Abstract
In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (meet) and implication, using finite duality for distributive lattices and universal models. We give a description of the finitely generated universal models of this fragment and give a complete characterization of the up-sets of Kripke models of intuitionistic logic which can be defined by meet-implication-formulas. We use these results to derive a new version of subframe formulas for intuitionistic logic and to show that the uniform interpolants of meet-implication-formulas are not necessarily uniform interpolants in the full intuitionistic logic.
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Notes
- 1.
In less recent literature, these are also called Brouwerian semilattices.
- 2.
Essentially the same argument as the one sketched in this paragraph can be used to give a description of an arbitrary, not necessarily finitely generated, free distributive lattice, but we will not need this in what follows.
- 3.
For an equivalent characterization of descriptive general frames as the ‘compact refined’ general frames, cf. e.g. [5, Definition 2.3.2, Theorem 2.4.2].
- 4.
The canonical frame and model are also known as the Henkin frame and model.
- 5.
Recall that a model \(M\) is called image-finite if, for each \(w\in M\), the set of successors of \(w\) is finite.
- 6.
- 7.
We use the usual convention that \(\bigvee \emptyset = \bot \) and \(\bigwedge \emptyset = \top \).
- 8.
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Acknowledgements
We are thankful to Mai Gehrke for many inspiring discussions on this paper. We also thank the referees for many useful suggestions. The first listed author would also like to acknowledge the support of the Rustaveli Science Foundation of Georgia under grant FR/489/5-105/11.
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Bezhanishvili, N., Coumans, D., Gool, S.J.v., Jongh, D.d. (2015). Duality and Universal Models for the Meet-Implication Fragment of IPC. In: Aher, M., Hole, D., Jeřábek, E., Kupke, C. (eds) Logic, Language, and Computation. TbiLLC 2013. Lecture Notes in Computer Science(), vol 8984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46906-4_7
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