  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.
