The image pattern of an unweeded garden exemplifies Hamlet’s ..

This thorny question must have been further exacerbated for Shakespeare's audience by the play's allusions to Purgatory. The word itself is not actually mentioned inHamlet (it is named elsewhere twice, in Romeo and Juliet and Othello), but when the Ghost describes to Hamlet how he is "Doomed for a certain term to walk the night, / And for the days confined to fast in fires" until the "foul crimes" done in his "days of nature / Are burnt and purged away" (1.5.11-14), he seems unmistakably to be talking about Purgatory. Roman Catholic doctrine had evolved that concept partly as a way of explaining what must happen to souls of those who depart from mortal life without having received Last Rites or Extreme Unction. Since Church teaching held that salvation was not possible without forgiveness of sins, the souls of all those persons dying without forgiveness were thought to be held in a place not unlike what the Ghost describes to Hamlet, where they would penitently suffer until they had made sufficient satisfaction for their sins. Presumably in King Hamlet's case the "foul crimes" that he had committed were not violent crimes like murder, for only those who were deemed worthy of eventually receiving God's grace were granted the dispensation of Purgatory rather than hell. King Hamlet's "foul crimes" must have been the pride, anger, envy, covetousness, gluttony, slothfulness, and lechery constituting the Seven Deadly Sins to which all humans have been incessantly prone since the fall of Adam and Eve in the Garden of Eden. Dante devoted the middle of three books of his Divine Comedy to a vivid description of what Purgatory could be like.

and then slammed his hand angrily on the table to exclaim “Fie” at the world’s unweeded garden
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Dramatic art, for Hamlet, should aim at a very high cultural standard. It must not cater to the "groundlings," the "unskillful," the iunjudicious, who, in Hamlet's austere view, are for the most part "capable of nothing but inexplicable dumb shows and noise" (3.2.11-26). Groundlings were those who stood around the platform stage in theaters like the Globe, and were indeed an important part of Shakespeare's audience, so that the insult here seems a bit jarring, but it may be that Hamlet, in speaking directly to them, is warning them of what "groundlings" are all too capable of and is thereby congratulating those among them who will choose to be discriminating and judicious rather than philistine.

>Compares Denmark to “unweeded garden” ..

Denmark is to him "an unweeded garden / That grows to seed" ..
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As he readied himself for the business of murdering his royal brother, Claudius must also have calculated his chances very carefully about Gertrude. Would she agree to marry him? We never learn if he broached his scheme to her before the event, though it seems doubtful. Later Hamlet does confront his mother briefly with this question when he says to her that his own mistaken bloody deed of killing Polonius is "almost as bad, dear mother, / As kill a king, and marry with his brother" (3.4.29-30). But her quick answer, incredulously repeating his words, "As kill a king!" evidently satisfies him that she knew nothing of Claudius's murderous plans, for Hamlet turns instead to the other more plausible accusation that she has been faithless and wanton. The Ghost of Hamlet's father apparently regards her as guilty of lesser crimes than those of her new husband, for the Ghost instructs Hamlet in his revenge to punish Claudius with death but to leave Gertrude "to heaven / And to those thorns that in her bosom lodge / To prick and sting her" (1.5.87-9). Even if she was not complicit in murder, however, she may have unintentionally encouraged Claudius by a tangible warmth of affection. He must have known that he could have her if her husband were suddenly out of the picture. Certainly, in Hamlet's view, she has violated all sense of decency by joining Claudius so quickly in bed.

>Hamlet asks her if she is honest and fair because if she …

Perhaps this irony is well suit fited to this profoundly ironic play. Hamlet has fitfully been viewed as one who might rule Denmark ably, even brilliantly. But that appears to be possible only in another world, one in which Denmark is no longer "an unweeded garden" (1.2.135). Claudius, during his brief reign, has proved to be just the sort of monarch that Denmark pragmatically longs for: militarily strong against invasion, suave in diplomatic efforts to keep those enemies at bay, and politically united under a king who is a superb rhetorician in the art of persuasion, which after all is rhetoric's main purpose. Polonius loves this king, as do Rosencrantz and Guildenstern. Hamlet is the chief opposing figure, but he despises politics and makes no attempt to flatter persons he regards as toadies like Polonius. Idealizing as he does his warlike father, Hamlet is no person to conduct the kinds of diplomatic trade-offs that Claudius handles so adroitly. Fortinbras is another kind of leader, one who practices a more direct and ruthless control, wholly unlike what we think Hamlet might choose, but ruthlessly effective in the world as it exists. Hamlet is too noble, too idealistic, too intellectual, for the job. As he dies, he sees that and accepts the consequences of this, as indeed he has few if any other options. The tragedy of Hamlet is as much a tragedy of the fallen state of the world as it is of its protagonist.