“In seceding the Southern states were giving up all claims to the Western territories and thus all chance to spread slavery there.”
Not Quite. In July of 1861 Davis ordered General Sibly to organize a force to enter the Arizona Territory and take control of it from The U.S. IMO that Davis’s objective was the abundant natural resources in the territory, not so much as to expand slavery.
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Yes, quite. You are talking about a time AFTER Lincoln initiated hostilities. Once two sides are fighting in a war, all bets are off. Each will do whatever is militarily expedient to win. When the Southern states seceded, they made no claim to the western territories of the US - only to their own sovereign territory. They were literally giving up any chance to spread slavery to those territories by the very act of secession. So obviously, their main complaint was not that they were prevented from spreading slavery - else they would not have done so.