How are the core values of integrity, honesty, loyalty, and moderation presented in Shakespeare's Richard III?
Of the values listed—integrity, honesty, loyalty, and moderation—the protagonist, Richard himself, exhibits none. What's more, he tells us in his opening soliloquy that he isn't going to: "I am determined to prove a villain / And hate the idle pleasures of these days."
Throughout most of the play, characters that do exhibit these values either do so at their peril, or do so in warped ways. For example, Queen Margaret is savagely honest in her appraisal of almost everyone else in the play, including Richard, but her honesty comes from bitterness, and she is far from honest about herself and her own actions. Buckingham exhibits loyalty to Richard almost to the end, but this is in expectation of riches and lands as rewards, and he abandons Richard when it becomes clear he has fallen from favor with the newly made king.
It's left to the highly idealized figure of Henry Richmond, the eventual historical Henry VII, to display integrity and moderation. Shakespeare deliberately contrasts Richmond and Richard's orations to their troops to show that Richmond is a paragon of knighthood, invading the kingdom only to bring law, order, and justice back to the land, while Richard is as bloodthirsty and duplicitous as ever. As virtually all scholars observe, Henry Richmond had to be portrayed in this way, since his ascension to the throne had to be seen as legitimate. He becomes the first Tudor king, father of Henry VIII, and thus grandfather to the reigning monarch, Elizabeth I.
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