AFFIRM: Web definitions for vow make a vow;
promise; "He vowed never to drink alcohol again"
wordnet.princeton.edu/perl/webwn
VOW: An earnest promise to perform a specified
act or behave in a certain manner, especially a
solemn promise to live and act in accordance with.
www.answers.com/topic/vow
OATH: A solemn, formal declaration or promise to
fulfill a pledge, often calling on God, a god, or
a sacred object as witness. The words or formula
of such a declaration or promise. Something declared
or promised. http://www.answers.com/topic/oath?method=6
But we're asking about the word, "affirm".
Using that site you linked to, the definition is,
af·firm (ə-fûrm')
v., -firmed, -firm·ing, -firms.
v.tr.
1. To declare positively or firmly; maintain to be true.
2. To support or uphold the validity of; confirm.
v.intr. Law.
To declare solemnly and formally but not under oath.
In fact, this is the exact same definition as the other links.
Nowhere in there does it say anything about 'affirm' means to take an oath . . .