Pulitzer Prize winning philosopher/cognitive scientist [wiki]Douglas Hofstadter[/wiki] (co-authored with Emmanuel Sander) On Sandwiches
Book is Surfaces and Essences: Analogy as the Fuel and Fire of Thinking
Anyone who is interested in these sorts of categorization questions should find the book rather insightful.
I cannot readily accept this philosophical definition of a sandwich as being anything that is placed between two like things. While I recognize sandwich has entered the vulgate to mean exactly that, this definition is problematic in the technical culinary field as it is simultaneously too broad and too narrow. It would include an ice cream sandwich as a sandwich making appears as though ice cream is modifying sandwich when it is the other way around. Similarly it would likely place the hot dog in the realm of the sandwich when the hot dog is not a sandwich (more on this later). At the same time, the use is too narrow as it would exclude open face sandwiches from being recognized as sandwiches.
This is not to say that the definition is wholly unhelpful. Certainly having a composure of A-B-A is at least indicative of sandwichness. That is helpful, but is not dispositive in itself.
Expectation of Sandwichness
Instead of attempting to develop a uniform definition of sandwich as a first principle, it seems better to first discuss the experiential definition of a sandwich. To this end, whether or not something is a sandwich is informed by the following hypothetical:
If you asked me for a sandwich and I offered you X foodstuff, would that appease you desire for a sandwich?
Where the answer is yes then the thing is a sandwich, or likely to be one. Where the answer is no then the thing is not a sandwich.
Sandwiches and bread (slice or split bun)
There are a number of things that are undeniably sandwiches: French dips, Monte Cristos, PB & Js, Ruebens, etc. These all have the notable quality of being composed of material between two slices of bread.
Then there are the myriad of grinders, hoagies, heroes, po'boys, cheesesteaks, subs, etc. These are often, but not exclusively, composed of a long bun spilt not quite all the way through so there we have examples of sandwiches that are between one piece of bread.
The Strange Case of the Hot Dog
There are things that are not sandwiches despite having qualities similar to the sandwich. Most notable is the hot dog. Like most (but not all!) sandwiches, the hot dog consists of savory meat contained by bread, in this case a split roll. Despite Gori's statement to the contrary, a hot dog is not a sandwich.
One curious factor about the hot dog is that it is dominated by a single foodstuff, the hot dog sausage itself. In contrast, most sandwiches are a combination of several foodstuffs in the bread. A Rueben isn't a Rueben w/o corned beef AND kraut. In contrast, the hot dog remains a hot dog even if there isn't any onions with it (although why you would have a hot dog sans onions is beyond me).
(As an aside, there is a notable exception in the grilled cheese sandwich. However, the grilled cheese sandwich is made with two slices of bread rather than a split roll, and the grilled cheese is an obvious simplification of other things also recognizable as sandwiches. Of course the quesadilla, which is almost the exact same thing, is not a sandwich, but the loaf of French bread with brie is.)
(As a further aside, the hot dog raises the interesting question of whether or not a lobster roll is a sandwich. Like the hot dog, the lobster roll is dominated by a single foodstuff and in a split roll, but unlike the hot dog (and like most sandwiches) lettuce and other secondary ingredients are required to make the ur-lobster roll.)
Burrittos and Wraps
We have the burrito which is a composition of many things contained within a single piece of tortilla. It is not a sandwich.
The wrap provides us with a poser. The wrap is likely to be composed of the same things as a sandwich save that it is wrapped in a single piece of bread. Based upon the number of wraps in "sandwich platters" I've had at catered meetings, it seems likely that the wrap does fulfill the criteria of whether or not what is received satisfies the recipient as a sandwich. That said, if you had a platter of sandwiches and wraps in front of a person and asked her to pick which items were sandwiches and which were not, the wrap would likely fall into the latter category.
Back to the Burger
Viewing this all, a simple list of attributes that qualifies one thing as a sandwich does not appear to be helpful because for every attribute listed above, a counterexample that breaks the rule can readily be cited. It is probably better to instead assess the sandwichness of a thing as a analysis of all the relevant factors with no one thing necessarily outweighing another.
Bringing the inquiry to the hamburger: if you asked for a sandwich and were given a burger, would you feel your request had been satisfied?
This, I think, is the best way to view the question at hand.