Question: Who is the county that borders the county with Mount Carmel named after?

full_context: Document 1: Ohio Township, Clermont County, Ohio: Ohio Township is one of the fourteen townships of Clermont County, Ohio, United States. The 2010 census reported 5,192 people living in the township, 2,610 of whom were in the unincorporated portions of the township.
Document 2: List of the most common U.S. county name etymologies: Twenty three of the twenty six Jefferson Counties in the United States are directly named for Thomas Jefferson, the third President, and the remaining three are indirectly named for him.
Document 3: Washington Township, Clermont County, Ohio: Washington Township is one of the fourteen townships of Clermont County, Ohio, United States. The 2010 census reported 2,278 people living in the township, 1,993 of whom were in the unincorporated portions of the township.
Document 4: Hamilton Township, Franklin County, Pennsylvania: Hamilton Township is a township in Franklin County, Pennsylvania, United States. The population was 10,788 at the 2010 census, up from 8,949 at the 2000 census.
Document 5: Franklin Township, Clermont County, Ohio: Franklin Township is one of the fourteen townships of Clermont County, Ohio, United States. The 2010 census reported 4,188 people living in the township, 3,307 of whom lived in the unincorporated portions of the township.
Document 6: Miami Township, Clermont County, Ohio: Miami Township is one of the fourteen townships of Clermont County, Ohio, United States. The population at the 2010 census was 40,848, up from 36,632 in 2000. The township's students are served mostly by Milford Exempted Village Schools.
Document 7: Hamilton County: Hamilton County is the name of ten counties in the United States of America, eight of them named for Alexander Hamilton, first United States Secretary of the Treasury:
Document 8: Clermont County Public Library: The Clermont County Public Library is a public library in Clermont County, Ohio, located east of Hamilton County and within the greater Cincinnati area. There are ten library branches in the system: Amelia, Batavia, Bethel, Felicity, Goshen, Milford, New Richmond, Owensville, Union Township, and Williamsburg.
Document 9: Mount Carmel, Ohio: Mount Carmel is a census-designated place (CDP) in Clermont County, Ohio, United States. The population was 4,741 at the 2010 census.
Document 10: Alexander Hamilton Institute: The Alexander Hamilton Institute is a former institute for business education in New York City founded in 1909, and dissolved in the 1980s. The Alexander Hamilton Institute was a corporation engaged in collecting, organizing and transmitting business information.
Document 11: Mount Carmel, Pennsylvania: Mount Carmel is a borough in Northumberland County, Pennsylvania, United States. The population was 5,869 at the 2010 census. It is located 88 miles (141 km) northwest of Philadelphia and 71 miles (114 km) northeast of Harrisburg, in the Anthracite Coal Region. It is completely encompassed by Mount Carmel Township.
Document 12: Jackson Township, Clermont County, Ohio: Jackson Township is one of the fourteen townships of Clermont County, Ohio, United States. As of the 2010 census the population was 2,980.
Document 13: United States ten-dollar bill: The United States ten - dollar bill ($10) is a denomination of U.S. currency. The obverse of the bill features the portrait of Alexander Hamilton, who served as the first U.S. Secretary of the Treasury. The reverse features the U.S. Treasury Building. All $10 bills issued today are Federal Reserve Notes.
Document 14: Williamsburg Township, Clermont County, Ohio: Williamsburg Township is one of the fourteen townships of Clermont County, Ohio, United States. The 2010 census reported 5,746 people living in the township, 3,256 of whom lived in the unincorporated portions of the township.
Document 15: Wayne Township, Clermont County, Ohio: Wayne Township is one of the fourteen townships of Clermont County, Ohio, United States. The 2010 census reported 4,885 people living in the township, 4,493 of whom were in the unincorporated portions of the township.
ground_truth: Alexander Hamilton
sub_queries: Mount Carmel >> located in the administrative territorial entity Answer: Clermont County
Clermont County >> shares border with Answer: Hamilton County
Hamilton County >> named after Answer: Alexander Hamilton