dog fight--who should pay the vet bill?

A neighbor came onto our property uninvited with his dog on a leash. Our dogs were out in the yard off leash with my husband, and the dogs were involved in a three-second scuffle. The neighbor's dog was cut on the hip. Our dogs have not injured anyone before. We agreed to pay half of the vet bill, but the neighbor wants us to pay the full amount. There is an easement across our property going to the two properties beyond ours, and one of those properties was recently purchased by the town for a nature preserve (it was privately owned before, when the easement was created). The town has not formally established a public right of way through any agreement with us and it is questionable whether the public has a right to use the easement, and whether dogs are allowed on the easement. The neighbor saw that our dogs were loose on our property, yet proceeded to enter anyway. Are we responsible for the full vet bill?