In the starkest terms, a state is defined by a bureaucratic hierarchy that monopolizes the use of force over a specific geography. Ideally, nobody need fear the authorities except those who break the law. And because the authorities monopolize violence, nobody need fear his fellow man. Of course, tyrannical states induce general fear among much of the population. And weak states have a difficult time monopolizing the use of force -- the reason why they are weak in the first place. By these standards, many states in the world are weak. And Libya has gone from being a tyrannical state to being barely a state at all.
Given the calls for intervention in Syria, let's consider Libya, where a modest intervention was tried.
The authorities in the capital of Tripoli openly acknowledge the fact that they do not monopolize the use of force and have wisely opted for compromise and arbitration in eastern Libya (the Benghazi region) and in the far-flung Sahara to the south. It is difficult to predict whether Libyan affairs will carry on in the form of a benign and relatively mild anarchy (with some institutions working and others not) or will advance in the direction of a more coherent democratic state. Of course, a descent into worse chaos cannot be ruled out.
Libya's fundamental problem is that rather than comprising a compact cluster of demography like the Nile Valley, it is but a vague geographical expression -- a monumentally vast desert and coastal region between historic Egypt and Greater Carthage (Tunisia). Because Morocco, Tunisia and Egypt are geographically associated with specific knots of civilization going back to antiquity, they did not require suffocating forms of tyranny to hold them together like Libya, and to a lesser extent like Algeria, which for decades during the height of the Cold War had a radical socialist regime. For Libya, Moammar Gadhafi's regime was, in fact, anarchy masquerading as tyranny.
Therefore, it should surprise no one that the toppling of Gadhafi brought about the veritable collapse of the state. Libyan authorities do not govern so much as negotiate the terms of geographic control. If anyone doubts the fact that the Libyan state barely exists, they should investigate the situation on Libya's borders. In Libya, borders -- with their connotation of specific, legal lines characterized by passport and security surveillance -- have given way in the direction of frontiers, a term implying overlapping movements of gangs, militias and tribes. Modern states have borders; weak and failed states have frontiers.
For example, the collapse of Gadhafi's regime brought about the second-order effect of war and anarchy in nearby Mali. Ethnic Malian Tuaregs who had backed Gadhafi fled Libya en masse, taking with them large-scale caches of weapons upon the Libyan leader's demise. The Tuaregs headed back to Mali, where they wrested control of the desert north of that country from a government located far to the south in the capital of Bamako. After the Tuareg rebellion was co-opted by jihadists, there were reportedly almost 2,000 deaths and wholesale raping and looting, in addition to the sacking of world heritage sites. The French government subsequently intervened with troops. Now there are multiple patches of sovereignty in a confused battlefield all across the Sahel and Sahara. The stability of regimes in places like Mauritania and Niger are somewhat more in doubt than before Gadhafi's collapse. Libya, for that matter, is now an ungovernable space in significant parts of the country where al Qaeda can very possibly find refuge. The killing of the American ambassador in Benghazi was indicative of the terrors that a chaotic, post-Gadhafi Libya can offer up.