Sense and denotation

Assignment for
this week

Read chapter one carefully. Except for Section 1.2.1. Read Section 1.2.1 carelessly. Try not to remember any of the terminology introduced there.

Do the exercises introduced in this lecture with Exercise heading.

Sense vs

An expression like the president uttered today denotes a particular individual in the world, George Bush.

Uttered in 1993 it denoted Bill Clinton.

Nevertheless the meaning or sense of the expression the president hasn't changed. Modulo the elimination of a few voters in a few Florida and Ohio counties, the way of determining what the expression the president denotes is the same.

We say the sense determines the denotation. (though we'll modify this slightly below).

Ways of

Although in the previous discussion we focused on the expression the president, we can narrow our attention to the lexical item president. What exactly the word the does we're not ready to discuss yet, but clearly the word president bears primary responsibility for the fact that the expression the president picks out George Bush.

Since we're interested in meaning, let's talk about how the meaning of a word like president is defined:

  1. Ostensive (pointing) definition. An act of ostension is an act of pointing. A definition by ostension is one performed by pointing at the denotation or part of the denotation of a word.
      Pointing at George Bush to define the word president
    Of course if we are really explaining the word to someone who has no idea what the word president means, looking at George Bush may not be enough to give her the idea. So we may need a list of examples:
      Washington, Adams, Jefferson, Monroe, Madison.... George Bush
    There is often something a little unsatisfying about ostensive definitions. We are left a little unsure as to whether someone who knows all the list members could still recognize something or someone not on the list who falls under the definition. This despite the fact that the list may be factually complete.

    For example someone looking at this list might well conclude that being male was part of the definition of being preseident, but this should definitely not be part of the definition. More on this later...

  2. Definition by paraphrase Giving another word or phrase that has the same sense.
      president: the executive head of an organization or a state, usually elected.
    Of course this raises other issues, such as what exactly is an "executive" head? Also have we really identified all the necessary and sufficient conditions of being a president?

Note: A word may have many senses, even when we control for syntactic category. The verb bank has several senses:

  1. We bank at the Savings and Loan.
  2. We're banking on Drew Breeze playing well.
  3. The plane banked hard left.

Both kinds of definition have their problems and won't really play a role in our theory.

However they are useful in identifying some of the intuitions that go with sense and denotation. Ostensive definitions try to pick out parts of the denotation of a word; definitions by paraphrase try to pick out the sense.

Also we saw that in some sense definitions, of either kind, aren't complete. But it's not clear that complete definitions are the business of semantics. This is the subject of exercise one:

Exercise one:

Consider the following principle of Generative linguistics.

    Goal of theoretical linguistics: Characterize what speakers know when they speak and understand a language.

Part A: Are speakers are able to speak and understand without knowing complete definitions of the words they use? Give 3 plausible examples of successful communications in which speakers aren't able to produce complete definitions of the words they use. Hint: There are lots of ways to do this but use at least one simple example using what are called natural kind terms. A natural kind term denotes some natural kind of thing in the real world. Examples are dog, cat, tree, dolphin, Border Collie. You might also consider using some non-natural kinds in your discussion. Non-natural kinds are kinds that are human-made.Chair, telephone, computer are examples. Don't use any of these exact words as your examples. Also, don't use technical language. In other words, I don't know the definition of the term chondromalacia (something that can happen to a knee) but it's also pretty clear that me trying to communicate with this word is fraught with peril. So what we're interested in is cases where you don't know the exact definition of a common word but you can still communicate with it effectively.

Part B: Address the following issue in your discussion. When we do syntax we concede that speakers are not able to articulate what a subject or a constituent is, but we argue that they have tacit knowledge of these things on the basis of the evidence of their grammaticality intuitions. Thus we argue that speakers know that the book about Lincoln is a constituent in

    John lost the book about Lincoln.
on the basis of the fact that they agree that
    It was the book about Lincoln John lost.
is grammatical. And we argue that speakers know that the book on the table is not a constituent:
    John put the book on the table
on the basis of the fact that they agree on the oddity of:
    * It was the book on the table John put.

Can we argue the same about complete definitions? That is, can we argue that speakers DO have complete definitions in their heads but that the knowledge is tacit and they are unable to articulate these definitions?


Above we used the term denotation to pick out what a noun phrase referred to. In the special case of a noun phrase this is also called its reference. We use the term denotation for more than just noun phrases. For all kinds of expressions, the part of reality tthe expression picks out is its denotation.

    Expression Type Denotation
    the cat Noun Phrase a particular feline entity, say, Garfield, the one referred to on this occasion of utterance
    cat Common Noun a set of animals, all of them felines
    dog Common Noun a set of animals, all of them canines
    walk Verb a set of actions, entailing movement and use of the legs
    pass the salt Verb Phrase a set of actions, not greatly more or less complicated than walking as actions go, but requiring more linguistic parts to identify

We said above sense determines denotation.

This can't be quite right.

First of all the expression the president denotes different persons on different occasions of use.

So it's at least the case that sense + time of utterance determines the denotation. In other words, at least one other thing besides the sense is required to determine the denotation.

But also we have the following, which is perfectly fine even I utter it right now:

    In the summer of 1862 the war was going badly. The president knew...
And here sure as heck the expression the president uttered right now can denote Abraham Lincoln.

So context can fill in the time of the presidency we are talking about.

Of course context has been filling more stuff than time in all the examples we've been talking about:

  1. the president of the United States of America.
  2. the president of General Motors
Here the of-phrase specifies the organizational entity which the president heads, which was understood in our previous examples to be the United States. In this case something that was previous being filled in by context is now filled in by linguistic expressions. All along we understood ourselves to be talking about the president of the U.S., but we can make that linguistically explicit if we want, or even choose a different organizational entity for the president to be president of. We can also make the time explicit:
  1. the president of the United States of America in 1862
  2. the president of General Motors in 1960
  3. the president of the United States of America in 1860
And sometimes even when we fill in a particular attribute linguistically, that isn't enough to nail things down. The expression the president of the United States of America in 1860 is actually ambiguous since we may be referring to either James Buchanan or Abraham Lincoln depending on whether we mean the one before or after inauguration day.

So we have the following:

    Sense + Discourse Context + Linguistic Context Determines Denotation
Different sense
Same denotation

Expressions with very different senses can have the same denotations and this is largely unpredictable linguistically, because it has to do with the weird and wonderful way the world turned out.

Some expressions with different senses and identical denotations:

    The Boston Red Sox the winner of the 2004 World Series
    George Bush the president of the U.S. in 2005
    the United States the U.S.
    the president of the United States in 2005 the president of the U.S. in 2005
    the morning star Venus
    Venus the evening star
    the morning star the evening star
    Mr. Universe 1970 the governor of California 2005
    1970 the year of the great tsunami
    the governor of California 1970 the governor of California the year of the great tsunami

Speculation: Substitutivity of Identicals
Replacing an expression E with denotation D with another expression E' with the same denotation D does not change the denotation of the larger expression in which E occurred.

Sense or

The preceding examples should pretty much have convinced you that what's important for the theory of semantics is sense, not denotation. Denotations vary at the whim of the electorate and the Gods of baseball. What matters linguistically is senses!

The trouble with this conclusion is that there's lots wrong with senses as well.

  1. senses pick out classes of things in the world that may have vague or poorly defined boundaries. (bald, young, smart), so there is no paraphrase that captures a sense.
  2. family resemblances: Sometimes senses don't come with clear set of defining conditions; instead they come with a variety of sets of conditions, each set similar to, but not identical with, the last. Wittgenstein's example of the word game: soccer, chess, ring-around-the-rosie, pretend games, solitaire
  3. The family resemblance problem is a special case of a general problem with senses: Determining how many senses a word has isn't always easy, as any lexicographer (dictionary writer) knows.

Denotations, despite their limits, are clearer, For example, the principle of substitutivity gives us a very powerful handle on denotations. We have clearer intuitions about what they are, how they change.

Still there is the problem of the evanescence of denotations.

Consider dog again. We said its denotation is:

    a set of animals, all of them canines
Yet we don't want a theory that says the meaning of the word dog changes whenever a puppy is born...
A theoretical

Solution: We're going to extend the way we use the term denotation by means of a theoretical construct.

Consider dog. At any instant of time there is a certain set of individuals that is the set of dogs. If the world were only slightly different than it is, there would be a slightly different set of dogs.

Thus the denotation depends not only on the sense but also on the way the world is, or as we shall say, on what world we are in.

So our theoretical construct is this: there is a set of worlds we call the set all possible worlds, and the denotation of dog varies from world to world:

    w1 Fido, Bowser, Argus, Rex
    w2 Bowser, Argus, Rex
    w3 Fido, Argus, Rex
    w4 Ashes, Bowser, Argus, Rex
We call this table the intension of dog. We call the value of the table at each world the extension of dog at that world. We will call the the intension the denotation of dog in our new theory.

So denotations are double-layered. A denotation is an intension that picks out an extension at each possible world.

These examples are incomplete. The set of all possible worlds really needs to cover all possible ways the world might differ from the way it is. With respect to the word dog the set of possible worlds needs to provide with all possible individuals that might be a dog, and at least one world for each distinct possible set of dogs. That's a lot of worlds....

for sentences
  What should the denotation of a sentence be?

First, what should the extension at a particular world be?

To answer this question we will return to the principle of substitutivity of identicals.

A sentence is a complex expression with a denotation. Let's see what happens when we substitute different expressions with the same denotation into it:

  1. The governor of California in 2005 is grinning.
  2. Arnold Schwarzenegger is grinning.
  3. Mr. Universe 1970 is grinning.
According to the principle of substitutivity of identicals the denotation of these three sentences should be the same.

But what is it that is the same? Not the meaning. In a different world in which the same guy wasn't Mr Universe 1970 and the governor of California 2005, these sentences might have different truth-values. But the meaning would still be the same.

But here is the key point: In any world in which the denotations of

  1. The governor of California in 2005
  2. Arnold Schwarzenegger
  3. Mr. Universe 1970
are the same, then the truth-values of all 3 sentences must be the same.

So it is the truth-value of a sentence that functions like a denotation . It is the truth-value of a sentence that stays the same when expressions with identical denotations are substituted into it. [Frege's argument]

So the extension of a sentence will be a truth-value (either true or false); and the intension will be a table that tells us the truth value in each world. The extension at each world will depend on the extension of expressions inside it at that world.

    Intension of "Arnold Schwarzenegger" Intension of "the governor of CA 2005" Intension of "Mr. Universe 1970"
    w1 Arnold
    w2 Arnold
    w3 Arnold
    w4 Arnold
    w1 Arnold
    w2 Gray
    w3 Cruz
    w4 Arnold
    w1 Arnold
    w2 Brett
    w3 Steve
    w4 Arnold
Point to note: In w1 and w4 the extensions of all 3 expressions are the same.

And consistent with these we have the following intension:

    Arnold Schwarzenegger grins The governor of CA 2005 grins Mr. Universe 1970 grins
    w1 true
    w2 true
    w3 true
    w4 false
    w1 true
    w2 true
    w3 false
    w4 false
    w1 true
    w2 false
    w3 true
    w4 false
Point to note: In w1 Arnold is grinning and in w4, he is not, but all 3 sentences have the same denotation in those worlds, because all 3 subject noun phrases have the same extensions there, following the principle of substitutivity. Since all three subjects denote Arnold in those worlds, either all 3 sentences are true or all 3 are false there. In w2 and w3, the 3 sentences can differ in truth, because the 3 noun phrases have different extensions.

Note: I differ slightly from the text in saying that intension of a sentence is a table giving the truth value at each world. The text just says the intension of a sentence is the set of possible worlds at which the sentence is true. These ideas are equivalent: You can construct the table from the set, and vice versa. More on this later.


The wrong theory: the meanings of the words determine the meaning of a sentence.


    The A that Xed the B Yed the C

Pick from among the following nouns for A,B,C:

    dog, cat, rat

Pick from among the following transitive verbs for X,Y.

    chased, bit

Many sentences with MANY different meanings:

  1. The dog that chased the cat bit the rat
  2. The rat that chased the cat bit the dog
  3. The cat that chased dog bit the rat.
All use the same words. So words alone can't determine meaning.

Meaning = Lexical Meaning + Structural Meaning

A job for

Determine structural meaning!

That's what it's for. That's why there IS such a thing as syntax!

You finally know.


A compositional semantics for a (fragment of a) language is a formal account of how the meaning of the whole is composed of the meaning of the parts.

Given what we just noted about structural meaning, this means that a compositional semantics combines lexical meaning and structural meaning to compute the meaning of sentences. It does this by providing rules for interpreting strcutures.

More technically, we have just committed to a denotational theory, where denotation means intension. Thus our theory needs to account for how the intension of the whole is composed out of the intension of the parts.

But in working out the details, it quickly becomes clear that not every word has a natural intension:

  1. and
  2. or
  3. every be
These words don't seem to have extensions. Thatis, there aren't naturally parts of the world picked out by such words (roughly the classical linguistic category of function words). These words will NOT be given intensions. They will be given what is called a syncategorematic treatment:
    A syncategorematic expression is one that does not have a denotation of its own but which has a special interpretive RULE associated with it. Thus syncategorematic meaning can be thought of part of structural meaning.

For example, the extensional rule for and (discussed in Section 1.3.4, (38), (39)) is:

    'A and B' is true if and only if 'A' is true and 'B' is true.
This is a rule for a assigning the extension to a certain type of sentence. The extension of a sentence is a truth value.

Thus, all the rule has to do is tell us under what circumstances the sentences is true or false (in terms of the truth values of the sentence's parts). A similar rule for or:

    'A or B' is true if and only if 'A' or true or 'B' is true.

The intensional treatment of and has to define an intension; that is, for every world, it has to tell us under what circumstances the conjoined sentence is true in that world, and under what circumstances it is false:

    'A and B' is true at a world w if and only if 'A' is true at w and 'B' is true at w.
Exercise 2:

A small compositional semantics.

The exercise is in the section labeled "Exercise".


  1. Sense vs denotation
    1. Sense: a meaning or definition.
    2. Denotation: The part of the world a word picks out
  2. Problems with both:
    1. Senses are hard to define or even count
    2. Denotations change as the world changes
  3. Solution: Model meaning with an abstraction (called an intension) that tracks changes in denotation across changes in the world.
  4. The role of Structural Meaning: Combine with lexical meaning to give meaning.
  5. The role of Syntax: In this course Syntax is the handmaiden of semantics. The role of syntax is to determine structural meaning.
  6. Compositional Semantics: A formal theoretical account of how structural meaning combines with lexical meaning to give the meaning. It consists of rules that assigns denotations to structures.
  7. For us meaning is always denotational. Denotations are intensions; denotations at a possible world are extensions.
  8. Conclusions
    1. In some sense Chapter 1 of our text proposes an answer to the question "What is meaning?", but it doesn't go about trying to answer the question as if it was an empirical question.
    2. It's a not a common sense question with an answer drawn out of experience
    3. Meaning is a theoretical concept.
    4. The criterion we'll use for whether we've given the question a good answer is whether it helps us give account of what speakers of a language know when they speak and understand utterances of that language in contexts of real communication.