TITLE: Prove that a set of connectives is inadequate
QUESTION [12 upvotes]: It is relatively easy to prove that a given set of connectives is adequate. It suffices to show that the standard connectives can be built from the given set. It is proven that the set $\{\lor, \land, \neg\}$ is adequate, and from that set it can be inferred (applying De Morgan laws and such) that $\{\lor, \neg\}$, $\{\land, \neg\}$ and $\{\to, \neg\}$ are also adequate.
Nevertheless, I'm stuck trying to understand how to prove that a given set of connectives is inadequate. I know I have to prove that a standard connective can't be build using only the connectives of the given set, but I can't figure out how to do it.
FYI, I'm trying to prove that $\{\lor, \land\}$ and $\{\leftrightarrow, \neg\}$ are inadequate sets of connectives.
Thanks in advance.

REPLY [7 votes]: There is a result in Robert Reckhow's thesis that characterizes the adequate sets of connectives. The result says that for a set of connectives to be complete one needs the following:

F and T (or formulas with these values),
an odd connective (a connective is called odd if some fixing of its input variables with T and F except two input variables has odd number of Ts in its truth table),
a non-monotone connective (a connective that turning an F to a T will make its value change from T to F).

These are necessary and sufficient conditions for a set of connectives to be adequate. If a set of connectives is not adequate then it lacks one of these. If you have an odd connective, you can fix the inputs with T and F to get either $\vee$ or $\wedge$. If you have a non-monotone connective, you can fix the inputs with T and F to get $\lnot$. Any boolean function can then be constructed from these as a CNF.
Note that even connectives are closed under composition as well as monotone connectives. So to prove that a set of connectives is inadequate, you typically need to show that either

all of the connectives are monotone, or
all of the connectives are even.

For your examples, the first one is a set of monotone connectives, the second one is a set of even connective. So they cannot be adequate.