Fully homomorphic encryption schemes allow one to evaluate any arbitrary computation over encrypted data. Intuitively this seems to be too weak, irrespective of how we achieve this.
An adversary who has access to the cipher text only could do variety of operations. For example, duplicates in the cipher text could be found easily. No separate trapdoors are required apart from Public Key. Variety of further operations could be performed on the given corpus of cipher text, so identifying prefixes, suffixes etc. among the cipher words by simply running their respective circuits.
Also with additional information like if an adversary could guess if the column in the encrypted database is an integer. He can do operations like $<$ or $>$ among the encrypted values to further deduce interesting information.
Is this true with homomorphic schemes ? or am i missing some thing ?