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Modal auxiliaries are verbs which are characteristically used with other verbs to express mood or tense. In English, the modal auxiliaries are can, may, must, ought, shall, should, will, and would, this according to the American Heritage Dictionary. Have to, ought to, and be going to act as modals too, and can often be used interchangeably with must, should, and will, respectively. I will do so when valid. Comparative adjectives are words which compare nouns which each other, such as better, sillier, and more wonderful. The problem addressed here is a linguistic one, dealing with the structure of languages. Consider the following situation:

  1. The fence has to be between four and six feet high.
  2. The fence ought to be between four and six feet high.

There's not much difference between these two sentences right now, is there? Perhaps sentence 1 is a bit more imperative than sentence 2. However, when we construct a conditional situation:

  1. The fence is higher than it has to be.
  2. The fence is higher than it ought to be.

Oops, what happened? Now, sentence 1 denotes a fence of at least four feet, while sentence 2 only refers to fences of more than six feet. To analyze this, let us take a closer look at the two operative verbs.

Have to (or must) is pretty strong as modals go, setting down a requirement. By this token, however, it is decidedly weak in the negative, as in the sentence: "You don't have to do anything you don't want to." If you're mathematically inclined, imagine that each modal lies on a line which measures "strength;" negating a verb flips it around the middle point. In this model, our have to comes in way on one side, so that not have to gets mirrored all the way across to the other end.

On the other hand, Ought to (or should) is not quite so commanding, setting down more of a preference; likewise, negation doesn't change its punch quite as much. In fact, "you should go" and "you shouldn't go" have pretty much identical urgency, which puts this verb around the middle of the line in the mathematical visualization.

Anyway, as to the question at hand, let us express the two sentences again, this time without the confusing spans:

  1. The fence must be at least four feet high; the fence must not be more than six feet high.
  2. The fence should be at least four feet high; the fence should not be more than six feet high.

If the fence is higher than it must be, that means that it does not have to be as high as it actually is. And how high does it have to be? At least four feet, as seen in the above sentence 1. So this leads to our conclusion that the fence is more than four feet high.

However, if the fence is higher than it ought to be, it should not to be as high as it is. And according to sentence 2, it should not be more than six feet high. Therefore, this fence is more than six feet high.

So, to sum it up, the problem caused by comparative adjectives seems to stem from the relative emphases of the verbs have to and ought to, and their corresponding negative forms.

--I encountered this problem, but not the analysis, several months ago on the Internet. Unfortunately, now that I finally want to write it up, the website seems to have vanished. If anyone comes across a site which mentions this problem, I'd be more than happy to cite it here.

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