Failure of Demand
Trying to argue any complex idea with the committed opponents is a usually a fool's errand. Thus it is with Keynsian economics. Like relativity, it contains a concept or two which is unintuitive, and most people lack the imagination or willingness to challenge their common sense - what Einstein called the collection of prejudices we have acquired by age 18.
The first unituitive idea of Keynsianism is the so-called paradox of thrift. The idea is that if everybody with money starts becoming afraid to spend or lend it, everyone gets poorer. With nobody spending money, those who owe money are unable to find work, and can't pay back their creditors, making them poorer as well. With fewer people working, society produces less, so their are fewer real goods to go around. This is a second notion that some claim to find paradoxical - failure of demand.
At this point economists go their separate ways. Rational expectationists appear to claim that such a failure can't happen, since it would violate some axiom of their perfect market theory. Austrians say great, there can never be too much destruction in the creative destruction cycle - it's an optimal time to grind up and spit out those underqualified workers and uncompetitive businesses. Monetarists say - drop money from helicopters - that should stimulate demand, though in practice, their helicopters always seem to be hovering over banks. Keynsians say, OK, let government be the employer of last resort, and hire people to do useful things like repairing roads and schools. This is feasible if, as in the US today, the government can borrow money very cheaply to hire those people.