A slightly different form of this is to say " Sorry, I have a previous engagement." This stresses that you have already made a commitment to someone else.
"Sorry, I have a previous engagement." Here, does 'engagement' mean an agreement between two people to marry?
It has different meaning from the one in discussion, hasn't it? (I have another engagement = I have another plan vs. I have a previous engagement = I'm engaged)
"Sorry, I have a previous engagement." Here, does 'engagement' mean an agreement between two people to marry?
Basically, to have an engagement means to have an appointment or commitment. In other words, I am busy with something/somebody. Thus, in a sense it can refer to both 'being busy' an