Types of Nonsense
Natural languages, such as English and Armenian, are highly nonsense-tolerant. Formal languages, such as lambda calculus and first-order logic, are not. I just called languages "tolerant" and you somehow made sense of it. If I told a programming language interpreter that Graham's number is big, it would halt and catch fire (not a bad show, check it out after you read this). Historically, the direction of influence has been from natural to formal languages. This is reflected in the names of the logical constants (in the case of declarative languages) and the reserved words (in the case of algorithmic languages) of formal languages (e.g. "while", "for each", "if...then"). But formal languages have a lot to teach us about natural languages. Here we'll use insights from formal language theory to identify nonsense in natural languages.
Due to limitations of time, imagination and intelligence, I'll focus on five lessons in this article. They have only been tested with English, so expect no more than medium-sized miracles when applying them to other languages. The outline of the article is the following. We begin (Introduction) by distinguishing between meaningful and meaningless sentences. In §1, we discuss the notion of alphabetical meaninglessness by looking at a sentence that includes symbols from a foreign alphabet. In §2, we discuss the more familiar notion of grammatical meaninglessness by looking at a sentence that violates the rules of grammar. In §§3-4, we look at sentences that use expressions that don't refer to anything. Lastly, in §5, we discuss the notion of non-well-typedness by looking at a sentence that mixes apples and oranges. Cliche, I know. And yes, the accent is missing for a reason (cf. §1).
There are two types of sentences: meaningful and meaningless. Meaningful sentences communicate ideas, thoughts, feelings, by expressing propositions (things that are true or false). "April is the cruellest month" and "to love is to possess" are meaningful sentences. Sentences that say nothing true or false fail to express a proposition, so we call them "meaningless" or "nonsense." "Square circles are round" and "colorless green ideas sleep furiously" are examples of meaningless sentences.
To account for interrogative sentences (or questions), let us posit, for the duration of this article, that a question is meaningful if and only if it can be answered by a meaningful sentence. "What else do Tiersen and Kaczmarek have in common?" is an example of a meaningful question, because it can be answered by the meaningful sentence "their first names are pronounced the same in English."
Our problem is that of distinguishing meaningful sentences from nonsense. To solve it, we need to specify the conditions both sufficient and necessary that sentences have to satisfy to be considered "meaningful."
Sufficient: a sentence is meaningful if it meets conditions x, y, and z.
Necessary: a sentence is meaningful only if it meets conditions x, y, and z.
In this article we'll be concerned only with necessary conditions, the "only if" part of the definition of meaningfulness.
We'll look at five conditions that meaningful sentences must satisfy. Meeting these five conditions does not guarantee the sentence will be meaningful. Not meeting those five conditions, however, does guarantee the sentence will be meaningless.
Let's begin by considering the following sentences:
- Վիկտոր Համբարձումյան co-founded theoretical astrophysics.
- Is a poet great Fernando Pessoa.
- The integer between three and four is even.
- Where is the center of a triangle?
- T.S. Eliot is a prime number.
These sentences are meaningless, because each fails to meet a particular necessary condition of meaningfulness. In the rest of this article we will look at each in turn, identifying the condition it violates and considering a way to make it meaningful.
§1 — Alphabet
"Վիկտոր Համբարձումյան co-founded theoretical astrophysics."
Every language has an alphabet, a set of symbols from which its words (or terms) are constructed. The English language is based on the Latin alphabet. The Armenian language, however, is not. So Armenian letters "ի", "ո", "ն" don't belong to the English alphabet and English letters "i", "n", "j" don't belong to the Armenian one.
Sentence (1) belongs neither to English (because it contains non-English letters) nor to Armenian (because it contains non-Armenian letters). So it's meaningless both in English and in Armenian. If we wanted to give a name to this kind of nonsense, we could call it alphabetically meaningless.
Advice. Unless the letters appear in quotation marks or other kinds of opaque contexts, make sure they belong to the alphabet of the language you're working with. Sentences involving the words "naïve" and "façade," for instance, do not belong to the English language. In most cases, adequate alternatives exist (e.g. "naive" and "facade"). If they don't, feel free to make them up!
§2 — Grammar
"Is a great poet Fernando Pessoa."
This next sentence is alphabetically meaningful, i.e. every letter in it belongs to the English alphabet, but it's still meaningless in that it defies the rules of grammar. In English, the copula "is" functions as an infix operator (it appears between the noun and the verb phrases), but here it appears prefixed. The grammatical form of the sentence is the following: VP + NP, where the VP ("verb phrase") position is occupied by "is a great poet" and the NP ("noun phrase") position by "Fernando Pessoa." We can transform the sentence into a grammatical one by swapping the VP and the NP:
"Fernando Pessoa [NP] + is a great poet [VP]."
We are able to do these sorts of transformations in our heads, so we find no difficulty in understanding what's being claimed in (2), but, strictly speaking, it's grammatically meaningless.
Advice. Get acquainted with the rules of grammar.
§3 — Existence
"The integer between three and four is even."
This one is both alphabetically and grammatically meaningful, but it still turns out to be meaningless once we understand the meaning of "integer." So let's understand that meaning.
There are different kinds of numbers. We have the natural numbers 1, 2, 3, ..., the weirdo 0, the negative integers -1, -2, -3, ..., the rational numbers 1/2, 2/3, 3/4, the irresponsible, eh, irrational numbers π, e, and so on. The number systems get pretty fancy after real (the set of rational and irrational numbers combined) and complex (special pairs of reals) numbers. For our purposes it'll be sufficient to define the meaning of "integer": an integer is either a 0, a natural number (e.g. 1, 2, ...), or a negation of a natural number (e.g. -1, -2, ...).
Let's now see whether the integer between three and four is even or odd. The integer between three and four is some integer greater than three and less than four. How about 3.5? 3.5 is exactly in the middle of three and four, but the problem is that it's not an integer! 3.5 is a rational number. There is an infinity of numbers between three and four (so close, and, yet, so far apart), but none is an integer. A good rule for checking whether a number is an integer is to see if you can reach it by incrementing (adding one to) or decrementing (subtracting one from) zero.
Because there is no such thing as "an integer between three and four," the sentence fails to express a meaningful proposition. We can call this type of nonsense meaningless due to a lack of a referent. The "referent," in our case, is an integer that doesn't exist.
Note. For this same reason numerical expressions of form (m/0), where m ≠ 0 are said to be "meaningless." Recall how fractions are defined: (m/n) denotes the unique number k such that k * n = m. Let's consider the value of (m/0), for some arbitrary m different from 0. It refers to the unique number k such that (k * 0) = m. Adding the premise (that m is different from 0) we obtain the equation:
(k * 0) ≠ 0.
But no k satisfies this equation, because given any k, (k * 0) evaluates to 0. Because such a number k doesn't exist, (m/0), where m is different from 0, is said to be "meaningless." This is half the reason division by zero is undefined.
Advice. Simply defining something doesn't guarantee that an object meets the condition of the definition. A successful definition must be justified by two proofs: one showing that at least one object meets the criteria, and the other showing that no more than one does.
§4 — Uniqueness
"Where is the center of a triangle?"
This sentence is the cousin of the previous one, but it's slightly trickier to identify. The reason it's meaningless is that, unlike circles, triangles don't have a unique property that can be called their "center." There are different definitions of centrality of triangles, so the expression "center" in that sentence is ambiguous. We can, for example, define "the center of a triangle" as the point of the intersection of its angle bisectors (this type of a center is called an "incenter"). But we can also define "the center of a triangle" as the arithmetic mean of all its points (this type of a center is called a "centroid"). There are many other equally-acceptable definitions.
Because there is not a unique thing called "the center of a triangle," the sentence fails to express a meaningful proposition. We can call this type of nonsense: meaningless due to a lack of a unique referent.
Note. In the previous note we looked at the reason expressions of form (m/0), where m ≠ 0, are meaningless. Here, we look at the case where m = 0, viz. (0/0). Note first that the previous argument doesn't apply to this one: there exists a number k such that k * 0 = 0. That number is zero itself! But (0/0) is still undefined. Why? Because (0/0) is the unique number k such that k * 0 = 0. But all numbers k satisfy that equation, so there is no unique number k that satisfies it. Because of this lack of uniqueness, (0/0) also turns out to be meaningless. This is the other half of the reason division by zero is undefined.
Advice. See the one for §3.
§5 — Type
"T.S. Eliot is a prime number."
Like all great things, this too comes last.
First, note that the sentence suffers from none of the aforementioned flaws: it's constituted entirely of symbols from the Latin alphabet (unlike sentence 1), it's a grammatical, declarative sentence (or statement) (unlike sentence 2), the noun phrase "T.S. Eliot" refers to one (unlike sentence 3) and only one (unlike sentence 4) individual, and "is a prime number" is a well-defined predicate expression.
What, then, is wrong with this sentence? It's not well-typed. Before you start searching for typos in the sentence, I should clarify the sense in which I use the word "type." From the mathematical perspective, types are a less degenerate kind of sets (don't switch to the Netflix tab yet).
We have already seen some types in this article. Remember those VPs and NPs? Those are syntactical types: they indicate the syntactical category to which an expression belongs. We say, for instance, that "T.S. Eliot" has the syntactical type "NP," it's an expression that eventually denotes an individual (the individual doesn't have to be a human being, by the way). The expression "is prime" has the syntactical type "VP," meaning that it denotes a function from individuals to true or false. Apply "is prime," for instance, to "2" and you'll get an affirmative answer to the question "is there an even prime number?"
Beyond syntactical types, there come logical types. Syntactically-speaking, all is well with the sentence "T.S. Eliot is a prime number." It's of the form: NP + VP, and since VPs map individuals to truth-values and NPs are individuals, the whole sentence reduces to either true or false, depending on the primality of T.S. Eliot. But the sentence is still meaningless because we have a logical type mismatch. To see what exactly doesn't line up, let's recall the definition of "is prime:" a positive integer is prime if and only if its divisors are itself or 1. Assuming that we know what "positive," "integer," "divisor," "equality", and "1" mean, this definition allows us to decide whether a given positive integer (or natural number) is prime or not. What it does not tell us is whether an arbitrary thing, be it a number or not, is or is not prime. In other words, "is prime" is defined here (and in general) only for natural numbers, so it's meaningless to ask what the truth-value of "T.S. Eliot is prime" is. We can call this type of nonsense meaningless due to a type mismatch.
Advice. When predicating a concept of an object, make sure that the concept is well-defined for that particular type of objects. As we saw, "is prime" is not well-defined for anything but natural numbers, so applying it to human beings results in nonsense.
Formal languages are smaller and less expressive but more precise and less ambiguous than natural languages. They lack metaphors and ambiguity, have a strict typing diet and many other features that limit their expressive power. This makes formal languages a convenient sandbox within which to explore the confusions and paradoxes of natural languages. In this article, we've looked at five problems with meaning that go unnoticed in English and, taking cues from formal languages, have suggested ways of resolving them.
- Many thanks to people saying and writing all sorts of nonsense.
- Version 0.1 was inspired by Carnap, deadlines, and the iced triple americano at Cafe Milano.
- Version 0.2 benefited from a Cancun conversation with Eric Heltzel, Goni Dubnov, and Erskine Wilson.
- Version 0.3 owes much to the generous advice and the criticisms of Erskine, Ela Provost, Don McQuade (v0.3.53), and Matthew Ramirez (v0.3.42).
- Version 0.4.7 is the one printed here.
- Thank you Chris Ramirez for making beautiful things. Thank you Don for encouraging me to make the article more concrete. Thank you Matthew for your insightful comments and for telling me that the first version didn't suck. Thank you Ela and Erskine for your challenging questions, patience, humor, and editorial help.
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To read the next installment in this series on sense and nonsense, see Erskine's post "Types of Sense."