Finitely monotone properties

A characterization of definability by positive first order formulas in terms of Fraisse-Ehrenfeucht-like games is developed. Using this characterization, an elementary, purely combinatorial, proof of the failure of Lyndon´s Lemma (1959) (that every monotone first order property is expressible positively) for finite models is given. The proof implies that first order logic is a bad candidate for the role of a uniform version of positive Boolean circuits of constant depth and polynomial size. Although Lyndon´s Lemma fails for finite models, same similar characterization may be established for finitely monotone properties, and we formulate a particular open problem in this direction