We are talking about abandoned property for the most part such as a person dying with no will or relatives. Lost quarters or $100 bills are not part of the article in MHO.

By distinguishing a sub-category of rights as that of property, there is an inference that there exists something known as property. You cannot just act freely, a right, with respect to reality without acting with respect to something. In the case of property rights, that something is property. In the case of your usual dear intellectual property, it is not just some brain patterns, thought, that is the property but the actual writings or useful objects that are the property.

Salta explained this above: "Once [property] is owned it does not have to be "used" at all. The productive use requirement applies only to making the first claim on unowned [property], for example when homesteading the frontier."

Please use the term property rights. There is no such thing as property.

If something is unowned, you can only properly obtain property rights in something by making it useful. It may only be useful to you, but unless you do this you do not have property rights in the object and other can take it.

You are confusing cause and effect.

As Rand pointed out, a right is a moral principle which sanctions and defines one's freedom of action in a social context. Not sure that was an exact quote, so left off the quotation marks.
One need not make something useful to have a right in its use. A property right would be a freedom to act with respect to certain existents without obtaining permission from others. Usefulness is not a defining term for property, only an individuals opinion as to some of his property. I would sure hate to have you come to my property and decide whether each bit of it is useful by your standards. I have about 1500 books, with more than half of them not having been useful in several decades. Should they be removed from my house so that someone else could find usefulness in them? By your standards, the extra car that was licensed, insured, and runnable, which I lost because I had not used it for a couple of months, so was declared a junk vehicle, and required to be junked, was justly taken from me due to it not having some proper usage at the time? Should one be free to keep property without demonstrating the usefulness of it to others?

Or found a quarter lying in a parking lot... No problem. I think the question has to do with larger abandoned property, not accidental losses. In a society where everything is privately owned, or by partnerships or large numbers of participants, it is unlikely that someone leaves a car by the curb or out in a field forever. Things to be discarded would be recycled or sold for scrap or consigned to dealers. Where private property is so important, one's possessions would always be designated for a future owner.

I like it. If I found $100 laying on the ground and there was nobody around...well? What would be the difference?

Should be just like a tree in a forest of unclaimed land.

Maritime law is rich in rules for this.

and the proceeds should apply to reducing taxes that the government collects.

Private property rights are the most trampled of all in our society. Owners of property are out numbered by the masses for whom the elected strive to get all they can to keep up with the outrageous entitlements and giveaways.

LOL! Thanks, I needed that.

Probate law should extend to finding absolutely any "nearest of kin," no matter how remote, to any deed holder of record who dies without making a will or putting the property into a trust.

If the deed holder is a company, and not an individual or sole proprietor, the property should pass to the shareholders of record.

There are ways to establish ownership of property that might appear unclaimed. A government that just takes it is, to say the least, lazy, and to say the most, deliberately stealing.

Great question, CBJ. In general societies in the past have created laws dealing with the way such are to be disposed that usually begin with default rights falling to government to then dole out according to Probate courts. I don't have a problem with the government being charged with the duty to find and resolve any claims upon the land and even an auction of the land in question.

Look a little harder for the true owner before confiscating. I don't know.

Good question

First of all there is no such thing as property. You have property rights in things. When we say something is property we are using short hand. For instance, you have property rights in land, but land is not property.

With that in mind we need to rephrase your question. If an object that was formerly owned by someone and that person loses or abandons the object who owns it? You obtain property rights in things by making them useful. As a guiding principle it would be the next person that found the object and made it useful. So if the object has not be made useful by someone the answer would be no one. However under the law we often make rules for practical reasons such as recording deeds makes evidence of ownership more clear and avoids disputes. One place where we can see this is in estates law. When a person dies they abandon their property – a dead person cannot have property rights. Under a pure philosophical response we would say the next person to make, for instance, the land the person who died had property rights in, would be the owner. However, this would be chaotic and lead to fights. As a result, we decide that when a person dies he can say who has the first shot at acquiring property rights in the dead person’s object or we have a statute that does the same thing.

Finders keepers?

It should be put up for auction. No money, cars, property or anything else should be taken by the government. Private auction companies should handle it.

Once land is owned it does not have to be "used" at all. The productive use requirement applies only to making the first claim on unowned land, for example when homesteading the frontier.

I am struggling with Rand's "productive use" argument for acquiring land. There are large areas of land that someone currently owns but are not putting into productive use. Does this mean that anyone can come in and put it to productive use? How much productivity do you have to get from your property before no one is allowed to simply take it over?

Give it to me. I'm a nice guy.

As I understand Rand's brief comments on that, she spoke about the organization of the frontier into territories and then states by registering the ownership of farms to those who worked the land.

By extension, in a "libertarian" society, in which the dominant implicit philosophy is Objectivism, several options are moral. It depends on the actual property in question.

An abandoned lot of land can be occupied and worked and the deed be registered.

An abandoned intellectual property would just advance to the commons. Anyone and everyone could use it. Unlike land, intellectual property is non-rival and non-exclusive. Title comes entirely from the government. It has no "natural" foundation. So, there is no practical means of securing it without a primary owner. If the owner dies intestate, then it has no new owner.

The question of movable property might be closed off by considering that everything on Earth is literally on the Earth. If you find an unowned property with an automobile on it, the automobile is "chattels" on the land, and part of its ownership.

However, it is possible that an objective (or "Objectivist") legal system could delineate moveable property ("chattels" - originally "cattles") from the land itself and allow primary ownership of the land separate from other stuff on it. And a case in point would be an automobile abandoned on a public street. ... or a forklift in an abandoned factory.