It is well known that the literature on judgment aggregation

inherits the impossibility results from the aggregation of preferences

that it generalises. This is due to the fact that the typical judgment

aggregation problem induces an ultrafilter on the the set of individuals,

as was shown in a model theoretic framework by Herzberg and

Eckert (2009), generalising the Kirman-Sondermann correspondence and

extending the methodology of Lauwers and Van Liedekerke (1995). In the

finite case, dictatorship then immediately follows from the principality

of an ultrafilter on a finite set. This is not the case for an infinite set

of individuals, where there exist free ultrafilters, as Fishburn already

stressed in 1970. The main problem associated with free ultrafilters in the

literature on aggregation problems is however, the arbitrariness of their

selection combined with the limited anonymity they guarantee (which

already led Kirman and Sondermann (1972) to speak about invisible

dictators). Following another line of Lauwers and Van Liedekerke's (1995)

seminal paper, this note explores another source of impossibility results

for free ultrafilters: The domain of an ultraproduct over a free ultrafilter

extends the individual factor domains, such that the preservation of the

truth value of some sentences by the aggregate model - if this is as

usual to be restricted to the original domain - may again require the

exclusion of free ultrafilters, leading to dictatorship once again.