# On the expressive power of first-order modal logic with two-dimensional operators

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## Abstract

Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages.

## Keywords

First-order modal logic Expressive power Two-dimensional operators Actually Fixedly Vlach Operators Bisimulation## Notes

### Acknowledgments

Many thanks to Melissa Fusco, Wes Holliday, and two anonymous reviewers for all their feedback on this paper. Thanks also to those who participated in UC Berkeley’s dissertation seminar in the Spring of 2016 for all their valuable comments and suggestions on an earlier draft of this paper.

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