Hybrid Logic as extension of Modal and Temporal Logic

Daniel Álvarez Domínguez


Developed by Arthur Prior, Temporal Logic allows to represent temporal information on a logical system using modal (temporal) operators such as P, F, H or G, whose intuitive meaning is “it was sometime in the Past...”, “it will be sometime in the Future...”, “it Has always been in the past...” and “it will always Going to be in the future...” respectively. Valuation of formulae built from these operators are carried out on Kripke semantics, so Modal Logic and Temporal Logic are consequently related. In fact, Temporal Logic is an extension of Modal one. Even when both logics mechanisms are able to formalize modal-temporal information with some accuracy, they suffer from a lack of expressiveness which Hybrid Logic can solve. Indeed, one of the problems of Modal Logic consists in its incapacity of naming specific points inside a model. As Temporal Logic is based on it, it cannot make such a thing neither. But First-Order Logic does can by means of constants and equality relation. Hybrid Logic, which results from combining Modal Logic and First-Order Logic, may solve this shortcoming. The main aim of this paper is to explain how Hybrid Logic emanates from Modal and Temporal ones in order to show what it adds to both logics with regard to information representation, why it is more expressive than them and what relation it maintains with the First-Order Correspondence Language.


Arthur Prior; First-Order Correspondence Language; Temporal Representation; Nominals; Translations

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Areces, C. and ten Cate, B. (2007). Hybrid Logics. In P. Blackburn, J. van Benthem and F. Wolter (eds.). Handbook of Modal Logic, Vol. 3, pp. 822-863. Amsterdam: Elsevier.

Areces, C., Blackburn, P., Huertas, A. and Manzano, M. (2014). Completeness in Hybrid Type Theory. Journal of Philosophical Logic, 43(2-3): 209-238.

Areces, C., Blackburn, P., Huertas, A. and Manzano, M. (2011). Hybrid Type Theory: A Quartet in Four Movements. Principia: An International Journal of Epistemology, 15(2): 225-247.

Blackburn, P. (2000). Representation, Reasoning, and Relational Structures: a Hybrid Logic Manifesto. Logic Journal of the IGPL, 8(3): 339-365.

Blackburn, P. (2006). Arthur Prior and Hybrid Logic. Synthese, 150(3): 329-372.

Blackburn, P. and van Benthem, J. (2007). Modal Logic: A Semantic Perspective. In P. Blackburn, J. van Benthem and F. Wolter (eds.), Handbook of Modal Logic, Vol. 3, pp. 2-79. Amsterdam: Elsevier.

Findlay, J. (1941). Time: A Treatment of Some Puzzles. Australasian Journal of Psychology and Philosophy, 19: 216-235.

McTaggart, J. (1908). The Unreality of Time. Mind, 17(68): 457-474.

Øhrstrøm, P. and Hasle, p. (1995). Temporal Logic. From Ancient Ideas to Artificial Intelligence. Dordrecht: Springer Science+Business Media Dordrecht.

Prior, A. (1955). Diodoran Modalities. The Philosophical Quarterly, 5(20): 205-213.

Prior, A. (1957). Time and Modality. Oxford: Oxford University Press.

Prior, A. (1967). Past, Present and Future. Oxford: Oxford University Press.

Prior, A. (2010). Papers on Time and Tense. Oxford: Oxford University Press.

Reichenbach, H. (1947). Elements of Symbolic Logic. New York: Dover Publications.

Rescher, N. and Alasdair, U. (1971). Temporal Logic. New York: Springer.

van Eijck, J. (2006). Bisimulation. In https://homepages.cwi.nl/~jve/courses/lai0506/LAI11.pdf.

DOI: https://doi.org/10.22370/rhv2019iss13pp34-67

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