Finally, a post for me. I've actually spent a considerable amount of time trying to come up with a basic definition of a sandwich. I ran into some problems too, but different ones than the ones you mentioned. The solution I found most satisfying was defining a sandwich as "a set of ingredients contained between two distinct 'halves' of a bread product." This was the best (and simplest) definition to me for these reasons:

* It excludes wraps, burritos, gyros, and other similar products (violates the 'distinct halves' section)

* It includes sub sandwiches and other sandwiches that have partially connected halves (still 'distinct')

* It excludes toast, flatbreads, and other nonsandwich bread-based foods.

The main problem with my definition is that it excludes open-faced sandwiches. I don't personally consider an open-faced sandwich to be a sandwich, but people that do would probably want to create their own definition. I also initially took issue with defining burgers and hot dogs as sandwiches, but I resolved that by making "burgers" and "hot dogs" subsets of the main "sandwich" set.

I didn't bother to define what bread was. Maybe that should be the next thing I try to do.