Establishing Negative Scientific Claims

Some of the more famous (and more important) of the laws of science involve a statement of what could not happen. For example, thermodynamics states that no substance could ever go below a certain temperature, and special relativity states that nothing could ever move faster than a certain velocity. As far as I know, however, there is nothing in Dr. Peikoff's induction theory that can deal directly with these cases (his theory deals with establishing positive causative connections). How, then, should these claims be reached?

Following Dr. Peikoff's pattern, let's look at the history of science for a clue. According to Wikipedia (which I am going to assume is correct for this post, since the particular historical facts aren't fundamental here), the first person to come up with the idea of an absolute zero in temperature was Guillaume Amontons. Amontons established (using the methods already outlined by Dr. Peikoff) a positive (in the mathematical sense) relationship between pressure and temperature, which would later be be shown to be a linear relationship. He then realized that this relationship, if it held true at extremes, implied a limit to temperature because pressure would eventually reach zero. If pressure were allowed to reach below zero, that would have to mean that somehow decreasing temperature causes the pressure around the containter to increase beyond the air pressure at that point, which would be absurd. Thus, the idea of a minimum temperature is necessary.

I think this example opens the door to answering my initial question. In this case, and I think in all such cases, a relationship between two factors was established a la Peikoff (not explicitly a la Peikoff, of course), and then, as an implication of the relationship established, a certain condition of one of the factors was shown to lead to a contradiction or an absurdity in the other, thus necessitating a limit on the first factor. Any condition which would theoretically cause some quantity to become actually infinite, or some scalar quantity to become negative, would therefore have to be thrown out. An important precondition of using this method is apparent: the relationship established must be correct even in the context of the extremes being discussed. If, for example, Newton's laws seemingly led to a contradiction in one of their terms near the speed of light (I don't think they do, but as an example), then there would still be no necessity to limit one of the other terms since Newton's laws do not apply at such speeds.

If this method is correct, how was the speed of light as an upper limit established? What contradiction would be reached if something exceeded that speed?