Our dog escaped! You can't have shut the door properly!
Does the phrase 'You can't have shut the door properly!' mean 'You must have shut the door in a wrong way', or 'you haven't been able to shut the door properly'?
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Does the phrase 'You can't have shut the door properly!' mean 'You must have shut the door in a wrong way', or 'you haven't been able to shut the door properly'?
Since the sentence appears to refer to a single escape incident, "You can't have shut the door properly!" is more properly described by: "You must have shut the door in a wrong way" - which refers to a single incident- than by: "you haven't been able to shut the door properly", which implies more than one instance of gate-shutting incompetence.
You can write it differently, while keeping the "can't":
It can't be that you shut the door.
or
It's impossible that you shut the door.
So, s/he wasn't unable to shut the door, just didn't do it.
If you want to express the inability to shut it:
You can't shut the door, or
You couldn't shut the door.
This page on BritishCouncil.org called Modals - deductions from the past "focuses on making deductions about the past." It says:
We can use modal verbs for deduction – guessing if something is true using the available information. The modal verb we choose shows how certain we are about the possibility.
It also says:
We use can't have and couldn't have + past participle when we think it's not possible that something happened.
and provides these examples:
She can't have driven there. Her car keys are still here.
I thought I saw Adnan this morning but it couldn't have been him – he's in Greece this week.