Anything left behind is deemed abandoned and becomes your property to dispose of as you see fit.
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If he only wants some portion of his belongings, go along with it and discard the rest later. To deny him some and require he take "all or nothing "is only looking for trouble. Don't look for trouble.
My answer is for general purposes only and is not not intended to establish an attorney-client relationship, nor is it advice upon which you should act or rely. But, if you really want me to tell you something upon which you can actually rely: don't eat yellow snow.
The law is murky on this particular issue. The general practice is to permit the evicted tenant a single opportunity to come remove the belongings. If you can obtain the tenant's new address, it is a good practice to send a letter by certified and first class mail with proof of mailing stating that you will provide a single opportunity for the tenant to retrieve her belongings from storage on or before May 31 and any belongings not removed by that day will be considered abandoned and discarded.
Allow the tenant to recover the left behind personal property. If possible, invoice the items and prepare a receipt. Provide for the tenant to indicate the property he removed and the property he is abandoning. Make a duplicate and make sure the tenant signs on both forms at a place where the receipt is dated and the tenant agrees that the items marked as taken, he took, the items noted as abandoned he is allowing you to discard without any recourse and without liability.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.
My colleagues all make excellent points. Going forward, you may want to consider a few ideas to avoid this issue in the future. If you do not already do so, give some thought to using a written lease, and consider some kind of move-out or abandoned property clause.
This communication is intended only to provide general information. No attorney-client relationship is created.