Lecture Note
Semantics 1: Possible World Semantics for Attitudes
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University:
Massachusetts Institute of Technology -
Course:
24.910 | Topics in Linguistic Theory: Propositional Attitudes Academic year:
2024
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Views:
421
Pages:
8
Author:
dedalcatechizes
24.910, Spring 2009 (Stephenson)
1. Some Notes on the Reading
[Ch. 1-2 of von Fintel & Heim’s Intensional Semantics]
“intensional” / “intension”
¾ Spelled with an “s” – presumably on analogy with “extension” and to distinguish
it from “intention”
“Assignment functions”
These are used (among other things) to capture the interpretation of pronouns, and
shouldn’t be relevant to us.
¾ You can ignore reference to “assignment functions” and the parameter “g.”
¾ Read [[α]]w,g as [[α]]w
(In most of the cases in the reading, “g” is an idle wheel anyway.)
Lambda notation
¾ Quick and dirty version: This is a way of writing a function:
λ _______________ .
______________
domain variable
output of function
¾ For example: instead of writing f(x) = x2, we would write
f = λx.x2
or f = λx∈R.x2
[specifying that the domain is real numbers]
[Note: this also means that λx.x2 is simply the name of a function]
¾ One benefit: making it easier to write functions that take other functions as
arguments (very common in semantics).
¾ For example: instead of writing f(g(x)) = g(x) + 1, we would write
f = λg . λx . g(x) + 1
or f = λg∈R×R . λx∈R . g(x) + 1
Specifying types
¾ e = individuals [“entities”]
t = truth values
s = worlds [“situations” / “states of affairs”]
¾ xe = x (indicating that x is a variable ranging over individuals)
¾ <α, β> = function from things of type α to things of type β. For example:
¾ For example: = function from individuals to truth values (1-place predicate)
= function from worlds to truth values (proposition)
1 24.910, Spring 2009 (Stephenson)
Set-talk and function talk
¾ Many semantic values are functions from something to truth values.
¾ Functions of this kind have a natural mapping with sets:
The set characterizing f = {x: f(x)=1}
¾ Thus <α,t> can also be thought of as the type of a set of objects of type α
This relationship is so familiar to semanticists that we often go back and forth between
set-talk and function talk without making this explicit.
1.1.
Selected Exercises
Exercise 1.1 (p. 6)
¾ [[A famous detective lives at 221B Baker Street]]w7
[Note: The subject takes the VP as an argument instead of vice versa because the subject
is a quantifier; this makes the exercise needlessly complex for our purposes, but may help
give some practice with the system assumed in the reading]
= [[A famous detective]]w7 ( [[lives at 221B Baker Street]]w7)
¾ [[lives at 221B Baker Street]]w7 = [[lives at]]w7 ( [[221B Baker Street]]w7 )
[Substituting in lexical entries from (13c) and (15c) in Ch. 1 of von Fintel & Heim]
= [λx . λy . y lives at x in w7] ( 221B Baker St.)
[Substituting 221B Baker St. for x]
= [λy . y lives at 221B Baker Street in w7]
¾ [[famous detective]]w7
[Using a rule of “predicate modification” introduced in H&K. This rule basically says
that multiple modifiers can be put together semantically using set intersection.]
= [[famous]]w7 ∩ [[detective]]w7
= λy . y is famous and y is a detective in w7
In sets: {y: y is famous and y is a detective in w7}1
¾ [[A famous detective]]w7 = [[a]]w7 ( [[famous detective]]w7 )
[lexical entry for a/some from (14d)]
= [λf . λg . ∃x: f(x) = 1 and g(x) = 1] ( [λy . y is famous and y is a detective in
w7] )
1
Of course, being a famous detective (on one salient reading, at least) isn’t really the same thing as being
famous and being a detective – if Sherlock Holmes happened to moonlight as a waiter, we wouldn’t call
him a famous waiter. So famous should probably be treated as a non-intersective modifier (more generally,
a function from sets to sets). But this will not be our problem in this course.
2 24.910, Spring 2009 (Stephenson)
[substituting bolded argument for f]
= λg . ∃x: [λy . y is famous and y is a detective in w7](x) = 1 and g(x) = 1
= λg . ∃x: x is famous and x is a detective in w7 and g(x) = 1
Back to the main computation:
¾ [[A famous detective lives at 221B Baker Street]]w7
= [[A famous detective]]w7 ( [[lives at 221B Baker Street]]w7)
[from above]
= [λg . ∃x: x is famous and x is a detective in w7 and g(x) = 1]
( [λy . y lives at 221B Baker Street in w7] )
[substituting bolded argument for g]
= ∃x: x is famous and x is a detective in w7 and [λy . y lives at 221B Baker
Street in w7] (x) = 1
= ∃x: x is famous and x is a detective in w7 and x lives at 221B Baker Street in w7
2. Introducing Intensional Semantics
We can set up a lot of the mechanics of an intensional semantics using a simplistic but
useful example of a fictional world.
2.1.
(1)
Preliminaries
A famous detective lives at 221B Baker Street.
¾ Extension of (1) at a given world w:
[[a famous detective lives at 221B Baker Street]]w
= 1 iff a famous detective lives at 221B Baker St. in w
¾ Intension of (1):
λw' . a famous detective lives at 221B Baker St. in w'
[We won’t need to deal with the internal composition of (1).]
(2)
In the world of Sherlock Holmes, a famous detective lives at 221B Baker Street.
¾ Assume in the world of S.H. is a sentential modifier
¾ We can give a semantics for this modifier using an intensional system.
For convenience, let’s pretend that:
¾ w9 = the world depicted in Sir Arthur Conan Doyle’s stories about Sherlock
Holmes
¾ w6 = the actual world
3 24.910, Spring 2009 (Stephenson)
2.2.
Version I: “The world of Sherlock Holmes”
Informal idea
The modifier signals that the sentence is to be evaluated at world w9 (the world of
Sherlock Holmes) rather than at the actual world
More formally:
(3)
[[In the world of Sherlock Holmes, φ]]w = [[φ]]w9
[i.e., in the world of Sherlock Holmes, φ is true in any world w iff φ is true in w9.]
More compositionally:
(4)
[[In the world of Sherlock Holmes]]w = λp . p(w9)
Evaluating (2) at world w6:
(5)
[[In the world of Sherlock Holmes, a famous detective lives at 221B Baker
Street]]w6
[the meaning of the modifier (at the actual world) applied to the proposition that there is a
famous detective at 221B Baker St.]
= [[in the world of Sherlock Holmes]]w6
( [λw . a famous detective lives at 221B Baker St. in w] )
= [ λp . p(w9) ] ( [λw . a famous detective lives at 221B Baker St. in w] )
[substituting the argument for p]
= [λw . a famous detective lives at 221B Baker St. in w] (w9)
[substitute w9 for the variable w]
= 1 iff a famous detective lives at 221B Baker St. in w9
2.3.
Version II: The world of Sherlock Holmes as
presented by Sir Arthur Conan Doyle in world w
¾ A problem with the idea in 2.2:
It’s a contingent fact that Sir Arthur Conan Doyle wrote the Sherlock Holmes
stories the way he did. He might have (in some other world) set things up so that
Sherlock Holmes lived on Abbey Rd (and no detective lived on Baker St.)
¾ Improvement: Use a function from worlds to worlds:
sher (w) = the world of Sherlock Holmes as set down in w
so in particular, sher(w6) = w9
4 24.910, Spring 2009 (Stephenson)
Perhaps:
w6 Æ w9
w7 Æ w12
w8 Æ w11
…
Implementing this:
[[In the world of Sherlock Holmes, φ]]w = [[φ]]sher(w)
(6)
Again, more compositionally:
[[In the world of Sherlock Holmes, φ]]w = λp . p(sher(w))
(7)
= λp . the world w' as it is described in the Sherlock Holmes stories as written
in w is such that p(w') = 1
2.4.
Version III: The set of worlds compatible with what is
presented by Sir Arthur Conan in world w
¾ A problem with the idea in 2.3: Sir Arthur Conan Doyle didn’t actually make
every aspect of his fictional world explicit. For example, we don’t know whether
Holmes had an odd or even number of hairs on his head the day he met Watson
(and in some sense Sir Arthur Conan Doyle doesn’t know either!)
So:
¾ It makes more sense to talk about a set of worlds compatible with what is written
in the Sherlock Holmes stories.
¾ Recall: it’s still a contingent fact that they were written the way they are
¾ So the function sher should now yield a set of worlds:
sher(w) = {w': w' is compatible with the world as depicted in the Sherlock
Holmes stories as written in w}
Finally:
[[In the world of Sherlock Holmes, φ]]w = 1 iff ∀w'∈sher(w), [[φ]]w' = 1
(8)
Again, more compositionally:
[[In the world of Sherlock Holmes, φ]]w = λp . for all w'∈sher(w), p(w') = 1
(9)
3. Semantics of Attitude Predicates
Recall: the function sher:
¾ sher(w) = {w': w' is compatible with the world depicted in the Sherlock Holmes
stories, as written in w}
We can define a similar function for a person’s beliefs (and other attitudes):
(10)
Belx, w = {w': w' is compatible with what x believes in w}
5 24.910, Spring 2009 (Stephenson)
[Note: We haven’t said much of anything about what it means to be “compatible with
what x believes” – more on this later.]
Using this function:
(11)
[[x believes φ]]w = 1 iff ∀w'∈Belx,w : [[φ]]w' = 1
Breaking it down:
(12)
[[believe]]w = λp . λxe ∀w'∈Belx,w : p(w') = 1
= λp . λxe ∀w': w' is compatible with what x believes in w: p(w') = 1
Of course, we can do something parallel for other attitudes:
(13)
[[know]]w = λp . λxe ∀w': w' is compatible with what x knows in w: p(w') = 1
(14)
[[suspect]]w
= λp . λxe ∀w': w' is compatible with what x suspects in w: p(w') = 1
(15)
[[imagine]]w
= λp . λxe ∀w': w' is compatible with what x imagines in w: p(w') = 1
(16)
[[want]]w = λp . λxe ∀w': w' is compatible with what x wants in w: p(w') = 1
Obviously we’ll have more to say than this…
3.1.
Accessibility Relations
Q. what does it mean to be “compatible with” a person’s knowledge, beliefs, etc.?
We won’t really answer this, but we can say a little bit more about knowledge, belief, etc.
[and thus about the lexical semantics of know, believe, etc.]
One thing that helps: hold the subject and type of attitude constant, and consider mental
states as relations between worlds:
Another notation:
(17)
w RxBel w'
= w' is compatible with x’s beliefs in w
When the subject and attitude type are understood, we might write wRw'.
Some terminology:
¾ R’s of this type are called accessibility relations.
¾ wRw' can be read as w' is accessible to w / (sometimes) w sees w'
We can say something more about propositional attitudes by talking about the properties
of these various accessibility relations
6 24.910, Spring 2009 (Stephenson)
3.1.1. Some properties of relations
Reflexivity
R is reflexive iff for all x in the domain of R, xRx
Transitivity
R is transitive iff whenever xRy and yRz, it’s also the case that xRz
Symmetry
R is symmetric iff whenever xRy, it’s also the case that yRx
3.1.2. Accessibility Relations for know
¾ Reflexive
(This reflects the intuition that you can only know things that are true)
¾ Transitive?
Maybe… This depends on whether we want to assume that if you know
something, then you know that you know it
¾ Symmetric?
Probably not … This depends on whether we want to assume that if something
happens to be true in the actual world, then you know that it’s compatible with
your knowledge
3.1.3. Accessibility Relations for believe
¾ NOT reflexive
(because you can believe things that are false)
¾ Transitive
If you believe something, then you believe that you believe it
¾ NOT Symmetric
Something can be the case in the actual world which you do not believe to be
compatible with your beliefs
3.1.4. Accessibility Relations for want
¾ NOT reflexive
(because you can want things to be the case that are not the case)
¾ NOT Transitive
(because presumably you can want something without wanting to want it)
¾ NOT Symmetric
7 24.910, Spring 2009 (Stephenson)
Something can be the case in the actual world which is not compatible with what
you want yourself to want
[Obviously there’s a lot more to say about these relations than these three properties, but
this gives us a framework]
3.2.
Selected Exercises
[Possibly work through in class depending on time]
Exercise 2.1 (p. 19)
Exercise 2.2 (p. 19)
Exercise 2.3 (p. 20)
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