Can the modal auxiliary verbs be considered to be in the present tense (form, not meaning)?

According to Practical English Usage by Michael Swan, modal auxiliary verbs do not normally have past tenses:

The modal auxiliary verbs are will, would, shall, should, can, could, ought, may, might, and must. Their grammar is different from that of other verbs: for example, they have no infinitives, participles or past tenses [...] Modal verbs do not have infinitives or participles (to may, maying, mayed do not exist), and they do not normally have past forms.

So can each modal auxiliary verb be considered to be in the present tense (in form, not meaning)? Since the modal auxiliary verbs are finite and they do not have past tense forms (according to Michael Swan's Practical English Usage), each must be in a present tense form.

Or are will, can, may and shall the present tenses (in form, not meaning) and would, could, might and should the past tenses (in form, not meaning)?

What about the modal auxiliary must?

Getting back to the original question (asked 3 years, 8 months ago by someone who's left the group since), the actual answer is that whether you should consider modal auxiliary verbs to be inflected for tense depends entirely on what your grammatical system's definition of "inflected for tense" is, and how your grammatical system uses tense inflection.

For instance, some grammars of English insist that every sentence have a main tensed clause. Since modal auxiliaries can occur in simple sentences, these would have to be assigned a tense (typically Present, by default) by that grammar. On the other hand, your grammar might simply except modals from that requirement, or it might not have that rule at all, using something else instead.

No modern grammar that I know of distinguishes the historical present and past forms of modals (may, can, shall, will vs might, could, should, would, must) in any useful way, so the various final -t's and -d's are merely etymological curiosities, like the a-'s in awake, atop, aloft, and asleep.

A semantic past-present difference shows up with can/could in the ability sense, and will/would in the deontic 'willing to' sense. Both past descriptions refer to repeated behavior:

• When I was young, I could do 100 pull ups; now I can only do 99.
• When he was young, he would talk by the hour; now he won't say a word.

But that's about it; there's no tense or meaning difference between He might do it and He may do it -- both are possible, and that's it. Should has a special meaning as a weak must, both deontic and epistemic:

• She should/must go to the ball. (weak/strong obligation to go)
• This should/must be the place they mentioned. (weak/strong likelihood)

and, in the US at least, shall is not used for the future like epistemic will, but rather is restricted to two very special and limited deontic question constructions:

1. First person singular question - an offer to do a favor for the addressee

   • Shall I open the window?

2. First person plural question - an invitation to do something with the addressee

   • Shall we dance?

And, as you note, there is no historic present tense for must, as there is in German, where modal auxiliary verbs are not defective verbs like English modals, but real strong verbs, with all their principal parts, inflected regularly for everything.

So, if your grammar (and every linguist creates their own grammar of their own language) requires English modals to have tense, Voila! they do. If not, not. If you prefer, some may have tense and others may not. Questions like this are not matters of fact; they're arbitrary, so you might as well take whatever answer you like. It's your grammar, after all.