Transcript PowerPoint

CAS LX 502
Semantics
2a. Reference, Compositionality,
Logic
2.1-2.3
Meaning as truth conditions
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We know the meaning of p if we know
the conditions under which p is true.
conditions under which p is true = which are
the possible worlds in which p holds
 possible worlds = ways things might be
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The meaning of p: A specification of
possible worlds.
Recall the trick we can do
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Homer stands.
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True iff Homer stands.
How do we arrive at truth
conditions?
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Homer stands. Marge stands.
True iff Homer stands.
 True iff Marge stands.
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Two parts of Homer stands
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Homer stands. Marge stands.
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Homer stands. Bart’s father stands.
The Homer part and the stands
part
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Homer stands.
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We use the name Homer to
refer to that guy.
Homer stands is true when
that guy has the property
that he stands (being
upright on his feet).
Other things/people can
stand, and we feel that
standing should be
basically the same
regardless of who we say
holds that property.
Unsaturated propositions
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A proposition with a “hole” in it is
called an unsaturated proposition.
It’s something that, once we fill in the
hole, will be true or false (of a given
possible world).
Portner draws
these like so:
 true
 false
Unsaturated propositions
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A proposition with a “hole” in it is
called an unsaturated proposition.
Thus:
 true
 false
Unsaturated propositions
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Although perhaps we could come up
with a better picture, the idea is that:
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Homer stands: the possible worlds in which
Homer stands
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Stands: (unsaturated) Given a referent x, the
possible worlds in which x stands.
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Homer:
 true
 false
Meaning is compositional
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It seems that there is something common
across all the propositions we might express
using Homer.
And something in common across all the
propositions we might express using stands.
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Homer stands. Homer snores. Marge stands.
Given that each word seems to have a
consistent contribution to the meaning, (to
some extent) regardless of the sentence in
which it appears…
Meaning is compositional
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We hypothesize this:
Meaning is compositional
The meaning of a sentence is formed
from the meanings of its parts, and the
way in which they are arranged.
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Homer strangles Bart. Bart strangles Homer.
Meaning is compositional
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And it really has to be compositional. We
after all know what the world has to look like
in order for a sentence to be true, even if we
haven’t heard the sentence before—and
have to compute the meaning.
So the project here is really:
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Understand the pieces of meaning
Understand how they combine to form larger units of
meaning
Where are we so far?
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In the set of things that we’ve been
considering as part of meaning:
Possible world: A state of affairs.
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One special possible world is the actual world,
w0.
Individuals: Referents, like Homer.
Propositions: Sets of possible worlds
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In which the proposition is “true.”
Unsaturated propositions
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We’ve added the idea of an
unsaturated proposition, which would
be a proposition, but for the lack of an
individual.
Given an individual, it would be a set
of possible worlds.
It’s “waiting for an individual.”
It, in a sense, turns individuals into sets
of possible worlds.
Limiting our attention to wk
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For simplicity in presentation, let’s stop
thinking about sets of possible worlds briefly,
and limit out focus to specific possible
worlds.
One good candidate would be w0, but it
doesn’t have to be that one necessarily.
If we do this, we can consider a proposition
to be either true or false.
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Though in the back of our minds, we know that this is
in a particular possible world.
Semantic type
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The entire semantics that we are creating
here depends on two types of things,
individuals and truth values.
We can label individuals as being of type
“e” (traditional, think “entity”), and truth
values as being of type “t”.
In these terms, names like Homer are of type
<e>, and sentences like Homer stands are of
type <t>.
A formal system
of semantic types
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<e> is a basic type.
<t> is a basic type.
If a and b are types, <a,b> is a type.
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<a,b> is a function that takes something of
type a and returns something of type b.
<e,t> is a type. <<e,t>,<e,<e,t>>> is a
type.
<e,t,e> is not, nor is <<e,t>>.
Functions
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A function transforms one thing into another.
We can define the squaring function as a
function that takes a number and gives
back that number multiplied with itself
Square(n) = n  n
This is a function from numbers to numbers. It
takes a number, it gives back a number.
Functions
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A function doesn’t need to give back the
same kind of thing it gets. Usually, the thing it
gives back depends on the thing it gets, but
it doesn’t need to be of the same type.
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Change-machine($n-bill) = 4  n quarters.
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This is a function from bills to quarters.
<e,t> functions
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An intransitive verb like stands can be
viewed as a function from individuals to truth
values. Given an individual x, it will return
true if x is boring, or false if x is not boring.
Stands(x) = true if x stands; false otherwise.
This is a function from individuals (type <e>)
to truth values (type <t>). That is, it has type
<e,t>.
Enter the l
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There is a way to write functions that we will get some
experience with as the semester progresses, using
lambda notation. Here’s a first introduction
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The structure of a function written in lambda notation is:
l argument [ return value ]
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So, for the meaning of stands, we might write this:
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l x [ x stands (in wk) ]
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Type <e,t>
l argument [ return value ]
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Change-machine($n-bill) = 4  n quarters.
Change-machine = l $n-bill [ 4  n quarters ]
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Square = l n [ n  n ]
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Not very complicated, just a short way to
write “that function f such that, given
argument, returns return value.”
value l argument [ return value
]
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Square = l n [ n  n ]
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Square(3) = 3  3 = 9
Square(4) = 4  4 = 16.
To evaluate a function, we take the value
and substitute it in for the argument within
the return value. If we give it a 3, and the
argument is n, then we replace all of the ns
with 3s and evaluate the return value.
value l argument [ return value
]
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Strictly speaking, there’s an intermediate step,
which is written like so:
Square = l n [ n  n ]
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Square(3) = 3 l n [ n  n ] = 3  3 = 9
Square(4) = 4 l n [ n  n ] = 4  4 = 16.
What value l argument [ return value ] means is:
Replace every instance of argument within return
value with value, then evaluate return value.
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This operation goes by the name lambda conversion.
value l argument [ return value
]
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One last piece of terminology: Instances of
argument within return value are said to be
variables that bound by the lambda operator.
Triple = l n [ 3  n ]
Lambda operator
Bound variable
Desiderata for a theory of
meaning
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A is synonymous with B
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A entails B
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B is part of the assumed background against which A is said.
A is a tautology
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A is inconsistent with B
A presupposes B
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If A holds then B automatically holds
A contradicts B
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A has the same meaning as B
A is automatically true, regardless of the facts
A is a contradiction
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A is automatically false, regardless of the facts
Intuitions about logic
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If it’s Thursday, ER will be on at 10.
It’s Thursday.
ER will be on at 10.
Modus Ponens
Logic is essentially the study of valid argumentation
and inferences.
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If the premises are true, the conclusion will be true.
Truth out there in the world
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A statement like It’s Thursday is either true
(corresponding to the facts of the world) or it is false
(not corresponding to the facts of the world).
Same for the statement ER is on at 10.
It turns out that modus ponens is a valid form of
argument, no matter what statements we use. Let’s
just say we have a statement—we’ll call it p. The
statement (proposition) p can be either true or
false. And another one, we’ll call it q.
Modus ponens
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So, whatever p and q are:
If p then q.
p.
q.
Granting the premises If p then q and
p, we can conclude q.
An invalid argument
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Incidentally, some things are not valid
arguments. Modus ponens and modus
tollens are. This is not:
If it is Thursday, then ER is on at 10.
It is not Thursday
*ER is not on at 10.
Other forms of valid argument
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If it is Thursday, then ER is on.
If ER is on, Pat will watch TV.
If it is Thursday, the Pat will watch TV.
Hypothetical syllogism
If p then q.
If q then r.
If p then r.
Other forms of valid argument
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Pat is watching TV or Pat is asleep.
Pat is not asleep.
Pat is watching TV.
Disjunctive syllogism
p or q.
q.
p.
Logical syntax
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A proposition, say p, has a truth value. In light of the
facts of the world, it is either true or false. The
conditions under which p is true is are called its truth
conditions.
We can also create complex expressions by
combining propositions. For example, q. That’s true
whenever q is false.  is the negation operator
(“not”).
Logical connectives
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We can combine propositions with
connectives like and, or. In logical
notation, “p and q” is written with the
logical connective  (“and”): p  q; “p
or q” is written with  (“or”): p  q.
p  q is true whenever p is true and q is
true. Whenever either p or q is false, p
 q is false.
Truth tables
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p
T
F
We can show the effect of logical operators
and connectives in truth tables.
p
F
T
p
q
pq
p
q
pq
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
Or v.  v. e
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The meaning we give to or in English (or any other
natural language) is not quite the same as the
meaning that of the logical connective .
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We’re going to South Carolina or Oklahoma.
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You will pay the fine or you will go to jail.
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Seems odd to say this if we’re going to both South Carolina and
Oklahoma.
Seems a bit unfair if you get put in jail even after paying the fine.
We will preboard anyone who has small children or needs special
assistance.
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Doesn’t seem to exclude people who both need special assistance
and have small children.
Or v.  v. e
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There are two interpretations of or, differing in their
interpretation with respect to what happens if both
connected propositions are true.
Exclusive or (e) is “either…or…but not both.”
Inclusive or (disjunction; ) is “either…or…or both.”
p
q
pq
p
q
peq
T
T
F
F
T
F
T
F
T
T
T
F
T
T
F
F
T
F
T
F
F
T
T
F
Material implication
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The logic of if…then statements is
covered by the connective .
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If it rains, you’ll get wet.
(pq, where p=it rains, q=you’ll get wet)
p
q
pq
T
T
F
F
T
F
T
F
T
F
T
T
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What is the truth value of
If it rains, you’ll get wet?
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Well, it’s true if it rains and
you get wet, it’s false if it
rains and you don’t get wet.
But what if it doesn’t rain?
Truth and the world
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In most cases, the truth or falsity of a statement has to do
with the facts of the world. We cannot know without
checking. It is contingent on the facts of the world
(synthetic).
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Sometimes, though, the very form of the statement
guarantees that it is true no matter what the world is like
(analytic).
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John Wilkes Booth acted alone.
Either John Wilkes Booth acted alone or he didn’t.
John Wilkes Booth acted alone and he didn’t.
The first is necessarily true, a tautology, the second is
necessarily false, a contradiction.
Limits of propositional logic
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There are some kinds of logical intuitions that are
not captured by propositional logic. For example:
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All men are mortal.
Socrates is a man.
Socrates is mortal.
Try as we might, we can’t prove this logically with
only p, q, and r to work with, but it nevertheless
seems to have the same deductive quality as other
syllogisms (like modus ponens).
Predicate logic
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Propositional logic is about predicting the truth and
falsity of propositions when combined with one
another and subjected to operators like negation.
What we need for the All men are mortal case is
something like:
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For any individual x, if x is a man, then x is mortal.
That is, we need to be able to look inside the
sentence, to refer to predicates (properties) not just
to truth and falsities of entire propositions.
Predicate logic
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Predicate logic is an extension of
propositional logic that allows us to do
this.
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Mortal(Socrates)
True if the predicate Mortal holds of the
individual Socrates.
Individuals have properties, and just
like we labeled our propositions p, q, r,
we can label properties abstractly like
A, B, C.
Predicate logic
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Thus:
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Man(x)  Mortal(x)
Man(Socrates)
Mortal(Socrates)
A(x)  B(x)
A(S)
B(S)
Note: This is not exactly in the right form yet, but it’s close. The
right form of the first premise is actually
x[Man(x)Mortal(x)]. More on that later.
Entailment
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From the standpoint of linguistic knowledge of
meaning (intuition), there are sentences that stand
in a implicational relation, where the truth of the first
guarantees the truth of the second.
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The anarchist assassinated the emperor.
The emperor died.
It is part of the meaning of assassinate that the
unlucky recipient dies. So, the first sentence entails
the second.
Entailment
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This is the same relationship as pq from before. If we
know p is true, we know q is true—and if we know q is
false, we know p is false.
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The anarchist assassinated the emperor.
The emperor died.
At the same time, knowing q is true doesn’t tell us one
way or the other about whether p is true—and knowing
p is false doesn’t tell us one way or the other about
whether q is false.
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We take entailment relations to be those that specifically arise from
linguistic structure (synonymy, hyponymy, etc.).
Synonymy
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For a paraphrase to be a good one, and accurate
rendering of the meaning, the sentence should
entail its paraphrase and the paraphrase should
entail the sentence.
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The dog ate my homework.
My homework was eaten by the dog.
This kind of mutual entailment (like  from earlier) is
a requirement for synonymy.
Truth and meaning
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A young boy named Rickie burned down the library at
Alexandria in 639 AD by accidentally failing to
extinguish his cigarette properly.
True? Well, we’ll pretty much never know
(though perhaps we can rate its likelihood).
But knowing whether it is true or not is not a
prerequisite for knowing its meaning.
Rather, what’s important is that we know its
truth conditions—we know what the world
must be like if it is true.
Truth and meaning
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If we know what a sentence means
we know (at least) the conditions
under which it is true.
On that assumption, we proceed in
our quest to understand meaning in
terms of truth conditions.
Understanding how the words and
structures combine to predict the truth
conditions of sentences.
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