Implication and Satisfiability Apologies if this is not an appropriate place to ask a question like this, but I was just wondering why it is that an unsatisfiable sentence implies every other sentence. Please let me know your thoughts. Thanks.

Remember how implication works: $\varphi$ implies $\psi$ if, in any model where $\varphi$ is true, $\psi$ is true. If $\varphi$ is never true, then certainly $\psi$ is true in all the models where $\varphi$ is true, since there aren't any! That is, the implication $\varphi\implies\psi$ is vauously true.

The above is true for classical logic. It is quite reasonable to take issue with this definition of implication; there are lots of logics out there where "ex falso quodlibet" (a false statement implies everything) or similar principles, such as the law of the excluded middle, don't hold, and they're quite interesting. (Google "intuitionistic logic" to get started.) But in classical logic, we do indeed have that unsatisfiable statements imply everything.
