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).

But: what is a sense?

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 the expression picks out is its denotation.

We elaborate this idea below.

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...

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

3. the president of the United States of America in 1862
4. the president of General Motors in 1960
5. the president of the United States of America in 1860

So we have the following:

Sense + Discourse Context + Linguistic Context Determines 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.

What is sense?

We tried to get at word senses through word definitions.

Definitions as understood by a lexicographer aren't going to play much of a role in our theory of semantics. We decided that people really don't have definitions in their heads.

We looked at two kinds of definitions:

1. Ostensive
2. Definitions by paraphrase

We found numerous problems with word definitions:

1. Hard to get right
2. definitions, to the extent that they are precise, don't seem to capture a speaker's smantic competence.
3. Knowing a prototype and a rought set of clear cases may qualify us for kn owing the meaning of a word.
4. Definitions dont tell us anything about how simple meanings are composed to make complex ones.

These do have a rough correlation 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.

• Describing how the truth conditions of closely related sentences are related
  1. John drives a Buick
  2. John doesnt drive a Buick.
• Describing the semantic relation of systematically related ambiguities (e.g., scope ambiguities)
• Describing entailment relations among sentences:
  1. John drives a Buick.
  2. There is a Buick (A buick exists).

Describing linguistic knowledge that has a systematic status in the grammar.

How we know the meaning of a complex but isolated concept like alimony may be related to various complex cognitive and social abilities.

How we know the meaning of the comparative morpheme is a linguistic question that cuts across many concepts.

How we know how to build complex meanings out of simple ones is a linguistic question that cuts across over many concepts.

• Encylopedic knowledge about words
• Complete definitions of words
• Complete truth-conditions for words (couch, oak, bronze, ...)
• A complete set of semantic features that distinguishes all meanings (NOT like Chopmsy-Halle phonology, and maybe phonology isn't even like that, anymore)

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

So meanings can't just be denotations.

We're going to try get at meaning top-down.

In this way, we hope to get at semantic ideas that get us closer to talking about semantic competence.

Instead of starting with word meanings, we're going to start with sentence meanings.

Let's start with denotation. What is the denotation of a sentence? Sentences (at least many of them) have the property that they can be true or false on each occasion of utterance (just the way the president refers to a particular person on each occasion of utterance).

We will assume that the denotation of a sentence is its truth value. We will assume that the sense of a sentence is a proposition. Propositions are claims about reality that may be true or false on different occasions.

We will assume that the denotations of words are their contributions to the truth of a sentence.

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

Expression Type	Definition	Example
term	an expression that denotes an individual.	Fido
predicate	an expression that denotes a set	dog

Can we get more specific about senses by getting more specific about propositions?

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 might be a different set of dogs. Then the claim Fido is a dog might turn out to have a different truth value. In general, the truth of a proposition depends on the particular way the world is, the particular sets and individuals that instantiate the predicates and terms that express the proposition.

Thus the denotation of a sentence 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 truth value of Fido is a dog varies from world to world, and the way that happens is that 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

So senses are double-layered. A sense 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....

We need a theory of semantics that covers all expressions. Thus in particular, many kinds of expression should have senses/intensions.

Start: What should the intension of a sentence be?

First, what should the extension at a particular world be? A truth-value.

What about the extension of a predicate like dog? A set. The set of dogs.

How do we decide this in general?

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

Substitutivity of Identicals

The denotation of a complex expression remains the same when a sub-expression is replaced with an expression of identical denotation.

A sentence is a complex expression with a denotation, which we decided was a truth-value.

What happens when we substitute different expressions with the same denotation into a sentence:

1. The governor of California in 2005 is grinning.
2. Arnold Schwarzenegger is grinning.
3. Mr. Universe 1970 is grinning.

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

Just as we would like, the truth-value of a sentence functions like the denotation of a noun phrase. 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.

Now let's think about predicates. Consider dog. Assume the proper name Fido denotes the same individual in every world (just as we assumed for the proper name Arnold Schwarzenegger). Now if the denotation of Fido is a dog is going to change, it is going to have to so because the denotation of some other expression in the sentence does.

The natural candidate is the word dog. I the denotation o dog changes from world to world then so will Fido's doghood.

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

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

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.

Fido is a dog.

world	extension of Fido	extension of dog	extension of Fido is a dog
w1	Fido	{Fido, Bowser, Rin Tin Rin}	True
w2	Fido	{Fido, Bowser, Lassie}	True
w3	Fido	{Lassie, Bowser, Rin Tin Rin}	False
w4	Fido	{Fido, Lassie, Rin Tin Rin}	True

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

Problem:

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.

Meaning = Lexical Meaning + Structural Meaning

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

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 is:

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

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' is 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.

These are some notes for this homework exercise.

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 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.