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Hi, could someone explain what's the difference among those words? thanks.
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'Vow' and 'pledge' are the most formal activities; 'swear' is next, and 'promise' is the least formal activity. By 'formal', I do not mean register; I mean the activity itself.
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The difference is that some have particular uses, though they all can mean the same thing.

Vows are exchanged at a wedding.

Allegiance is pledged.

One swears to tell the truth in court.

In many other situations, though, they can be used almost interchangeably.