# Temporal logics

Chapter
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 12)

The logics that have been examined in previous chapters were defined in terms of languages containing a unique primitivemonadic operator, usually chosen in the set {□, ◊}, or also (for logics containing the axiom (T)), in the set {∆, ∇}. But, in principle, it is notdifficult to introduce, as primitives in the language, two ormore distinct modal operators or even an infinite number of them.

Logics having modal languages extended in this way are called multimodal logics. In Chapter 8 we shall study, in an abstract way, the properties of multimodal logics containing an arbitrary number of primitive modal operators. The logics studied up to now turn out to be just special cases of such multimodal logics and should be properly called monomodal logics.

To appraise the philosophical interest of multimodal logics and their prospective applications, it is enlightening to consider some simple examples of bimodal logics, that is, of logics whose languages have two primitive modal operators.

## Keywords

Temporal Logic Accessibility Relation Modal Language Hybrid Logic Tense Logic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Bibliograph1y

1. [AB05]
S. Artemov and L. Beklemishev. Provability logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 13, pages 229–403. Springer, Amsterdam, 2005.Google Scholar
2. [Ale95]
N. Alechina. Modal Quantifiers. PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, 1995.Google Scholar
3. [Aum76]
R. J. Aumann. Agreeing to disagree. The Annals of Statistics, 14(6):1236–1239, 1976.Google Scholar
4. [B0′5]
[B0′5] J.-Y. Béziau. Paraconsistent logics from a modal point of view. Journal of Applied Logic, 3:7–14, 2005.Google Scholar
5. [BdRV01]
P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic. Cambridge University Press, Cambridge, 2001.Google Scholar
6. [Ben83]
E. Bencivenga. Free logics. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 3, pages 373–426. Dordrecht, Reidel, 1983.Google Scholar
7. [Ben84]
E. Bencivenga. Free Logics. Bibliopolis, Naples, 1984.Google Scholar
8. [BG07]
T. Brauner and S. Ghilardi. First-order modal logic. In P. Black-burn, J. van Benthem, and F. Wolter, editors, Handbook of Modal Logic, pages 549–620. Elsevier, Amsterdam, 2007.Google Scholar
9. [BGM98]
M. Baldoni, L. Giordano, and A. Martelli. A tableau calculus for multimodal logics and some (un)decidability results. In H. de Swart, editor, Proceedings of Tableaux 98—International Conference on Tableaux Methods, pages 44–59. Springer, Berlin, 1998.Google Scholar
10. [Boc61]
I. M. Bochenski. A History of Formal Logic. University of Notre Dame Press, Indiana, 1961.Google Scholar
11. [Boo79]
G. Boolos. The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press, Cambridge, 1979.Google Scholar
12. [Boo93]
G. Boolos. The Logic of Provability. Cambridge University Press, Cambridge, 1993.Google Scholar
13. [BS85]
G. Boolos and G. Sambin. An incomplete systemof modal logic. Journal of Philosophical logic, 14:351–358, 1985.Google Scholar
14. [BS04]
B. Brogaard and J. Salerno. Fitch's paradox of knowability. In Edward N. Zalta, editor, Stanford Encyclopedia of Philosophy, volume http://plato.stanford.edu.Center for the Study of Language and Information—Stanford University, Stanford, 2004.
15. [BT99]
P. Blackburn and M. Tzakova. Hybrid languages and temporal logics. Logic Journal of the IGPL, 7:27–54, 1999.Google Scholar
16. [Bul66]
R. A. Bull. That all normal extension of S4.3have the finite model property. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 12:609–616, 1966.Google Scholar
17. [Bul68]
R. A. Bull. An algebraic study of tense logic with linear time. The Journal of Symbolic Logic, 33:27–38, 1968.Google Scholar
18. [Bur84]
J. P. Burgess. Basic tense logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 89–133. Dordrecht, Reidel, 1984.Google Scholar
19. [BvB07]
P. Blackburn and J. van Benthem. Modal logic; a semantic perspective. In P. Blackburn, J. van Benthem, and F. Wolter, editors, Handbook of Modal Logic, pages 1–84. Elsevier, Amsterdam, 2007.Google Scholar
20. [Car47]
R. Carnap. Meaning and Necessity. University of Chicago Press, Chicago, IL, 1947.Google Scholar
21. [Cat88]
L. Catach. Normal multimodal logic. In T. Mitchell and R. Smith, editors, Proceedings of AAAI′88— Seventh National Conference on Artificial Intelligence, pages 491–495. The AAAI Press, Menlo Park, CA, 1988.Google Scholar
22. [Cat89]
L. Catach. Les Logiques Multimodales. PhD thesis, Univérsité de Paris VI, France, 1989.Google Scholar
23. [Cat91]
L. Catach. Tableaux: a general theorem prover for modal logics. Journal of Automated Reasoning, 7(4):489–510, 1991.Google Scholar
24. [CC07]
W. A. Carnielli and M. E. Coniglio. Combining logics. In Edward N. Zalta, editor, Stanford Encyclopedia of Philosophy, volume http://plato.stanford.edu. Center for the Study of Language and Information—Stanford University, Stanford, 2007.
25. [CCG+07]
[CCG+07] W. A. Carnielli, M. E. Coniglio, D. Gabbay, P. Gouveia, and C. Sernadas. Analysis and Synthesis of Logics: How to Cut and Paste Reasoning Systems. Springer, Amsterdam, 2007.Google Scholar
26. [CCM07]
W. A. Carnielli, M. E. Coniglio, and J. Marcos. Logics of formal inconsistency. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 14, pages 1–93. Springer, Amsterdam, 2007.Google Scholar
27. [CD97]
W. A. Carnielli and I. M. L. D'Ottaviano. Translations between logical systems: a manifesto. Logic et Analyse, 40(157):67–81, 1997.Google Scholar
28. [Che80]
B. F. Chellas. Modal Logic: An Introduction. Cambridge University Press, Cambridge, 1980.Google Scholar
29. [Chi63]
R. M. Chisholm. The logic of knowing. Journal of Philosophy, 60:773–795, 1963.Google Scholar
30. [CM02]
W. A. Carnielli and J. Marcos. A taxonomy of C-systems. In W. A. Carnielli, M. E. Coniglio, and I. M. L. D'Ottaviano, editors, Paraconsistency - The Logical Way to the Inconsistent, volume 228 of Lecture Notes in Pure and Applied Mathematics, pages 1–94. Marcel Dekker, New York, 2002.Google Scholar
31. [Coc66a]
[Coc66a] N. B. Cocchiarella. A logic of actual and possible objects. The Journal of Symbolic Logic, 31:688–689, 1966.Google Scholar
32. [Coc66b]
N. B. Cocchiarella. Tense Logic: A Study of Temporal Reference. PhD thesis, U.C.L.A, LA, 1966.Google Scholar
33. [Con71]
J. H. Conway. Regular Algebras and Finite Machines. Chapman & Hall, London, 1971.Google Scholar
34. [Coo71]
S. Cook. The complexity of theorem proving procedures. In M. A. Harrison, R. B. Banerji, and J. D. Ullman, editors, Proceedings of the Third Annual ACM Symposium on Theory of Computing, pages 151–158. Shaker Heights, Ohio, 1971.Google Scholar
35. [Cop02]
J. Copeland. The genesis of possible worlds semantics. Journal of Philosophical logic, 31:99–137, 2002.Google Scholar
36. [Cre88]
M. J. Cresswell. Necessity and contingency. Studia Logica, 47:145–149, 1988.Google Scholar
37. [CZ97]
A. Chagrov and M. Zakharyaschev. Modal Logic. Clarendon Press, Oxford, 1997.Google Scholar
38. [Dug40]
J. Dugundji. Note on a property of matrices for Lewis and Langford's calculi of propositions. The Journal of Symbolic Logic, 5:150–151, 1940.Google Scholar
39. [EC00]
R. L. Epstein and W. A. Carnielli. Computability: Computable Funtions, Logic, and the Foundations of Mathematics, with Computability and Undecidability—A Timeline. Wadsworth/Thomson Learning, Belmont, CA, 2nd edition, 2000.Google Scholar
40. [EdC79]
P. Enjalbert and L. Fariñas del Cerro. Modal resolution in clausal form. Theorethical Computer Science, 61(1):1–33, 1979.Google Scholar
41. [Fey65]
R. Feys. Modal Logic. Nauwelaerts, Louvain, 1965.Google Scholar
42. [FH02]
J. M. Font and P. Hájek. On lukasiewicz's four—valued modal logic. Studia Logica, 26:157–182, 2002.Google Scholar
43. [FHMV95]
R. Fagin, J. Y. Halpern, Y. Moses, and M. Vardi. Reasoning About Knowledge. MIT Press, Cambridge, 1995.Google Scholar
44. [FHV92]
R. Fagin, J. Y. Halpern, and M. Y. Vardi. What can machines know? On the properties of knowledge in distributed systems. Journal of the ACM, 39(2):328–376, 1992.Google Scholar
45. [Fin70]
K. Fine. Propositional quantifiers in modal logic. Theoria, 36: 336–346, 1970.Google Scholar
46. [Fit63]
F. Fitch. A logical analysis of some value concepts. Journal of Symbolic Logic, 28:135–142, 1963.Google Scholar
47. [Fit83]
M. C. Fitting. Proof Methods for Modal and Tense Logics. Dordrecht, Reidel, 1983.Google Scholar
48. [Fit91]
M. C. Fitting. Many-valued modal logics I. Fundamenta Informaticae, 15:235–254, 1991.Google Scholar
49. [Fit92]
M. C. Fitting. Many—valued modal logics II. Fundamenta Informaticae, 17:55–73, 1992.Google Scholar
50. [Fit06]
M. C. Fitting. Intensional logic. In Edward N. Zalta, editor, Stanford Encyclopedia of Philosophy, volume http://plato.stanford.edu. Center for the Study of Language and Information—Stanford University, Stanford, 2006.
51. [Gab76]
D. Gabbay. Investigations in Modal and Tense Logics, with Applications to Problems in Linguistics and Philosophy. Dordrecht, Reidel, 1976.Google Scholar
52. [Gär73]
[Gär73] P. Gärdenfors. On the extensions of S5. Notre Dame Journal of Formal Logic, 14:277–280, 1973.Google Scholar
53. [Gar84]
J. W. Garson. Quantification in modal logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 249–307. Dordrecht, Reidel, 1984.Google Scholar
54. [Gär88]
[Gär88] P. Gärdenfors. Knowledge in Flux: Modeling the Dynamics. MIT Press, Cambridge, 1988.Google Scholar
55. [GHR94]
D. Gabbay, I. Hodkinson, and M. Reynolds. Temporal Logic: Mathematical Foundations and Computational Aspects. Oxford University Press, Oxford, 1994.Google Scholar
56. [GKWZ03]
D. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev. Many-Dimensional Modal Logics: Theory and Applications. North Holland, Amsterdam, 2003.Google Scholar
57. [Göd32]
[Göd32] K. Gödel. Zum intuitionistischen aussagenkalkül. Anzeiger der Akademie der Wissenschaften in Wien, 69:65–66, 1932. mathematisch-naturwissenschaftiliche Klasse.Google Scholar
58. [Gol75]
R. Goldblatt. Solution to a completeness problem of Lemmon and Scott. Notre Dame Journal of Formal Logic, 16:405–408, 1975.Google Scholar
59. [Gol93]
R. Goldblatt. The Mathematics of Modality. CLS Publications, Stanford, CA, 1993.Google Scholar
60. [Grz67]
A. Grzegorczyk. Some relational systems and the associated topological spaces. Fundamenta Mathematicæ, 60:223–231, 1967.Google Scholar
61. [GT75]
R. I. Goldblatt and S. K. Thomason. Axiomatic classes in propositional modal logic. In J. N. Crossley, editor, Algebra and Logic, volume 450 of Lecture Notes in Mathematics, pages 163–173. Springer, Berlin, 1975.Google Scholar
62. [Haa74]
S. Haack. Deviant Logic. Cambridge University Press, Cambridge, 1974.Google Scholar
63. [Har79]
D. Harel. First-Order Dynamic Logic, volume 68 of Lecture Notes in Computer Science. Springer, Berlin, 1979.Google Scholar
64. [Haz79]
A. Hazen. Counterpart-theoretic semantics for modal logic. The Journal of Philosophy, 76(6):319–338, 1979.Google Scholar
65. [HC68]
G. E. Hughes and M. J. Cresswell. An Introduction to Modal Logic. Methuen, London, 1968.Google Scholar
66. [HC84]
G. E. Hughes and M. J. Cresswell. A Companion to Modal Logic. Methuen, London, 1984.Google Scholar
67. [HC86]
G. E. Hughes and M. J. Cresswell. A companion to modal logic: some corrections. Logique et Analyse, 29:41–51, 1986.Google Scholar
68. [HC96]
G. E. Hughes and M. J. Cresswell. A New Introduction to Modal Logic. Methuen, London, 1996.Google Scholar
69. [HF98]
P. W. Humphreys and J. Fetzer. The New Theory of Reference, Kripke, Marcus, and Its Origins, volume 270 of Synthese Library. Kluwer, Dordrecht, 1998.Google Scholar
70. [HG73]
B. Hanssonand P.Gärdenfors. Aguideto intensionalsemantics. In Modality, Morality and Other Problems of Sense and Nonsense: Essays Dedicated to Sören Halldén, pages 151–167. CWK Gleerup Bokfürlag, Lund, 1973.Google Scholar
71. [HH91]
C. W. Harvey and J. Hintikka. Modalization and modalities. In T. Seebohm, D. Follesdal, and J. N. Mohanty, editors, Phenomenology and the Formal Sciences, pages 59–77. Kluwer, Amsterdam, 1991.Google Scholar
72. [Hin62]
J. Hintikka. Knowledge and Belief. Cornell University Press, Ithaca, NY, 1962.Google Scholar
73. [Hin63]
J. Hintikka. The modes of modality. Acta Phylosophica Fennica, 16:65–81, 1963.Google Scholar
74. [HKT00]
D. Harel, D. Kozen, and J. Tiuryn. Dynamic Logic. MIT Press, Cambridge, 2000.Google Scholar
75. [HM85]
J. Y. Halpern and Y. Moses. A guide to the modal logics of knowledge and belief. In A. K. Joshi, editor, Proceedings of the 9th International Joint Conference on Artificial Intelligence (IJCAI 85), pages 480–490. Morgan Kaufmann, Los Angeles, 1985.Google Scholar
76. [HM92]
J. Y. Halpern and Y. Moses. A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54:319–379, 1992.Google Scholar
77. [Hod01]
W. Hodges. Logic and games. In Edward N. Zalta, editor, Stanford Encyclopedia of Philosophy, volume http://plato.stanford.edu. Center for the Study of Language and Information—Stanford University, Stanford, 2001.Google Scholar
78. [Hum81]
L. Humberstone. Relative necessity revisited. Reports on Mathematical Logic, 13:33–42, 1981.Google Scholar
79. [Jan90]
R. Jansana. Una Introducciòn a la Lògica Modal. Tecnos, Madrid, 1990.Google Scholar
80. [JdJ98]
G. Japaridze and D. de Jongh. The logic of Provability. In S. R. Buss, editor, Handbook of Proof Theory, volume 137 of Studies in Logic, pages 475–546. Elsevier, Amsterdam, 1998.Google Scholar
81. [Kal35]
L. Kalmár. Über die Axiomatisierbarkeit des Aussagenkalküls. Acta Scientiarum Mathematicarum, 7:222–243, 1935.Google Scholar
82. [Kam68]
H. Kamp. Tense Logic and the Theory of Linear Order. PhD thesis, U.C.L.A, LA, 1968.Google Scholar
83. [Kau60]
R. Kauppi. Über die Leibnizsche Logik mit besonderer Berück-sichtigung des Problems der Intension und der Extension. Acta Philosophica Fennica, 12:1–279, 1960.Google Scholar
84. [KK62]
W. Kneale and M. Kneale. The Development of Logic. Clarendon Press, Oxford, 1962.Google Scholar
85. [KM64]
D. Kalish and R. Montague. Logic. Techniques of Formal Reasoning. Harcourt, Brace & World, New York, 1964.Google Scholar
86. [Koz79]
D. Kozen. On the representation of dynamic algebras. Technical Report RC7898, IBM Thomas J. Watson Research Center, October 1979.Google Scholar
87. [Koz80]
D. Kozen. A representation theorem for modelsof *-freePDL.In Automata, Languages and Programming — Lecture Notes in Computer Science, Lecture Notes in Computer Science, pages 351–362. Springer, Berlin, 1980.Google Scholar
88. [Kra93]
M. Kracht. How completeness and correspondence theory got married. In M. de Rijke, editor, Diamonds and Defaults. Kluwer, Dordrecht, 1993.Google Scholar
89. [Kri59]
S. Kripke. A completeness theorem in modal logic. The Journal of Symbolic Logic, 24:1–14, 1959.Google Scholar
90. [Kri63a]
[Kri63a] S. Kripke. Semantic analysis of modal logic I, normal propositional calculi. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 9:67–96, 1963.Google Scholar
91. [Kri63b]
[Kri63b] S. Kripke. Semantical considerations on modal logic. Acta Philosophica Fennica, 16:83–94, 1963.Google Scholar
92. [Kri65]
S. Kripke. Semantical analysis of intuitionistic logic I. In J. N. Crossley and M. A. E. Dummett, editors, Formal Systems and Recursive Functions—Proceedings of the Eighth Logic Colloquium Oxford, July 1963, pages 92–130. North-Holland, Amsterdam, 1965.Google Scholar
93. [Krö87]
[Krö87] F. Kröger. Temporal Logic of Programs. Springer, New York, 1987.Google Scholar
94. [Kuh80]
S. Kuhn. Quantifiers as modal operators. Studia Logica, 39: 145–158, 1980.Google Scholar
95. [Kuh89]
S. Kuhn. The domino relation: flattening a two-dimensional logic. Journal of Philosophical Logic, 18:173–195, 1989.Google Scholar
96. [Lem66a]
[Lem66a] E. J. Lemmon. Algebraic semantics for modal logics I. The Journal of Symbolic Logic, 31(1):44–65, 1966.Google Scholar
97. [Lem66b]
[Lem66b] E. J. Lemmon. Algebraic semantics for modal logics II. The Journal of Symbolic Logic, 31(2):191–218, 1966.Google Scholar
98. [Lem66c]
[Lem66c] E. J. Lemmon. A note on Halldén incompleteness. Notre Dame Journal of Formal Logic, 7:296–300, 1966.Google Scholar
99. [Len78]
W. Lenzen. Recent work on epistemic logic. Acta Philosophica Fennica, 30:1–219, 1978.Google Scholar
100. [Len79]
W. Lenzen. Epstemologische betrachtungen zu S4, S5. Erkenntnis, 14:33–56, 1979.Google Scholar
101. [Len80]
W. Lenzen. Glauben, Wissen und Wahrscheinlichkeit. Springer, Vienna, 1980.Google Scholar
102. [Lew69]
D. Lewis. Convention: A Philosophical Study. Harvard University Press, Cambridge, MA, 1969.Google Scholar
103. [Lin71]
L. Linsky. Reference and Modality. Oxford Readings in Philosophy, Oxford University Press, Oxford, 1971.Google Scholar
104. [LL32]
C. I. Lewis and C. H. Langford. Symbolic Logic. The Appleton-Century Company, New York, 1932. reprinted in paperback by Dover Publications, New York, 1951.Google Scholar
105. [Löb55]
[Löb55] M. H. Löb. Solution of a problem of Leon Henkin. The Journal of Symbolic Logic, 20(2):115–118, 1955.Google Scholar
106. [Lor55]
P. Lorenzen. Einführung in die operative Logik und Mathematik. Springer, Berlin, 1955.Google Scholar
107. [LS77]
E. J. Lemmon and D. Scott. An Introduction to Modal Logic. Blackwell, Oxford, 1977.Google Scholar
108. [Łuk70]
[Łuk70] J. Łukasiewicz. A system of modal logic. In L. Borkowski, editor, Jan Łukasiewicz's Selected Works. North-Holland, Amsterdam, 1970.Google Scholar
109. [Mag82]
[Mag82]R. Magari. Primi risultati sulla varietà di Boolos. Bollettino della Unione Matematica Italiana, 6:359–367, 1982.Google Scholar
110. [Mak66]
D. Makinson. On some completeness theorems in modal logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 12:379–384, 1966.Google Scholar
111. [Mak69]
D. Makinson. A normal modal calculus between T and S4 without the finite model property. The Journal of Symbolic Logic, 34:35–38, 1969.Google Scholar
112. [Mak70]
D. Makinson. A generalization of the concept of a relational model for modal logic. Theoria, 36:331–335, 1970.Google Scholar
113. [Man75]
A. B. Manaster. Completeness, Compactness and Undecidability. An Introduction to Mathematical Logic. Prentice Hall, Englewood Cliffs, NJ, 1975.Google Scholar
114. [Mar89]
N. M. Martin. Systems of Logic. Cambridge University Press, Cambridge, 1989.Google Scholar
115. [Mat89]
B. Mates. The Philosophy of Leibniz: Metaphysics and Language. Oxford University Press, New York, 1989.Google Scholar
116. [McG06]
C. McGinnis. Tableau systems for some paraconsistent modallogics. Electronic Notes in Theoretical Computer Science, 143: 141–157, 2006.Google Scholar
117. [Men64]
E. Mendelson. Introduction to Mathematical Logic. van Nostrand, Princeton, NJ, 1964.Google Scholar
118. [Mey03]
J. J. Meyer. Modal epistemic and doxastic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 10, pages 1–38. Kluwer, Dordrecht, 2003.Google Scholar
119. [MH69]
J. McCarthy and P. Hayes. Some philosophical problems from the standpoint of Artificial Intelligence. Machine Intelligence, 4:463–502, 1969.Google Scholar
120. [Mon70]
R. Montague. Universal grammar. Theoria, 36:373–398, 1970.Google Scholar
121. [MP92]
Z. Manna and A. Pnueli. The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer, Berlin, 1992.Google Scholar
122. [MT48]
J. C. C. McKinsey and A. Tarski. Some theorems about the sentential calculi of Lewis and Heyting. The Journal of Symbolic Logic, 13:1–15, 1948.Google Scholar
123. [Mug01]
M. Mugnai. Introduzione alla Filosofia di Leibniz. Einaudi, Turin, 2001.Google Scholar
124. [MV97]
M. Marxand Y. Venema. Multi-DimensionalModalLogic. Kluwer, Dordrecht, 1997.Google Scholar
125. [OM57a]
[OM57a] M. Ohnishi and K. Matsumoto. Corrections to our paper ‘Gentzen method in modal calculi I’. Osaka Mathematical Journal, 10:147, 1957.Google Scholar
126. [OM57b]
[OM57b] M. Ohnishi and K. Matsumoto. Gentzen method in modal calculi I. Osaka Mathematical Journal, 9:113–130, 1957.Google Scholar
127. [OM57c]
[OM57c] M. Ohnishi and K. Matsumoto. Gentzen method in modal calculi II. Osaka Mathematical Journal, 11:115–120, 1957.Google Scholar
128. [Par81]
D. Park. Concurrency and automata on infinite sequences. In P. Deussen, editor, Theoretical Computer Science, volume 104 of Lecture Notes in Computer Science, pages 167–183. Springer, Berlin, 1981.Google Scholar
129. [Pel00]
F. J. Pelletier. A history of natural deduction and elementary logic textbooks. In J. Woods and B. Brown, editors, Logical Consequence: Rival Approaches, volume 1, pages 105–138. Hermes Science Publications, Oxford, 2000.Google Scholar
130. [Pie06]
A.-V. Pietarinen. Peirce's conributions to possible—worlds semantics. Studia Logica, 82:345–369, 2006.Google Scholar
131. [Piz74]
C. Pizzi. La Logica del Tempo. Boringhieri, Turin, 1974.Google Scholar
132. [Pla74]
A. Plantinga. The Nature of Necessity. Clarendon Press, Oxford, 1974.Google Scholar
133. [Pop94]
S. Popkorn. First Steps in Modal Logic. Cambridge University Press, Cambridge, 1994.Google Scholar
134. [Pra65]
D. Prawitz. Natural Deduction. A Proof-Theoretical Study. Almquist and Wiksell, Stockholm, 1965. Second edition by Dover Publications, 2006.Google Scholar
135. [Pra76]
V. R. Pratt. Semantical considerations on Floyd-Hoare logic. In Proceedings of the 17th Annual IEEE Symposium on Foundations of Computer Science, pages 109–121. IEEE, 1976.Google Scholar
136. [Pra92]
V. R. Pratt. Origins of the calculus of binary relations. In Proceedings of the 7th Annual IEEE Symposium on Logic in Computer Science, pages 248–254. IEEE, 1992.Google Scholar
137. [Pri57]
A. N. Prior. Time and Modality. Oxford University Press, Oxford, 1957.Google Scholar
138. [Pri67]
A. N. Prior. Past, Present and Future. Clarendon Press, Oxford, 1967.Google Scholar
139. [Pri68]
A. N. Prior. Papers on Time and Tense. Clarendon Press, Oxford 1968.Google Scholar
140. [Rau79]
W. Rautenberg. Klassische und nichtklassische Aussagenlogik. Friedr. Vieweg & Sohn, Braunschweig, 1979.Google Scholar
141. [Rei47]
H. Reichenbach. Elements of Symbolic Logic. Macmillan, New York, 1947. Reprinted by Dover, 1980.Google Scholar
142. [Ren70]
M. K. Rennie. Models for multiply modal systems. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 16:175–186, 1970.Google Scholar
143. [RU71]
N. Rescher and A. Urquhart. Temporal Logic. Springer, Vienna, 1971.Google Scholar
144. [Ryb97]
V.V. Rybakov. Admissibility of Logical Inference Rules. Elsevier Science, Amsterdam, 1997.Google Scholar
145. [Sah75]
H. Sahlqvist. Completeness and correspondence in the first and second order semantics for modal logic. In S. Kanger, editor, Proceedings of 3th Scandinavian Logic Symposium, pages 110–143. North-Holland, Amsterdam, 1975.Google Scholar
146. [Sch60]
T. Schelling. The Strategy of Conflict. Harvard University Press, Cambridge, MA, 1960.Google Scholar
147. [Sch68]
K. Schütte. Vollständige Systeme modaler und intuitionistischer Logik. Springer, Berlin, 1968. Vol. 42 of Ergebnisse der Mathematik und ihrer Grenzgebiete.Google Scholar
148. [Sch89]
G. F. Schumm. Some compactness results for modal logic. Notre Dame Journal of Formal Logic, 30:285–290, 1989.Google Scholar
149. [Sch01]
G. Schurz. Rudolf carnap's modal logic. In W. Stelzner and M. Stöckler, editors, Zwischen traditioneller un moderner Logik. Nichtklassische Ansätze, pages 365–380. Mentis, Paderborn, 2001.Google Scholar
150. [Sco70]
D. Scott. Advice on modal logic. In K. Lambert, editor, Philosophical Problems in Logic. Some Recent Developments, pages 143–173. Reidel, Dordrecht, 1970.Google Scholar
151. [Scr51]
S. J. Scroggs. Extensions of the Lewis system S5. The Journal of Symbolic Logic, 16:112–120, 1951.Google Scholar
152. [Seg67]
K. Segerberg. Some modal logics based on a three-valued logic. Theoria, 33:53–71, 1967.Google Scholar
153. [Seg68]
K. Segerberg. On the logic of tomorrow. Theoria, 31:199–217, 1968.Google Scholar
154. [Seg70]
K. Segerberg. Modal logics with linear alternative relations. Theoria, 36:301–322, 1970.Google Scholar
155. [Seg71]
K. Segerberg. An Essay in Classical Modal Logic. University of Uppsala, Uppsala, 1971.Google Scholar
156. [Seg73]
K. Segerberg. Two-dimensional modal logic. Journal of Philosophical Logic, 2:77–96, 1973.Google Scholar
157. [Seg77]
K. Segerberg. A completeness theorem in the modal logic of programs. Notices of the American Mathematical Society, 4(6): 1–552, 1977.Google Scholar
158. [Seg82]
K. Segerberg. Classical Propositional Operators: An Exercise in the Foundations of Logic. Clarendon Press, Oxford 1982.Google Scholar
159. [Seg95]
K. Segerberg. Belief revision from the point of view of doxastic logic. Logic Journal of the IGPL, 3(4):535–553, 1995.Google Scholar
160. [Smo85]
C. Smorynski. Self-Reference and Modal Logic. Springer, New York, 1985.Google Scholar
161. [Smu68]
R. M. Smullyan. First-Order Logic. Springer, New York, 1968.Google Scholar
162. [Sol76]
R. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 25:287–304, 1976.Google Scholar
163. [Tap84]
B. Tapscott. Correcting the tableaux procedure for S4. Notre Dame Journal of Formal Logic, 25:241–249, 1984.Google Scholar
164. [Tar41]
A. Tarski. On the calculus of relations. The Journal of Symbolic Logic, 6(3):73–89, 1941.Google Scholar
165. [Tho70]
R. H. Thomason. Some completeness results in modal predicate calculus. In K. Lambert, editor, Philosophical Problems in Logic, pages 56–76, Reidel, Dordrecht, 1970.Google Scholar
166. [Tho72]
S. K. Thomason. Noncompactness in propositionalmodal logic. The Journal of Symbolic Logic, 37:716–720, 1972.Google Scholar
167. [Tho74]
S. K. Thomason. An incompleteness theorem in modal logic. The Journal of Symbolic Logic, 40:30–34, 1974.Google Scholar
168. [Urq81]
A. Urquhart. Decidability and finite model property. The Journal of Philosophical Logic, 10:367–370, 1981.Google Scholar
169. [vB76]
J. van Benthem. Modal Correspondence Theory. PhD thesis, University of Amsterdam, Amsterdam, 1976.Google Scholar
170. [vB78]
J. van Benthem. Two simple incomplete logics. Theoria, 44: 25–37, 1978.
171. [vB83a]
[vB83a] J. van Benthem. The Logic of Time. A Model Theoretic Investigation in to the Varieties of Temporal Discourse. Reidel, Dordrecht, 1983.Google Scholar
172. [vB83b]
[vB83b] J. van Benthem. Modal Logic and Classical Logic. Bibliopolis, Naples, 1983.Google Scholar
173. [vB84]
J. van Benthem. Correspondence theory. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 167–247. Reidel, Dordrecht, 1984.Google Scholar
174. [Ven93]
Y. Venema . Many-Dimensional Modal Logic. PhD thesis, Department of Mathematics, University of Amsterdam, Amsterdam, 1993.Google Scholar
175. [vOQ66]
W. v. O. Quine. The ways of Paradox and Other Essays. Random House, New York, 1966.Google Scholar
176. [vW57]
G. H. von Wright. Logical Studies. Routledge & Kegan, London, 1957.Google Scholar
177. [vW65]
G. H. von Wright. And next. Acta Philosophica Fennica, 18:293– 304, 1965.Google Scholar
178. [vW82]
G. H. von Wright. Wittgenstein. Blackwell, Oxford, 1982.Google Scholar
179. [Waj33]
M. Wajsberg. Ein erweiteter Klassenkalkul. Monatshefte für Mathematik und Physik, 40:113–126, 1933.Google Scholar
180. [Wil92]
T. Williamson. On intuitionistic modal epistemic logic. Journal of Philosophical Logic, 21:63–89, 1992.Google Scholar
181. [Wil00]
T. Williamson. Knowledge and Its Limits. Oxford University Press, Oxford, 2000.Google Scholar
182. [Wit01]
L. Wittgenstein. Tractatus Logico-Philosophicus. Routledge, London, 2001. David Pears and Brian McGuinness, translators.Google Scholar
183. [Xu88]
M. Xu. On some US-tense logics. Journal of Philosophical Logic, 17(2):181–202, 1988.Google Scholar
184. [Zem73]
J. J. Zeman. Modal Logic. Clarendon Press, Oxford 1973.Google Scholar