The Confederacy hoped to achieve the following with a major victory at Gettysburg:
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They hoped to convince European nations to recognize the Confederacy.
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They hoped to turn the tide of the war in favor of the South.
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They hoped to finally defeat the Army of the Potomac.
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They hoped to break the Union’s will to keep fighting.
All of the boxes apply to the Confederacy's hopes for a significant victory at Gettysburg.