Along the way to the official answer, though, a student gave the following characterization:
Sentences S_1 and S_2 are synonymous just in case the set of all sentences that follow from* S_1 is identical to the set of all sentences that follow from S_2.After class, I jotted down a proof-sketch showing that the student's characterization is equivalent to the "official" one. But I'm not 100% confident in it, so I'm just curious whether anyone can see a counter-example (i.e. are there any 2 sentences that meet one characterization but not the other?).
* EDIT: As Bryan made me realize in the comments, I should specify that 'follows from' here is semantic, not syntactic/ proof-theoretic; i.e. 'Conclusion C follows from premise P' means that every model (or world, or construction, or whatever it is that makes sentences true in your formal semantics) in which P is true is also a model in which C is true.