I couldn't resist writing an answer about Gracchus Babeuf and the Conspiracy of the Equals. Not only because they're the part of the Revolution I'm most interested in right now, but I think most would agree that the Equals were the most radical organized movement of the Revolution.

The conspiracy only lasted for about a month during the spring of 1796. It was formed out of various surviving parts of the revolutionary left, and its goal was to overthrow the government and abolish private property. They called the system they aspired to establish "common happiness". After being betrayed by one of their agents, the members who could be identified by authorities were put on trial. Gracchus Babeuf, a journalist who played a significant role in the conspiracy, was eventually convicted and executed.

According to Babeuf's defense speech, the movement's goals were justified because of the terms of the original social contract. Referring to the first article of the declaration of the rights of man and of the citizen from 1793, which states that "the goal of society is common happiness", Babeuf said that

That is the whole of the law and its prophets. I challenge you to say to me that men, uniting themselves in an association, would have another goal, another will than that of all being happy. I challenge you to claim that they would have consented to this union if they had been warned that institutions would be formed whose result had to be that soon the greatest number would bear all of the burden of labour, would sweat blood and die of hunger to support the delights and inaction of a handful of privileged citizens. But since this happened, as eternal laws don't expire, I have the right, as a man, to demand at any time the execution of the first pact, which, even if it were true that it were made silently, is found written by nature in ineffaceable letters deep in every heart. Yes, there is a voice that cries out to all: the goal of society is common happiness. There is the primitive contract; it didn't need terms to express itself, it is sufficiently extensive, because all institutions must derive from this source, and none can detract from it.

Another example would be Antoine Saint-Just's speech arguing for the execution of Louis XVI. The trial of the deposed king came with some legal difficulties, since the old constitution gave the king absolute immunity from prosecution. That was a problem the revolutionaries would have to solve if they wanted to punish Louis in a legal way. The idea of a contract as the basis of all law was essential to Saint-Just's solution.

Saint-Just argued that society is created from a pact between citizens. From contract law, he borrowed the principle that you could only be party to a contract if it imposed some obligation on you. For a normal citizen, the obligation is obvious. All citizens are bound by this contract through the law that emanates from it. Louis, however, was not. He had absolute impunity. Since he had no obligation according to the terms of the contract, he wasn't party to it. And if he isn't a part of the contract, the laws based on the contract don't apply. Therefore, Louis can be killed.