Giving a promise is a statement of assurance or commitment towards someone or something, which implies sticking to one's word. There are different synonyms for the phrase 'give promise,' including pledge, vow, guarantee, commit, and assure. Pledge and vow signify a solemn promise, while assure means to give confidence to someone. Guarantee is a term often used when one wants to give a warranty or assurance of something, while commit describes the act of dedicating oneself to a particular course of action. In conclusion, choosing the right synonym for the phrase 'give promise' will depend mainly on the context and the level of commitment and assurance one wants to convey.