Intension and Extension

Intension
A theoretical
Construct 		 

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....

Denotation
Extension 		 

Last week I was basically using sense as a synonym of meaning and denotation as a synonym for extension.

I did not talk about intension, because I want to reserve that term for the technical notion just introduced. A theoretical construct.

So let's talk about intensions and extensions for sentences now

Extensions
for sentences 		  What should the extension(denotation) of a sentence be?

Last week, we looked at TWO ANSWERS to this question:

  1. Frege's answer: a truth value (TRUE or FALSE, das Wahr order das ??)
  2. John Perry's answer: a described situation

This week we focus on Frege's answer, as our textbook does.

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: Since denotations of

  1. The governor of California in 2005
  2. Arnold Schwarzenegger
  3. Mr. Universe 1970
are in fact 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 For Frege 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.

Example:
Intensions 		 
    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 suppose that this is the way it is with grinning in our 4 worlds:
Intension of "grin"
w1 {Arnold, Cruz, Gray, Steve, Brett}
w2 {Arnold, Cruz, Gray, Steve}
w3 {Arnold, Gray, Brett, Steve}
w4 {Cruz, Gray, Steve, Brett}

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.

Sentence
Meaning 		 

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.
All use the same words. So words of a sentence can't alone determine its meaning.

Sentence Meaning = Lexical Meaning + Structural Meaning

This is the same point I made last week with the example of intersection semantics (only the words in the sentence mattered in that theory and that theory was wrong).

A job for
Syntax 		 

Determine structural meaning!

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

You finally know.

Compositional
Semantics 		 

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 an intensional theory. 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. That is, 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.

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.
This rules determines the extension of the sentence 'A and B" in terms of then extension of A and the extension of B. But there is no such thing as the extension of 'and'; 'and' is treated syncategorematically.

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:
Compositional
Semantics 		 

A small compositional semantics.

The exercise is in the section labeled "Exercise".

Summarizing  

  1. Solution: Model meaning with an abstraction Intension: a technical invention, but one way of approximating meaning. An intension is a table (a function) that tracks changes in denotation across changes in the world.
  2. The role of Structural Meaning: Combine with lexical meaning to give meaning.
  3. The role of Syntax: In this course Syntax is the handmaiden of semantics. The role of syntax is to determine structural meaning.
  4. 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.
  5. 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 make the right number of distinctions in keeping track of how the meanings of larger expressions are determined by their subexpressions.