Individuals would, of course, be 'allowed' to own nuclear, chemical, and biological weapons. However, as mentioned before, this would probably lead to very high insurance rates for individuals. Also, it is unlikely that anyone would be willing to deal with you if you kept nukes laying around the house.
On the other hand, private defense agencies and militias with a good reputation would probably own tactical nuclear and chemical weapons, as deterrents to large-scale invasions, without too much of a financial penalty. Biological weapons are rather more iffy, as (with current tech, this could change in the future) they are rather hard to control. Large-yield nukes (in excess of a few kilotons, really) would also probably be out of the question.
Again, all of this is completly voluntary. You *could* keep a 50-megaton hydrogen bomb in your basement for company. People just wouldn't buy from, sell to, or probably even talk to you.
"Excellence is an art won by training and habituation. We do not act rightly because we have virtue or excellence, but we rather have those because we have acted rightly. We are what we repeatedly do. Excellence, then, is not an act but a habit."
Aristotle, Nicomachean Ethics
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