I stuck at page 338, Longman Student Grammar of Spoken and Written English. Here's an excerpt,
Certainty adjective controlling subject-to-subject raising
1.[The government] is unlikely to meet the full cost.
. . . This meaning relationship becomes explicit if we paraphrase the structure with an equivalent that-clause. Two constructions are possible
1.a. That the government will meet the full cost is unlikely.
1.b. It is unlikely that the government will meet the full cost.
I don't understand the presence of will here, why is to meet.. rephrased to will meet... Does to refer to a future event?
Best Answer
This is not really about the word to or the verb meet. Rather, the phrase is unlikely combined with a to-infinitive (in this case, "to meet") tells us that the action of the infinitive verb will probably not happen in the future.
The words likely and unlikely are about how certain we are about something happening in the future. From Cambridge Dictionary's English Grammar Today1:
As noted in this explanation, if we say an event is [un]likely we are talking about how certain we are that the event will happen in the future. So in your example,
the sentence is talking about how confident we are now that the government will pay for the whole thing at some time in the future.
Note that if we used a different tense of the verb with unlikely, we could have a different meaning:
In this case, we've paired unlikely with the past tense of the verb "seem", so the sentence means that sometime in the past it looked like the government would be unable to pay. In that case, the expected failure to pay was at some time in the future of the time of "seeming" but not necessarily later than now. To make this clearer we can add a bit of context:
Similarly, we can use different infinitive verbs with these same likely and unlikely structures:
You can read more about the use of likely and unlikely in the article linked above.
EDIT:
As was pointed out in comments, unlikely can be paired with other forms of infinitive besides the simple infinitive (e.g. the continuous—is unlikely to be meeting—and so forth) which of course changes the meaning. As usual, the infinitive in this case takes its tense from the verb that precedes it, and contributes aspect—the relationship in time between the action of the preceding verb and the infinitive verb.
The key issue here is that the usual temporal relationship for simple to-infinitives is changed when we add likely or unlikely. With the simple infinitive, ordinarily the action of the infinitive and the action of the tensed verb are understood to happen at the same time. With (un)likely, however, the infinitive's action is shifted to happen after the tensed verb. Compare:
vs
With other forms of infinitive the relationship is more complicated, and outside of the scope of this Q&A, but generally the "action" that is shifted forward is finding out the truth or falsity of the infinitive.
vs.
You can learn more about the different aspects of infinitives in the Grammaring article linked above, or your favorite English grammar source.
1 Markdown formatting added; sorry it's so ugly, but I couldn't think of a better way of adding emphasis when the original is already using italics and bolding.