra and Logic 36, 2 (1997), 73–92.
  173. Yamamoto, K. Results in modal correspondence theory for possibility semantics. Journal of Logic and Computation 27, 8 (2017), 2411–2430.
  174. Zhao, Z. Unified correspondence and canonicity. PhD thesis, Delft University of Technology, 2018.
  175. Zhao, Z. Algorithmic correspondence and canonicity for possibility semantics. Journal of Logic and Computation 31, 2 (2021), 523–572.
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