 In 1969 Per Lindström used back-and-forth games to give some abstract characterisations of first-order logic in terms of its expressive power. One of his theorems says that if \(L\) is a language with a signature \(K, L\) is closed under all the first-order syntactic operations, and \(L\) obeys the downward Loewenheim-Skolem theorem for single sentences, and the compactness theorem, then \(L\) is reducible to the first-order language of signature \(K\). These theorems are very attractive; see Chapter XII of Ebbinghaus, Flum and Thomas for a good account. But they have never quite lived up to their promise. It has been hard to find any similar characterisations of other logics. Even for first-order logic it is a little hard to see exactly what the characterisations tell us. But very roughly speaking, they tell us that first-order logic is the unique logic with two properties: (1) we can use it to express arbitrarily complicated things about finite patterns, and (2) it is hopeless for discriminating between one infinite cardinal and another.