Perhaps some further explanation will help.
Where someone has been using a name for several years, and now wishes to confirm it in law, they can give a statutory declaration as to past use. That is attempting to describe historical fact, and is either true or false, and if false, the consequence of perjury can follow.
Where someone is attempting to change a name, there is no particular factual assertion involved. It is an act (a deed). If I do an act (say moving my arm to the left), that action will not normally have a characteristic of beig true or false.
It is worth noting that in either case, the legal essence is simply the use of a different/new name accompanied by a request to others to use that name also. Our law is very forgiving on this. No deed or stat dec is needed.
(Sorry, but the plural of deed poll is deeds poll)