If your tenant breaches the lease, then you can evict him. What he probably doesn't understand is that a lease is a contract and theoretically he would owe the remaining rent due under the term of the lease. However, in most states, including Washington, a landlord has a duty to mitigate damages. What this means is that if your tenant breaches the lease and you either evict him or he moves out, you must make reasonable efforts to re-rent the premises.
DISCLAIMER: Brandy A. Peeples is licensed to practice law in the State of Maryland. This answer is being provided for informational purposes only and the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship. For legal advice relating to your specific situation, I strongly urge you to consult with an attorney in your area. NO COMMUNICATIONS WITH ME ARE TO BE CONSTRUED AS ARISING FROM AN ATTORNEY-CLIENT RELATIONSHIP AND NO ATTORNEY-CLIENT RELATIONSHIP WILL BE ESTABLISHED WITH ME UNLESS I HAVE EXPRESSLY AGREED TO UNDERTAKE YOUR REPRESENTATION, WHICH INCLUDES THE EXECUTION OF A WRITTEN AGREEMENT OF RETAINER.
I agree with Mr. Peeples. Additionally, you might want to consider whether you want to bother trying to keep this tenant. If you just let him go, you can re-rent immediately (if the market is good) and keep you cash flow up. If you evict him, it will take most of a month to get him out, after which you'll have to sue him separately for the rent you lose, after which you'll have to try and collect.
If the market is good, accept his 20-day notice as long as he keeps current on his rent.
If the market is poor, consider whether the potential size of the judgment is worth the cost and effort of suing and collecting, and the lost-rent time while you evict him.
My comments are general statements of the law in Washington and are not meant to be legal advice. Specific legal advice requires an in-depth interview, review of past or ongoing court or agency actions or cases, and an evaluation of the special circumstances of each case.
I agree with the prior answer from Ms. Peeples. I would like to add that your tenant may be trying to get out of the lease by claiming you are harassing him. See RCW 59.18.575. If a tenant makes a report of unlawful harassment by his landlord, he can sometimes terminate the lease. Issuing a ten-day notice is not harassment because it is lawful. Act carefully, however, as this tenant may be trying to take advantage of you.