Can't have vs. couldn't have

Q: Can these two modal perfect forms be used interchangeably or is there any formal usage restriction in either case? I have found it difficult to explain this to my students.

A:
As you realize, references don't usually address can't have vs. couldn't have directly.
One that does mention it, however, is Betty Azar, in Understanding and Using English Grammar, 3rd ed. (Pearson longman, 2002). On page 181, she places can't have and couldn't have in a chart showing their status on a scale of probability. They are equal.

The title of the chart is "DEGREES OF CERTAINTY: PAST TIME," and the section of the chart is "PAST TIME: NEGATIVE," (distinguishable from the section above it called "PAST TIME: AFFIRMATIVE.")

In answer to the question "Why didn't Sam eat?":

• Sam wasn't hungry. (The speaker is 100% sure that this is the reason.)
• Sam couldn't have been hungry.
• Sam can't have been hungry. (The speaker believes - is 99% certain -that it is impossible for Sam to have been hungry.)
• Sam must not have been hungry. (The speaker is making a logical conclusion. We can say he's about 95% certain.)
• Sam may not have been hungry.
• Sam might not have been hungry. (The speaker is less than 50% certain, and is mentioning one possibility.)

So, can't have and couldn't have are equal in meaning when they express the impossibility of something.

Stylistically, couldn't have seems more formal than can't have.

In writing, could not have would probably be used, especially in referring to something historical:

• George Washington could not have known Abraham Lincoln - they lived at different times.
• The original inhabitants could not have eaten peanuts - peanuts were unknown at the time.

When couldn't have is used as a past conditional, can't have can not be substituted, in any style:

• The team could not/ couldn't [not can't] have won the game if they had not trained so intensively.
• I could not/ couldn't [not can't] have passed the course without your help.
• My father could not/ couldn't [not can't] have had the success he enjoyed had my mother not always encouraged him.

Q:Is there a difference in usage between can't have and couldn't have? 

A:
They are very similar in their degree of certainty, as described in Azar. But are they truly interchangeable, as Anonymous asks? They seem not to be fully interchangeable, not only in historical writing, as it was pointed out, but in other contexts as well.*

present contexts where the events being talked about are recent in time

Can't have seems to be used a great deal in present contexts where the events being talked about are recent in time. It expresses a judgement about a recent action or situation, a context in which the issues are still fresh and relevant to the present. One use is in present real conditionals, e.g., in sentence (a):

(a)
Welcome to my personal page about the eclipse of the sun, if you find any of this informative then I can't have written it right. (If A is true, then B is also true.)

simple belief in the impossibility of a (recent) past event or situation

Other uses of can't have express simple belief in the impossibility of a (recent) past event or situation, as in (b), (c) and (d):

(b)
Her favourite film is Kes, and she talks with awe about the moment when the boy finds the dead bird. She thinks, from the look on his face, that the child actor can't have known in advance that the bird was dead.

(c)
If he knew there were blacks in all the other countries, but he didn't know there were blacks in Brazil, he'd have to have thought Brazil was somehow an exception to the rule. But he can't have believed that. No sane person could. Brazil has the second-largest black population of any country in the world.

(d)
Please do let me know if you find problems [in the program] as I'm quite sure I can't have found every bug.

The underlying message of these assertions above — (b), (c), and (d) — is "It isn't possible (in the present) that..." It often expresses strong disbelief, or even refusal to believe something.

Or the speaker may have objective evidence to support the idea of impossibility, as in (e), (f) and (g):

(e)
Tom can't have written this because it is in French and he doesn't know French.

(f)
I've never written an Improv part. Therefore, I can't have written bad Improv.

(g)
From what you have said, you can't have known the new man very long.

In each of these cases — (e), (f) and (g) above — the speaker has a warrant for the assertion being made. And in every case so far, the topic under consideration is a recent situation or event rather than a remote past situation or series of events.

expressing a meaning that has nothing to do with possibility

Additionally, can't have is used to express a meaning that has nothing to do with possibility. Can't have is used for present (negative) obligation or permission. For example, in (h) below:

(h)
The Oscar rules state that the song must be recorded for "use in a film" prior to any other use...i.e., you have to write it for the film, you can't have written it before (even it was never recorded), and you have to record it for the film before it can be re-recorded, remixed, etc. on your own.

This use of can't have, which means "You are not allowed to have (written...)" is totally different in meaning from couldn't have, which is not used for negative permission in the past.

An interesting feature of can't have that was revealed in the search is that many instances of its use are in ES/FL lessons. This fact, together with the ubiquity of can't have in fiction writing, may say something about the practical usefulness of the form, suggesting that its use in everyday speech is probably rather limited. In fact, the proportion of instances of can't have in comparison with couldn't have is very small. For example, can't have known occurs 500 times, whereas couldn't have known occurs 14,500 times. Can't have been occurs 15,900 times, but couldn't have been occurs 229,000 times!

past impossibility of an idea being true

Couldn't have expresses past impossibility of an idea being true. It has a less complex range of meanings than can't have.

In addition to its use in full past hypothetical conditionals, as it was explained, it is found in incomplete conditionals, utterances in which the if- clause is merely implied. For example, in (i), (j), and (k) below:

(i)
I couldn't have done it without my family [if I hadn't had my family]

(j)
The timing couldn't have been worse [if circumstances had been different]

(k)
I couldn't have written a better script [if I'd tried to do it]

These ideas have no present tense counterpart with can't have.

impossibility in past time narratives or commentary

Non-conditional uses of couldn't have abound. It occurs to express impossibility in past time narratives or commentary, as in (l), (m), and (n) below:

(l)
[He] couldn't have known that just six hours later, he, the only male flight attendant on American Airlines Flight 11, would die suddenly when...

(m)
As a young woman who had never given birth before, she couldn't have known what the experience of childbirth would be like...

(n)
Athletic Director Edgar Johnson couldn't have known what he was getting into when he made that fateful decision to cut the Delaware wrestling program on June 18.

without the ability or skill 

With actions that take some ability or skill, it means didn't have the ability or skill to, as in (o), (p), and (q) below:

(o)
I wish I had written Nowhere Man, by Aleksandar Hemon, but I couldn't have written it, because no one can write like Hemon.

(p)
The imagination was there and so was the humour that was to appear later in my books, but my teachers didn't seem to notice, except for one who said, "You couldn't have written this."

(q)
... So, anyway, I could have possibly written about the incidents, but I couldn't have written about them effectively.

This use of couldn't have embodies the root meaning (ability or capability) of the modals can and could.

could have for present situations

Could have + past participle can refer to present situations which were possible but have not been realised.

• He could have been Prime Minister now if he hadn't decided to leave politics. 
• We could have spent today at the seaside, but we thought it was going to rain, so we decided not to.

could have for criticizing

This structure can be used to criticise people for not doing things:

• You could've helped me - why did you just sit down and watch?

To sum up: the two verb phrases share a similar degree of probability, as seen in Azar. But there are differences in usage between them:

1)
Can't have tends to be used more in present contexts or contexts about recent situations or events, while couldn't have is used more in definitely past contexts.

2)
Can't have is found in a lot of fictional writing, perhaps more than is couldn't have.

3)
Can't have is used to express negative permission, while couldn't have carries no such meaning.

4)
Can't have is used much less than couldn't have - a small percentage of usage. This fact may guide teachers in deciding how much time to spend teaching each of the forms.

*The example sentences have been taken from a Google on- line search.

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