Salad Theory

Background In our office, we sometimes enjoy pedantic conversations where we attempt to properly define ambiguous terms. Naturally, when the sandwich controversy took the world by storm around 2016[3], we were interested. We talked a lot about sandwiches, then went beyond. The sandwich debates spun off numerous side discussions regarding the categorization of other foods. Fatefully, one coworker raised the question of whether a salad could be called a pizza. The answer to that question shocked and appalled us. Inescapably, a pizza was actually a salad. Many heated lunchtime conversations ensued. We attacked the problem from different angles. We debated the merits of novel definitions. People got sick of it and avoided some of us at lunch. But the truth could not be avoided. Begrudgingly, the office pedants were forced to reach an agreement: All edible foods[1] are salads. We wrote up our conclusions in an internal document in May 2017. We were content to keep this dark knowledge to ourselves, but alas. Spurred by the release of food category theories which clearly misrepresent the nature of salads[4], the time has come to share our findings with the world. Derivation Assumptions Any sane definition of a salad must at least include foods which a typical English-speaking person would colloquially refer to as a “salad”. A typical English-speaking person would consider the following foods “salads”: caesar salad, fruit salad, chicken salad, pasta salad, tuna salad, taco salad, and an untossed caesar salad. A good salad definition should be concise. Fewer terms and conditions on the definition of a salad is more elegant (and by Occam’s razor, more likely to be correct). Considered Definitions and Restrictions Traditional A salad must be a mixture of greens. This definition is simple and restrictive, which is good. Unfortunately, it is too restrictive. We rejected this definition because it violates assumption (1) by excluding fruit salads, chicken salads, pasta salads, and tuna salads. Well-Mixed A salad must be a mixture of foods such that a typical person would not notice a difference if the relative position and orientation of its ingredients were randomized. This definition comes close, but it notably excludes an untossed salad, while still including things that are less salad-like than an untossed salad (e.g. a bowl of mixed nuts) which seems preposterous. Notably, by this definition, most foods (e.g., sandwiches) are still arguably “untossed” salads. Explicit Inclusions The set of things that are salads is the set of things listed in assumption (2). This definition violates assumption 3; merely enumerating a long list of things that are salads produces a lazy and cumbersome definition. It also plausibly includes things like a bunch of bananas (fruit salad), so it isn’t even very restrictive for all its arbitrary complexity. Explicit Exclusions A salad may not contain bread (etc etc). This condition is not a complete definition unto itself; it is an attempt to restrict the definition to exclude sandwiches and similar foods. This condition has two problems. First, it fails assumption (1). It plausibly excludes a caesar salad, and possibly pasta salad as well. Croutons are essentially stale bread, and pasta is closely related. Even if you claim that the exclusion is limited to fresh (soft) bread, that leads you to the conclusion that a piece of avocado toast is a salad, while a sandwich made with untoasted bread is not, which certainly seems suspect. Second, it fails assumption (3). The bread exclusion is arbitrary; there’s no clear reason to exclude bread but not a variety of other ingredients (e.g. nuts, candy, ice, jams, steaks, lasagna). Including an individual exclusion for each item clearly produces an impracticle, overly-complicated, and probably wrong definition. Solid Content A salad must be mostly composed of solid content. This definition sounds reasonable, but it turns out the devil is in the details of how you define “mostly”. At what point have you poured in so much balsamic vinaigrette that something stops being a salad? After much discussion, we ultimately concluded that we have no way of formally distinguishing between a “salad” and a “soup”, leading us to Theorem 1. Ingredient Count A salad must contain at least two ingredients. This definition satisfies all of our assumptions. It is also very broad. A coke might not be, but a coke with ice in it is. An individual apple is not a salad, but a pair of apples is a fruit salad. A head of lettuce isn’t a salad until you separate the leaves. This definition is ultimately the only self-consistent definition we could agree upon. Semantically, we refer to all collections of more than one ingredient a “salad”, and solitary food items as “low-entropy hyper-salads”/”hypo-salads”/”proto-salads”. See Theorem 2. Merriam-Webster 1 Merriam-Webster’s first definition of a salad is: Any of various usually cold dishes: such as a : raw greens (such as lettuce) often combined with other vegetables and toppings and served especially with dressing b : small pieces of food (such as pasta, meat, fruit, or vegetables) usually mixed with a dressing (such as mayonnaise) or set in gelatin This is actually a pretty strange definition. It includes things like a plate of mixed nuts, but excludes a caesar salad which you stuck in the microwave for two minutes. While people might object to serving a hot salad, it certainly seems bizarre and objectionable to exclude it from the definition simply based on temperature. We conclude that this definition does not meet the necessary rigor for a proper salad definition, and additionally it plausibly violates our assumption (3) by addition arbitrary limitations on temperature. Merriam-Webster 2 A green vegetable or herb grown for salad especially : LETTUCE In addition to being self-referential, this definition fails (1) by excluding tuna salad (among others), while still including things like a head of lettuce. Merriam-Webster 3 A usually incongruous mixture : HODGEPODGE This is actually broader than our definition: we restrict ours to edible ingredients. The Theory Theorem 1: Soup-Salad Duality All foods[1] are formally affine combinations of soups and hyper-salads. We describe this measure as a food’s “soupiness” coefficient, where a food with soupiness=0 contains no exterior liquid, and a food with soupiness=1 contains no solid components. Colloquially, foods which have soupiness above a certain value tend to be regarded as “soups” (with their liquid components being considered to be “broth”), whereas foods which have soupiness below a certain value tend to be regarded as “salads” (with their liquid components being considered to be “dressing”). With the possible exception of the boundaries (foods which have a soupiness of exactly 0 or 1), Salad Theory does not formally recognize a meaningful distinction between a soupy hyper-salad or a salady soup. That is: Theorem 1.1: Soupiness Lemma There does not exist a real number x, 0 < x < 1 such that all foods with a soupiness less than x are strictly salads, and all foods with a soupiness coefficient greater than x are strictly soups. Theorem 1.2: Beverage Lemma There is no formal way to distinguish a beverage from a soup with no solid components. Some might claim that substances commonly included in beverages are invalid as foods, but counterexamples to this postulate abound. Beverage substance Foods with same substance Water Almost everything, especially “soups”. Beer Beer batter, many poultry roasts. Milk Many sauces and batters. Soda Coca-Cola BBQ sauce Furthermore, it is impossible to even exclude beverages which “only” contain a single liquid substance, because doing so would exclude many colloquial foods (such as tomato soup). Given that “soups” and “salads” are not meaningfully distinct, we arbitrarily prefer the latter term in most formal discourse. Theorem 2: Salads, Hyper-salads, and Saladoids. A salad requires a certain minimum degree of entropy to be classified as such. Individual solid food items in isolation do not have enough entropy to be full salads, and are instead classified as low-dimensional hyper-salads (i.e. hypo-salads[2]), or low-entropy saladoids. Thus, a single apple is a low-entropy saladoid, but a set of apple slices is a fruit salad (or, more precisely, an apple salad). Theorem 3: Allowed Operations Salads are closed under union and addition (the union of a salad and anything else is a salad). Salads are not closed under subtraction (removing elements from a salad will either produce a salad or a low-entropy saladoid). Hyper-Salad Space Standard Projection All foods[1] live somewhere in Hyper-Salad Space[5]. Hyper-Salad Space is quite high dimensional; research is ongoing into the minimum number of dimensions required to uniquely identify every recognizable food. It tends to be useful to project such high-dimensional spaces down to few enough that humans can visualize them. Thus, we provide the Standard Projection of Hyper-Salad Space, which has just three particularly salient dimensions. Dimensions Soupiness This dimension is simply the value of the soupiness coefficient, as defined in Theorem 1. Ingredient Entropy This dimension is a measure of how many ingredients and how many kinds of ingredients a hyper-salad contains. Ingredient Entropy is computed as: $$i_e = lg(\text{number of ingredients}) \cdot (\text{number of unique ingredients})$$ Note that \(lg(x)\) is the base-2 log \(\frac{ln(x)}{ln(2)}\). For example, a glass of coke with 4 ice cubes has ingredient entropy: $$i_e = lg(5)\cdot 2 = 4.64$$ And a glass of water has ingredient entropy: $$i_e = lg(1) \cdot 1 = 0$$ This measure has a few nice properties: Monotonically increases as you add more ingredients. Adding ingredients of a distinct nature adds more entropy than adding another of the same ingredient (adding another ice cube is less influential than adding a lemon wedge).