Properties expressible in 2-CNF or 2-SAT How does one show that a certain property cannot be expressed in 2-CNF (2-SAT)?  Are there any games, such as pebble games?  It seems that the classical black pebble game and the black-white pebble game are unsuitable for this (they are PSPACE complete, according to Hertel and Pitassi, SIAM J of Computing, 2010).

Or any techniques other than games?

Edit:  I was thinking of properties that involve counting (or cardinality) of an unknown predicate (SO predicate, as finite model theorists would say).  For example, as in Clique or unweighted Matching.
(a) Clique: Is there a clique $C$ in the given graph $G$ such that $|C| \ge$ some given number $K$? $~$ (b) Matching: Is there a matching $M$ in $G$ such that $|M| \ge K$?

Can 2-SAT count?  Does it have a counting mechanism?  Seems doubtful.
A family of bitvectors is the class of solutions to a 2-SAT problem if and only if it has the median property: if you apply the bitwise majority function to any three solutions you get another solution. See e.g. https://en.wikipedia.org/wiki/Median_graph#2-satisfiability and its references. So if you can find three solutions for which this is not true, then you know it cannot be expressed in 2-CNF.