The Sandwich Alignment Chart doesn't work

The Sandwich Alignment Chart, shown below, is great. (I'm not sure where and when it originally came from; the earliest appearance I can find is this tweet, but I don't know if this is really where it comes from.)

As it turns out, however, it doesn't actually work—by which I mean, it cannot properly capture my intuitions about what a sandwich is.

An essay last year already has argued the point that it's hard to logically define the things that we have intuitions about, and thus that it's hard to nail down just what constitutes a sandwich. That essay, however, concluded that the Sandwich Alignment Chart is good; it suggested that, rather than having a simple yes-or-no debate about whether a hot dog is a sandwich, we should recognize there's a range of different ways to understand sandwich-hood, and that the Sandwich Alignment Chart expresses that range.

I would like to go further, though, and contend that even the Sandwich Alignment Chart doesn't get it right. It fails because of two reasons: entailment, and tradeoffs. Let's look at each of these in turn.

The entailment problem

The two dimensions of the Sandwich Alignment Chart are both based on what a person accepts as being a sandwich. One end of each scale is more strict (it accepts fewer things as being a sandwich), and one more permissive (it accepts more things). This is an entailment relationship. Entailment relationships are asymmetrical: for example, all dogs are mammals (being a dog entails being a mammal) but not all animals are dogs. In the Sandwich Alignment Chart, a "Structure Rebel" accepts everything that a "Structure Purist" accepts, but a "Structure Purist" does not accept everything that a "Structure Rebel" accepts. A Structure Purist only accepts something as a sandwich if it's in between two pieces of bread; for example, if we look at the left-hand column, a Structure Purist only accepts a BLT as a sandwich, and does not accept a sub or a chicken wrap. But a Structure Purist accepts all those things. If you accept a chicken wrap as a kind of sandwich (i.e., if you are a structure rebel), that entails that you also accept subs and BLTs as sandwiches.

This is fundamentally different from the traditional Dungeons & Dragons style alignment charts, which involve scales between two opposing options (good vs. evil, law vs. chaos) which are not in entailment relationships. In a traditional D&D alignment system, for example, neither good nor evil entails the other. A "good" character is one who works to help others, and an "evil" character is one who works to help themself. A good knight might throw herself in front of a rampaging ogre to protect a villager;* an evil knight would not do that (unless the evil knight saw some benefit for herself in doing so, for instance, to keep up her reputation or to get the ogre's loot). An evil thief would steal a treasured gold family heirloom from a poor farmer; a good thief would not. Good and evil characters would behave in diametrically different ways; it is not the case that an evil character would do all the things a good character would do, or vice versa. Thus, it should be clear that this is a different state of affairs than the Sandwich Alignment Chart, where a "rebel" would accept everything that a "purist" would accept (and more).

Let's look at what this means in terms of locating oneself, and the kinds of things one accepts as a sandwich, on the chart. Imagine you are someone who believes a hot dog is a sandwich, as shown in the chart below. (Note that I have modified the chart by removing the labels in the individual cell; here each cell is referring not to a person's own sandwich alignment, but to a category of food items that this person accepts as a sandwich. Accepting a BLT as a sandwich does not necessarily mean you are a hardline traditionalist, because everyone accepts that a BLT is a sandwich.)

The Sandwich Alignment Chart, with the center cell (hot dog) circled in red.

If you think a hot dog is a sandwich, then, because of entailment relationships, you presumably also think a chip butty is a sandwich: structurally, it is even more sandwich-like than a hot dog. Likewise, you presumably also think a sub is a sandwich, for the same sort of reason (the ingredients of a sub are even more sandwichy than the ingredients of a hot dog). And you certainly think a BLT is a sandwich: it's more sandwich-like than a hot dog in terms of both structure and ingredients. Thus, we can more accurately represent the class of things you accept as a sandwich with the chart below:

The Sandwich Alignment Chart, with the center cell (hot dog) and all cells above and/or left of it all circled in red.

A more general formulation of this is as follows: if you accept a given cell in the chart as a sandwich, then you can extend that to include the full range of cells from that one up to the upper left corner. That's how the entailment relationship works. For example, if you accept an ice cream sandwich as a sandwich, then you also accept everything to the left of it. If you accept a Pop Tart as a sandwich, then you also accept everything in the entire chart.

Interlude: what I think is a sandwich

So far, nothing we've seen about entailment is necessarily a problem for the Sandwich Alignment Chart. If people's intuitions about sandwich-hood really do follow the entailment patterns described above, then the Sandwich Alignment Chart is fine. But do people's intuitions follow that pattern? Here is my own feeling about which things in the chart are sandwiches:

The Sandwich Alignment Chart. Circled are the entire top row (BLT, chip butty, and ice cream sandwich), plus the leftmost cell of the second row (sub)

To me, everything in the top row (BLTs, chip butties, and ice cream sandwiches) is sandwiches, and subs are also sandwiches; hot dogs and ice cream tacos, however, are not sandwiches.

This is a problem. This pattern cannot be derived from the entailment relations described above. Or, to put it another way: if you follow my abovementioned strategy of selecting a cell and then extending the selection up and left, you cannot get this pattern.

Based on what I think is a sandwich, what am I? I can't say I'm ingredient neutral or an ingredient rebel; I accept weird ingredients if they're between two things (the top row), but not otherwise. I can't say I'm structure neutral; I accept a U-shaped container when it comes to subs, but not hot dogs or Choco Tacos. I also can't say I'm a hardline traditionalist, because I accept things outside the upper left cell. The biggest generalization that can be made is that I'm not a structure rebel: I don't accept anything in the bottom row. But that still leaves much of my pattern unexplained.

To explain my pattern, we have to adopt another element into the theoretical apparatus of sandwich-hood: that of tradeoffs.


Tradeoffs, or trading relationships, are common. For example, we can see them in politics. Pundits often assume a reductive sort of politics which places people into discrete categories and predicts their behaviour based on those categories (e.g., liberal people will do this, moderates will do that, Latinos will do this, whites will do that, etc.), while neglecting to consider that everyone has multiple identities. For instance, sure, a particular person might be Black, but maybe he's also a man and also a billionaire; his political behaviour (such as voting) might be driven more by one of these identities than the other. And such a person may be willing to compromise on one of these aspects if it's traded off by another aspect (e.g., in this hypothetical example, the Black billionaire might vote for a candidate who supports politics that are harmful to Black people but also plans to cut taxes for the ultra-rich).

Another place you see tradeoffs is in speech perception. (I think this is a better example than the political one, but is probably only comprehensible to fellow phoneticians, which is why I chose to include both.) Take, for example, the perception of certain consonants in Korean. It works a bit differently than in English. In English, if you want to hear the difference between an aspirated and an unaspirated consonant (e.g., a "t" vs. a "d"), the main cue you pay attention to is how long of a puff of air there is just after the consonant. You can notice this difference if you hold your hand in front of your mouth when saying "da" and "ta"; there is a bigger puff of air in "ta". Another way to do it is by holding a piece of paper in front of your mouth and seeing that it flutters more when you say "ta" than when you say "da". (I am oversimplifying and ignoring some details here.) In Korean, however, to hear the difference between the aspirated and the unaspirated (referred to as "lenis" in the study of Korean phonetics) sound, you have to pay attention to two cues: the puff of air, and how high-pitched the person's voice is. In Korean, you can't tell the difference between an aspirated and an unaspirated sound based on just one cue; the cues trade off against each other, as shown in the graph below. If you hear a sound with a medium-ish puff of air, you will hear it as aspirated if it is very high-pitched, but unaspirated if it is low-pitched. On the other hand, if the puff of air is longer, then it doesn't need to be so high-pitched for you to hear it as aspirated. (Here, too, I am oversimplifying things—such as the fact that Korean has a third whole sound category which English does not—but that's not relevant for understanding the Sandwich Alignment Chart.)

A graph in which the x-axis represents how big a puff of air is, and the y-axis represents how high pitch is. A red triangle labeled \

My intuition about sandwich-hood is clearly another case of a trading relationship. For me to consider something a sandwich, I care about both structure and ingredients. But I'm willing to be lenient with one, if the other stays more traditional. I will accept a bit of a weird structure as a sandwich, if it has traditional sandwich ingredients (e.g., the sub). I will also accept quite weird ingredients, if it has a traditional sandwich structure (i.e., the ice cream sandwich). But I can't tolerate too much lenience in both dimensions; if both the structure and the ingredients are nontraditional (e.g., a hot dog, or anything below or to the right of that), I won't accept it as a sandwich.

There's also an interesting asymmetry to my trading relationship: I am more tolerant of deviations from traditional sandwich ingredients than I am of deviations from traditional sandwich structure. This is demonstrated by the fact that I accept ice cream sandwiches, but not chicken wraps, as sandwiches.


As we have seen, the categories of the Sandwich Alignment Chart, while useful insofar as they identify two salient dimensions of what constitutes sandwichhood, do not actually capture how real people understand the category of "sandwich". The dimensions themselves are useful, but if we want to actually understand how humans make decisions and judgments and categorizations, we also need a theory of how the dimensions relate to one another, and how people weigh them when the dimensions come into conflict (i.e., when something is very sandwich-like on one feature but very un-sandwich-like on another). People don't sit in one hard-and-fast alignment box; instead, they have to reconcile multiple competing features using some kind of complicated mental process (Virgil Texas of the Bad Faith podcast has referred to this as mental "alchemy", in the context of discussing how individual voters weigh all their different concerns in order to arrive at a singular decision of whom to vote for in a given election). Fortunately, many social and behavioural sciences, including linguistics, psychology, and political science, have theoretical apparatus and research methods for tackling these issues. (For instance, in linguistics there is a whole theoretical framework, called Optimality Theory, which is all about how people reconcile conflicting-but-violable constraints.) Thus, to conclude, more research on the nature of sandwichhood is needed, and hopefully productive collaborations across these disciplines can help us take the next step in resolving these heretofore intractable debates in the sandwich world.

Finally, I must also acknowledge one alternative explanation for the observations described above. One could argue that the problem lies not in the Sandwich Alignment Chart, but with my own view of what a sandwich is. In other words, maybe the Sandwich Alignment Chart works just fine, but my own mind doesn't work and my idea of what a sandwich is is just weird. To which I say, 1) yes, that is logically possible, and 2) screw you!

*A very good knight might, while protecting the villager, also ask the ogre what are the conditions that make it need to rampage, and then help the ogres and the villagers to organize together to change those conditions through collective direct action and work towards transformative justice. But let's please not let nuances of socialist thought get in the way of my using simplified high-fantasy archetypes to illustrate a point.

by Stephen Politzer-Ahles. Last modified on 2021-05-11.