Neighborhood Semantics for Basic and Intuitionistic Logic

Morteza Moniri, Fatemeh Shirmohammadzadeh Maleki



In this paper we present a neighborhood semantics for Intuitionistic Propositional Logic (IPL). We show that for each Kripke model of the logic there is a pointwise equivalent neighborhood model and vice versa. In this way, we establish soundness and completeness of IPL with respect to the neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. Moreover, the notions of bisimulation and n-bisimulation between neighborhood models of IPL are defined naturally and some of their basic properties are proved. We also consider Basic Propositional Logic (BPL), a logic weaker than IPL introduced by Albert Visser, and introduce and study its neighborhood models in the same manner.


Intuitionistic Logic; Basic Logic; Kripke models; neighborhood models; bisimulation; modal logic; topological semantics

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Bezhanishvili, N., and W. Hoek, “Structures for epistemic logic”, Chapter 12 in Johan van Benthem on Logic and Information Dynamics, Springer. DOI: 10.1007/978-3-319-06025-5_12

Chellas, B., Modal Logic: An Introduction, Cambridge University Press, Cambridge, 1980.

Johnstone, P., Stone Spaces, Cambridge University Press, 1982.

Hansen, H.H., “Monotonic modal logics”, Master thesis, University of Amsterdam, 2003.

Kruszelnicka, M., “A note on bisimulations of finite Kripke models”, Bulletin of the Section of Logic, 41 (2012): 185–198.

Montague, R., “Universal grammar”, Theoria, 36, 3 (1970): 373–398. DOI: 10.1111/j.1755-2567.1970.tb00434.x

Ruitenburg, W., “Constructive logic and the paradoxes”, Modern Logic, 1, 4 (1991): 271–301.

Scott, D.S., “Advice on modal logic”, Chapter 7 in Philosophical Problems in Logic, K. Lambert (ed.), D. Reidel Publishing Company, 1970. DOI: 10.1007/978-94-010-3272-8_7

van Dalen, D., Logic and Structure, Fourth Edition, Springer, 2004. DOI: 10.1007/978-3-540-85108-0

Visser, A., “A propositional logic with explicit fixed points”, Studia Logica, 40, 2 (1981): 155–175. DOI: 10.1007/BF01874706

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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