We study FO+, a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is an FO-definable language that is monotone in monadic predicates but not definable in FO+. This provides a simple proof that Lyndon's preservation theorem fails on finite structures. We lift this example language to finite graphs, thereby providing a new result of independent interest for FO-definable graph classes: negation might be needed even when the class is closed under addition of edges. We finally show that the problem of whether a given regular language of finite words is definable in FO+ is undecidable.

@article{paperbot2203,
  title = {Positive First-order Logic on Words and Graphs},
  author = {Denis Kuperberg},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 19, Issue 3},
  year = {2023},
  doi = {10.46298/lmcs-19(3:7)2023}
}