# 6.1: Arbitrariness and Compositionality

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## Arbitrariness and Compositionality, from Sarah Harmon

### Video Script

With respect to arbitrariness and compositionality, there are a lot of the pieces that you're going to see below in what Catherine Anderson wrote that are exceptional, but I think they need a little bit of setup now. When we talk about semantics, one of the things that is kind of important to remember, is the fact that this is really like philosophy in some ways that we're going to talk about really abstract concepts. Meaning is harder to imagine sometimes than the structure of a phrase or how we put a lexicon together with different morphemes. To start off, let me start talking about a few things that are going to be closer to what you may hear in a philosophy class; there's a reason for that and I’ll get to it.

With respect to semantics, we always have to start off in the same place, and that is talking about an ontology. In linguistic terms, an ontology is pretty close to what you hear in a philosophy class: it is a box with all the possible meanings. Each language assigns a lexicon or a phrase to that meaning; that's what we call lexical meaning or lexical semantics. Let's unpack that a little bit, because that might be a little hard to understand. I like thinking of an ontology as a box, a box for each given language and, in many cases, you can say a specific dialect. You can say that that box come has an entire litany all of the possible phrases and lexicons and the meaning associated with it, so they are connected in some way. What happens is that as fluent speakers of a given language, whether it is our native language or one that we have learned, we pull from the items out of that box and we import it into our entailment, our mental lexicon. That mental lexicon that we talked about in morphology comes in here, because that is how we have built our language, we have pulled items from the ontology we have entailed certain other pieces to that and have built up our mental dictionary or mental lexicon. This really encapsulates a lot of what we do in semantics, because everything that we talked about with respect to semantics, and even pragmatics, which we'll get to later, has to do with understanding what are the items in that box, what are the lexicon and phrases, as well as the meaning in that ontology.

We have talked about this gentleman before in chapter one, as well as when we got to morphology: Ferdinand de Saussure. As you can see, he lived from the middle of the 19th century into the early part of the 20th century. We already talked about his Principle of the Arbitrary, but let's come back to that to refresh our minds a bit. It really is the cornerstone of semantics; you have a hard time understanding anything in semantics without understanding the arbitrariness of language. I talked about him briefly earlier; he was Swiss mathematician, logician, and philosopher that used these tools to look at language. He is a fascinating individual; at a different point in time, I might get into it.

For now, his Principle of the Arbitrary is what we need to focus on. I talked about earlier that you have, for example, sofa, couch, davenport; three different terms, and for most of us we use at least two of those to describe the same piece of furniture. Or, we talk about a groundhog or woodchuck and it's the same animal. Even how air, food and water, the three things that human beings need to live, that none of those terms are the same in any two languages. We use different lexicon in every single language to describe the same three things that we all need. That is proof of arbitrariness, that any given language arbitrarily assigned a lexicon or a phrase to a specific meaning. That's exactly what the Principal the Arbitrary is: connecting the reference to the meaning or the sense. Those were Saussure's terms and when he did this; he had a different aspect for us to think about. He described accurately that when we talk about the morning star and when we talk about the evening star. It's the same heavenly body. What does that mean? You may have heard those terms before, the morning star and the evening star and; if not, the morning star is the last heavenly body, the last thing we see in the sky before the sun comes up and makes it so bright that we can't see any stars. The evening star is the first thing we see once the sun goes down is the first heavenly body that we see. It isn't actually a star—it's the planet Venus—but even still, we use this these two terms to refer to the exact same thing. As Saussure described, that is arbitrary; we arbitrarily name it a different thing, depending on the time of day. That this happens in English or maybe used to happen (there aren’t too many people who say morning star and evening star), but arbitrariness does happen in every language.

Really what we're saying is that a reference or a lexicon or phrase that we use is really a set of truth values, meaning that, if it is correctly attaching or entailing to a certain meaning that it's a true reference: it's true for me, it's true for you. If it's not, then it's a false reference. Let me give you an example. If I hold up this device and call it my phone or my telephone or my mobile phone or my cell phone, all of those are true references, meaning that in any dialect of English this thing would be considered a ‘phone’. It could be any or all of those lexicons. But if I hold this thing up and I call it a pencil, that's going to be a false reference, because in no dialect of English does this apparatus get connected to or entailed to a ‘pencil’; it just doesn't happen. Let me give you another example. If I say, that is a true reference, because this thing that in English we call ‘sky’, it's the thing that is above us when we go outside right; get rid of the clouds and what you see is the sky. That thing up there that thing has a color to it, or at least that we perceive and that color is associated that hue is associated to the term blue, so that's a true reference. If I say the sky is green, for most all of you, that would be a false reference, because in no way does that color get correlated to the term ‘green’. I will say, though, if you are from an area or have spent time in an area that is prone to tornadoes, then you know what that phrase means. If you have the unfortunate instance of being in an area when a tornado is about to drop, the clouds look green; the light bounces off the clouds in such a way that it looks green, and so you say the clouds are green or the sky is green. In that specific reference, it would be true, but for most everybody else, it would be false.

If we talk about Saussure here and the Principle of the Arbitrary, then we have to include Gottlob Frege, who also came from Switzerland. He's also a mathematician, logician, and philosopher of language and he worked alongside Ferdinand de Saussure. He came he came up with the Principle of Compositionality and it is equally important when we talk about semantics; you can't talk about semantics without either the Principle of the Arbitrary or Principle of Compositionality. This principle is actually pretty straightforward and I probably don't need to explain it very much. It is that the meaning of the whole is determined by the meaning of the parts and the way they combine. The components help to form and shape the meaning of the overall. That pretty much makes sense, but I’ll give you a couple of examples. If we think about this morphologically, then certainly if we have a term unbelievable. All we need to do is look at the individual morphemes that are associated with that lexicon, note the meaning of the different morphemes, and we can put together the meaning of the full lexicon. In the term unbelievable, we have a root belief, we have a prefix un-, which means ‘not’ and it's a negator, and we have a suffix, a derivational suffix, that means ‘able to’, -able. ‘Not’ ‘believe’ ‘able’; ‘to not able to believe’—that's what unbelievable means.

It even helps us to understand the history of how certain terms or certain phrases get changed over time. For example, if I say holiday, you have a certain image in mind for most English speakers, that means just a day off. It may include a vacation; certain dialects of English equate a holiday to vacation. However, if I told you that, historically, that it was a sacred day or a holy day, that may not be the meaning of it now, but you can see the history of that, and you can understand how it could come to mean that. These are examples of the Principle of Compositionality: the parts and the way they combined inform the meaning of the whole.

We can also talk about when there is a lack of compositionality. We already have in the previous chapter, when I talked about grammaticality and I talked about the statement that Noam Chomsky came up with: colorless green ideas sleep furiously. We talked about how it is a grammatical sentence structurally; it follows the phrase structure rules of English. Why it doesn't make sense is because it lacks compositionality; the meaning of the individual pieces, that they don't work together. In semantics we call that anomaly; an anomaly is when there is a lack of compositionality. Most of the time, anomaly produces a lack of understanding; you don't understand what somebody is trying to communicate.

However, there are times that anomaly can be massaged a little bit; it can be understood, given a context. One example is, metaphors. If I say there is a fork in the road, what I’m saying has nothing to do with an eating utensil in the middle of the roadway. In that sense, there is anomaly, because there's a lack of compositionality. As an English speaker, you understand what I’m trying to say; I’m describing a situation in which a single road forks or splits into two or more subsequent roads. Because you are an English speaker, you understand what that metaphor is trying to say. That is an example of a bit of elasticity with this Principle of Compositionality, that we can play with this a little bit. This also includes idioms, and it includes something called collocations. A collocation is a combo: X and Y. They are understood as a unit, and they have a little bit of a metaphor quality to it. For example, if I was describing somebody in their late 40s through early 60s, and they are a dark haired individual but there's quite a bit of gray sprinkled throughout their hair. A term we might use in English is salt and pepper, that that person has salt and pepper hair. If I just said, salt hair or pepper hair, you wouldn't understand what I was trying to say. If I say, salt and pepper, together, that is understandable, as a chunk of meaning. It stretches that Principle of Compositionality just a little bit, but it's understood because of the context, because that phrase is part of the ontology of English. I'll give you another example. I used to have a neighbor who did not tell stories very well, and one of the things they used to do was not give very much in the way of detail. If you talked to them after a trip, they would say they went here and there, and did this and that. Here and there, this and that, those are two collocations. They are general phrases to explain that there was probably nothing important going on: they didn't go to anywhere that they felt was interesting, or they didn't want to tell us where they went. Either way, they used two different collocations to describe his vacation.

All of these expansions of the Principle of Compositionality and the Principle of the Arbitrary inform everything with respect to semantics. From this point going forward, both of those principles will have a role in every aspect of semantics and pragmatics.

# 7.11: Denotation

### Sense vs. denotation

So far in this chapter, we have spent a lot of time on lexical meaning: the meaning of words and other linguistic expressions you store in your mental lexicon. In doing so, we have been analysing meaning in term of the sense of the linguistic expressions. The sense of a word is what that word expresses; you store the sense of listemes in your mental lexicon.

Figuring out the sense of any given word is a difficult task. Let’s take a look at a summary of what we have been able to figure out about word senses so far.

1. It’s probably not just a list of necessary and sufficient conditions;
2. It’s probably tied to concepts in some way;
3. Verbs (and other predicates) specify how many arguments it takes, and what role these arguments play;
4. Nouns specify whether it’s count (bounded) or mass (unbounded);
5. Adjectives specify whether it’s stage-level (bounded) or individual-level (unbounded);
6. Some adjectives have a degree argument, some do not.

What we’ve done is identify some major patterns in word senses, but this of course doesn’t fully answer the question “what do words mean?”. If we ask right now what the sense of the adjective sour was, our best approximation would be ‘x is sour to degree d ‘. You may still be wondering, “but what does it mean for something to be sour, exactly, though?”. Similarly, we know that the lexical semantics of pencil says that it’s a count noun, but what makes a pencil a pencil?

Some linguists have proposed that word meaning encodes things like what parts it has (e.g., a pencil is something consists of graphite or a similar substance), what its purpose is (e.g., a pencil is something that is used for writing), and how it came into being (e.g., pencils are man-made, not found in nature) (Pustejovsky 1995). Much of this is still on-going research in linguistics.

Although words like pencil have a somewhat articulable meaning, there are other words like sour whose sense is actually quite hard to characterise, except that it’s, well… that sour taste. What about the word care? What does it actually mean for someone to care about something? What’s the different between a jacket and a coat? Are hotdogs sandwiches? If we focused on the sense of a specific word, we could write a whole book on it! Sense is fun to think about, but if we focus too much on a single word, we can lose sight of the bigger picture. Generally, lexical semanticists are not interested in “pinning down” the exact meaning of any particular word. Instead, they ask more general questions like: “What lexical meaning patterns do we see across different words?”, “What semantic classes of nouns, verbs, and adjectives are there?”, “What is the nature of lexical meaning in the lexicon?”, “How is lexical meaning represented in our minds?”, “How are lexical entries organised in our lexicon?”, and “How does lexical meaning relate to cognition?”.

There is another angle of analysing linguistic meaning: denotation. If you were really pressed by someone to say what the meaning of sour was, you might eventually grab a lemon and say, “Look, this is sour, OK! It’s whatever this is!”. This is a way to talk about the meaning of the word sour via denotation. The denotation of a linguistic expression is what that linguistic expression points to in the actual world.Sour points to all sour things in the world, including lemons, limes, grapefruit, vinegar, pickles, etc. The denotation of pencil would be the set of all things that are pencils in the actual world.

We just said that denotation is what the linguistic expression points to in the actual world, but it’s actually slightly more complicated than that. When we say that the denotation of a word is what it points to in the actual world, we run into problems when we want to analyse the meaning of words like unicorn, or names like Mario (the character from the Nintendo games). If denotations were what these words points to the actual world, then that would be nothing (what mathematicians call the empty set) — because unicorns and Mario do not actually exist in the real world. That means if we take word meaning to be denotations, then unicorn and Mario would be meaningless. But we don’t want to say that! Surely, they are meaningful still. So more accurately, denotation is what the linguistic expression points to is our cognitive representation of the actual world. So words point to the abstract representation of the world that we have in our cognitive faculty, and not the actual things out there, so to speak. Fictional things can certainly have a representation in our cognition, so if we think of denotation this way, even words like unicorn and Mario have “things” they point to in our cognitive faculty. For simplicity, we will say “in the actual world” when discussing denotations in this textbook, but keep in mind that technically, it’s the cognitive representation of the actual world in our minds.

Denotation and sense are related, which is why it is useful to talk about both. Sense is what you store in your head as the semantic information of that word. And whatever this internal information is, that‘s the knowledge that allows you to point to things out there in the real world and say, yes, that’s sour, no, that’s not sour, etc. So the two modes of meaning are connected in that the sense of a word is what you use to figure out the denotation of that word. So even without knowing the exact sense of a word, we can still get a lot of insight about how meaning works in language.

To understand the difference between sense and denotation, consider the following map. Let’s say that this map represents an actual town in Ontario.

Let’s say that you are explaining to someone where the post office is in this town. There are several ways to describe its location, including (1) and (2).

 (1) The building to the north of the bank (2) The building to the east of the library

The phrases in (1) and (2) have different senses: the phrases don’t contain identical words (e.g., north vs. east, bank vs. library), so the internal semantic content of each phrase would be different. However, (1) and (2) have the same denotation: the post office in this town. Although described differently, both phrases point to the same thing.

Consider also an expression like (3).

 (3) My house

Imagine that Kumiko said (3). If that were the case, the expression in (3) would point to Kumiko’s house in this town (represented by the bottom left box on the map). If Lev said it though, this same expression would point to Lev’s house (represented by the bottom middle box on the map). This means that my house has a different denotation depending on who says it. It has the same sense no matter who says it though: something along the lines of ‘the place of living that is associated with the speaker’.

## The denotation of words

Denotation gives us a fairly efficient way to talk about the meaning of words. For sour, you are essentially saying, “sour is whatever property that all of these sour things have in common.” You are pointing to that group of sour things. So, we can characterise the denotation of one-place predicates (predicates that only take one argument: some nouns, adjectives, and intransitive verbs) as sets of things. A set is a collection of things. In (4), ⟦x⟧ (x enclosed in double brackets) should be read as ‘the denotation of the linguistic expression x ‘.

 (4) a. ⟦sour⟧ = the set of all sour things in the actual world b. ⟦pencil⟧ = the set of all pencils in the actual world c. ⟦snore⟧ = the set of all things that snore in the actual world

You might find it unintuitive that verbs denote a set of individuals too. But think about it this way: if someone asked “Who snores?”, you can answer “they do” and point to the people that have this snoring property.

We will treat the denotation of proper names (like Nathan Sanders, the name of a linguist, who happens to be one of the co-authors of this textbook) to be the particular individual that that the name is pointing to in that context. In other words, the denotation of Nathan Sanders is the individual Nathan Sanders in the actual world. Sometimes the denotation of a name is abbreviated as the initial letter of the name (and bolded), like in (5b).

 (5) a. ⟦Nathan Sanders⟧ = the individual Nathan Sanders in the actual world b. ⟦Nathan Sanders⟧ = n

## Constants and variables

The symbol that stands for the denotation of Nathan Sanders (in this case the bolded lowercase n) is called a constant. A constant is different from a variable. Recall that a variable is a placeholder for other values; it’s called a variable because what it stands for can vary. A constant has a fixed value: what it stands for cannot be altered. So in (5), the constant n always stands for the individual Nathan Sanders. Any letter or symbol can be used as constants or variables, although traditionally, letters towards the beginning of the alphabet in English are typically used for constants (e.g., a, b, c) and letters towards the end of the alphabet tend to be used for variables (e.g., x, y, z). In this textbook, constants will be bolded, but variables won’t be.

When a linguistic expression denotes (points to) a unique individual, this can also be called the expression’s reference. For example, a name like Nathan Sanders points to a unique individual, so the individual Nathan Sanders is this name’s reference. We can also say that the name Nathan Sanders refers to that individual.

It should be re-emphasized here that words like sour, pencil, snore, and Justin Trudeau denote actual things in the actual world. When we say “⟦sour⟧ = the set of all sour things in the actual world,” we are not saying that the denotation of the word sour is the phrase (the linguistic expression) the set of all sour things in the actual world. We are quite literally saying that the word sour points to actual sour things that exist out there in the real world (or at least, our cognitive representation of them; see note above). The following representation may be helpful for imagining what denotation really is:

Because we don’t always have the time to find stock images to represent the denotation of a linguistic expression, in this textbook, we will use the convention in (4) and (5).

Thinking of meaning in terms of the linguistic expression’s denotation is useful in various ways. For example, how can we characterise the compositional meaning of the sentence Nathan Sanders snores? The denotation of snore(s) is the set of all things that snore in the actual world, and the denotation of Nathan Sanders is the individual Nathan Sanders in the actual world. What Nathan Sanders snores means, then, is that this individual Nathan Sanders is among this set of things that snore in the actual world.

## The denotation of sentences

We know what some words denote now; how about sentences? What do they denote? One property of a sentence is that it has a truth value. There are two truth values in language: the abstract truth value true (sometimes written as T or 1), or the abstract truth value false (sometimes written as F or 0). To say that a sentence has a truth value means that a sentence is always either true or false. Words and non-sentential phrases don’t have truth values. You can see if a linguistic expression has a truth value or not by embedding it in the construction “It is true/false that…”, like in (6).

 (6) a. It is true/false that Nathan Sanders snores. b. * It is true/false that Nathan Sanders. c. * It is true/false that snores. d. * It is true/false that lives in Canada. e * It is true/false that in Canada.

(6) shows that only sentences have truth values. Taking this observation, we can actually think of the denotation of sentences as its truth value: either the value T or the value F. In other words, sentences “point to” these abstract truth values. (7a) shows the denotation of a sentence that is true, and (7b) shows the denotation of a sentence that is false.

 (7) a. ⟦Nathan Sanders is bald⟧ = T b. ⟦Nathan Sanders is not bald⟧ = F

It is crucial to observe here that you need world knowledge in order for you to determine the denotation of a sentence. Oftentimes, we know enough about the world that we know whether a sentence is true or false (like the above examples). However, the more realistic view is that we don’t know everything about every single thing in the world. Take the sentence My neighbour’s cat’s liver weighs 326 grams for example: what’s the denotation of this sentence? It it true or false? The sentence surely has a truth value, but we don’t have the relevant world knowledge to actually determine the truth value. This is why we often express the denotation of a sentence as its truth condition (the circumstances that yield a certain truth value). The truth condition of My neighbour’s cat’s liver weighs 326 grams is shown in (8).

 (8) ⟦My neighbour’s cat’s liver weighs 326 grams⟧ = T if my neighbour’s cat’s liver weighs 326 grams in the actual world, F otherwise

It might help to think of it this way: if we really wanted to, we could figure out if the sentence in (8) is true or false. Whether we can or want to is another matter. What’s more relevant is that we know what circumstances would make the sentence true or false.

### If and only if

The denotation in (8) can also be written as “T if and only if my neighbour’s cat’s liver weighs 326 grams in the actual world”. If and only if means ‘provided that’: T provided that my neighbour’s cat’s liver weighs 326 grams in the actual world. This means that if my neighbour’s cat’s liver weighs 326 grams, then the truth value is T, and that if the truth value is T, then my neighbour’s cat’s liver weighs 326 grams. So, “if and only if” (sometimes also abbreviated as “iff“) means that the implication goes both ways. This is the same thing as what (8) says.

The formula for the denotation of a sentence should be seen as: truth condition + world knowledge = truth value. If we reframe (8) in terms of truth conditions, we get something like this:

 (9) a. ⟦Nathan Sanders is bald⟧ = T if Nathan Sanders is bald, F otherwise b. ⟦Nathan Sanders is not bald⟧ = T if Nathan Sanders is not bald, F otherwise

In this case, we can combine our world knowledge with each truth condition to get T in (9a) and F in (9b), which is what was shown in (7). If you do have the relevant world knowledge to determine the truth value of a sentence, you should write out the actual truth value, like in (7). If you don’t have the relevant world knowledge, you can write the denotation of the sentence as a truth condition.

In summary, intransitive verbs, adjectives, and nouns denote sets, proper names denote an individual, and declarative sentences denote truth values. In the rest of this chapter, we will look at various semantic phenomena that denotative meaning is equipped to explain.