Corrected entry: Kate is in the future and has named the dog Jack and informs Alex of this in a letter, which he receives in a magic time-travel mailbox. Alex adopts this same dog in the past, before the dog has ever met Kate. Jack tests out the name "Jack" on the dog, and the dog reacts to this name. The dog should not know this name as it hasn't yet met Kate. This is backwards. Alex should have been the one to tell Kate of a name he gave to the dog, and she should have been the one to test it on the dog.
Uncel Mark
20th Feb 2013
Correction: Jack the dog is magical, like the mailbox. Neither character knows where he came from. The dog is a sort of guardian. He runs to the past from the future. Re-watch the film, and this is clear. This interpretation is important for enjoying the film, if even more illogical than a magical Lake house without a magical dog.