These words all refer to the giving of help, hope or succour. Encourage is the most general term, and means to give hope, confidence or spirit, or to give active help: to encourage a young writer with praise; to encourage new industry by granting it tax concessions. A teacher who encourage his students to form their own opinions.

To embolden is to give someone confidence to undertake something. Success with a short story may embolden (or encourage ) someone to try his hand at a novel. To hearten is to renew someone’s spirit, especially to the point of giving him fresh courage to pursue his course of action: a victory that heartened the weary army; a visiting dignitary heartened by the unexpectedly warm reception he was accorded. To inspire , literally to breathe in, is to infuse with confidence or resolution, or to fire with enthusiasm, as though a new and vibrant spirit were breathed into one’s being: The example set by their valiant commander inspired the troops to take the hill despite heavy enemy fire.

To foster is, literally, to nurture, and by extension has come to mean to encourage by extending aid: to foster (or encourage) growth in the economy by lowering taxes. But even in extended senses, foster often retains the idea of gradual cultivation: Too rigid parental control fosters rebellion in children. In this sense foster is often used in contexts suggesting an unwise or at least controversial kind of help, and is sometimes akin to cherish, at other times to instigate: to foster the illusion that peace can be preserved by preparing for war; overzealous arguments that only fostered dissent and division.

Support , more than any of these words, implies that without the assistance offered the thing helped might founder and fail. It depends on the help given to survive or grow: to support a dying industry by granting it government subsidies. In another sense support means to endorse: a lawyer’s opinion that supported a litigant’s claim. Or support may mean imply to do what one can to help: to support the economy by buying government bonds. Back up is an informal phrase meaning to support or endorse the action or thinking of some party or person, but it has the additional sense of giving physical help if required: I’ll back up his criticism of the management; Will you back me up if I challenge him?

To promote is to encourage the growth or success of something: tariff revision that served to promote world trade; a good example serving to promote good manners in the young. Promote implies an aggressive and deliberate kind of assistance, often with a specific aim in mind: to promote one’s own welfare at the expense of others.