It’s not just statutes, but the Kentucky Constitution as well
Section 233A
Only a marriage between one man and one woman shall be valid or recognized as a marriage in Kentucky. A legal status identical or substantially similar to that of marriage for unmarried individuals shall not be valid or recognized.
http://www.lrc.ky.gov/legresou/constitu/233A.htm
“It’s not just statutes...”
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You are exactly right. Beshear would almost certainly find it impossible to change either the KY Constitution or the KY statutes. That’s why he turned to 0bama’s playbook and chose to circumvent the legislature and the KY constitution by issuing an illegal executive order.