 Button (2011) has argued that, although this kind of technical criticism has teeth against the version of Putnam's argument which explicitly invokes the downward Löwenheim-Skolem theorem, there are alternate formulations of Putnam's argument which can evade the criticism. In particular, Button notes that even very weak theories can prove theorems like: “if ZFC is consistent, then ZFC has a countable model.” Since any proponent of ZFC must accept that ZFC is consistent, these weak theories are enough to get several variants of Putnam's argument off the ground.  See Button 2011 for development of this point.  See Bellotti 2005 and Bays 2007a for discussion of a somewhat similar point. 