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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.

user275106
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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.

Noah Schweber
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    Your answer is nice, but it might have some mistaken point. Vacuous implication also holds in intuitionistic logic, though it might not hold in minimal logic. – Hanul Jeon Sep 29 '15 at 06:14
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    @tetori I didn't mean to imply that vacuous implication fails in intuitionistic logic - I just thought that for the OP, beginning with intuitionistic logic might be the best way to ease into non-classical logic (I'm assuming they're coming from a classical logic perspective, but statistically speaking that's probably a sound assumption). But you're quite right, I wasn't careful with my phrasing - I've edited to make it better. – Noah Schweber Sep 29 '15 at 07:47