I sent a friend a question in the form:
Just trying to remember how we know each other?
That is a "statement" without the question mark at the end. Is there a term for this? Since it does not begin with a "How", "Why", "Where" etc..
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Cambridge calls them statement questions.
Although it is not a "real" definition, Urban Dictionary calls it a "Question-Statement".
A statement said in the form of a question.
This is simply a question. However, the meaning behind it and what you're asking becomes very unclear when you add the question mark.
What you're saying is confusing me.
This is a statement. You are telling someone they are confusing you.
What you're saying is confusing me?
The statement is now a question that could be interpreted a few different ways.
To apply this to the question you provided, by adding the question mark you are saying you don't know if you're trying to remember how you two know each other. You mostly likely weren't striving to say this.
So in the end, adding the question mark doesn't break the sentence grammatically—it just completely changes what you're trying to say. People who live and breathe correct grammar won't take your question lightly.
What you're describing is a transcription of a relatively recent phenomenon in spoken English called, colloquially, "uptalk." Oxford Dictionaries, for instance, defines it as "[a]manner of speaking in which declarative sentences are uttered with rising intonation at the end, as if they were questions" (http://www.oxforddictionaries.com/definition/american_english/uptalk).
More formally, in linguistic circles, uptalk is known as "HRT," or "high rising terminal."