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When auxiliary verbs are used, must tense agreement always hold? For example, which of the following is grammatically correct?

Yesterday, I realized that I can solve puzzles.

or

Yesterday, I realized that I could solve puzzles.

I have this question because I was working on an SAT Writing problem, but came across the problem

In the fifteenth century, Leonardo da Vinci suggested that defective vision can be corrected by placing a lens in direct contact with the eye. No error

and the answer key says that the "can" should be "could".

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Both examples are of reported speech. Yesterday you said

"I can solve puzzles."

In the 1400s, Leonardo said

"Defective vision can be corrected by placing a lens in direct contact with the eye."

Thereby anticipating the contact lens industry by centuries. When we report these words later without quotes, we have to correct, where required, for person, place, and tense.

For tense, reporting may be "back shifted" to the past. You actually said (perhaps to yourself) the words "I can," but that was yesterday, which you may indicate by using the past tense:

Yesterday, I realized that I could solve puzzles.

Notice that I said that you "may" back shift. It's not always required, and for things that continue to be true in the present, and especially those things that always remain true, you may choose to keep the verb in the present tense. Thus,

Yesterday, I realized that I can solve puzzles.

means that you could solve puzzles yesterday when you realized that you had that talent and that you can still solve puzzles today.

Likewise for Leonardo's statement. The laws of optics haven't changed since Leonardo's time and they're not like to change ever, and this is implied by the statement:

In the fifteenth century, Leonardo da Vinci suggested that defective vision can be corrected by placing a lens in direct contact with the eye.

Contact lenses would have worked in Leonardo's time, they work today, and they will continue to do.

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  • So in the case that something would have worked in the past but not the present, we would have to use "could" rather than "can" correct? – Tae Hyung Kim Sep 16 '15 at 5:21
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    @thkim1011 Yes. Let me illustrate all three shifts. Suppose I say today (an election day) at the polling place, "I can vote here today." The next day, when somebody else somewhere else reports my speech about what has become a past event, he'll say "He said that he could vote there yesterday." The "I" becomes "he" because the reporter is talking about me as a third person; "here" become "there" because the reporter isn't at the polling place; "today" becomes "yesterday" and the tense changes from present ("can") to past ("could") because the event is past and confined to the past. – deadrat Sep 16 '15 at 6:03

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