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PROFESSOR: The
logic of predicates

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is a basic concept in
mathematical language

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as well as being a
topic on its own.

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In particular, I'm going to talk
now about the idea of the two

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so-called quantifiers.

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For all-- that's
the upside-down A.

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And exists-- that's
the backward E.

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So what's a predicate?

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Basically, a predicate
is a proposition,

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except it's got variables in it.

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Here's an example.

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P of x, y is the predicate
that depends on x and y.

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And let's say it's defined
to be x plus 2 equals y.

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Now, in order to figure
out whether or not

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a predicate is
true, I need to know

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the values of the variables--
in this case, x and y.

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So if I tell you that x is 1
and that y is 3, guess what?

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P of 1 and 3-- P of x and
y when x is 1 and y is 3--

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is true, because in fact,
1 plus 2 is equal to 3.

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If I tell you that x is 1 and
y is 4, then since 1 plus 2

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is not equal to 4, P
of 1 and 4 is false.

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On the other hand, since
P of 1 and 4 is false,

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that makes not P
of 1 and 1 true.

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That's the easy part.

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Now the quantifiers are read
as "for all" and "exists."

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But they control a variable.

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So I write "for all
x," it means-- sorry.

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I write upside-down Ax and
I read it as "for all x."

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I'm so used to reading it
that I forgot that I needed

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to read symbols literally.

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And backwards Ey is read
as "there exists some y."

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So let's see how
that would work.

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The upside-down A, for
all, acts like an AND.

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To understand what it means,
let's look at this example.

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Let's let a variable s range
over the staff members in 6.042

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at this term, of whom there are
about 30 counting the graders.

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Let's define a predicate
that depends on the variable

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s called P of s that says
that s is pumped about 6.2042.

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They're enthusiastic
about being on the staff.

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OK.

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If I tell you for
all s, P of s, that's

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exactly the same as saying
that P of Drew is true,

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and P of Peter is true,
and P of Keshav is true,

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and a whole bunch more
ANDs down to P of Michaela.

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They'll be 29 ANDs if
there are 30 staff members.

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Similarly, the backwards E--
there exists-- acts like an OR.

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If I tell you that t is now
ranging over the 6.042 staff

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just as s was, and I write
B of t, the predicate of t

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means the staff member
t took 6.042 before,

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then if I tell you that there
exists a t B of t, what I'm

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telling you is that B of Drew--
either Drew took it before,

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or Peter took it before,
or Deshav took it before,

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or Michaela took it before.

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This statement is
true, because in fact,

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several of the staff members
took 6.042 from me before.

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And I'd like to think that the
previous one-- that everybody's

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pumped up on the staff--
is true, although I

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can't guarantee that.

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So let's do a little practice
with existential quantifiers.

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So let's agree that
the variables x and y

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will range, for this example,
over the non-negative integers.

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And let's consider the
following predicates

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about y that says that there's
some x that's less than y.

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Q of y is there exists an
x that x is less than y.

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Let's see what happens.

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Well, what about Q of 3?

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Q of 3 is saying that
there is an x such

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that x is less than 3.

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Well, an example
of such an x is 1.

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So that means there is
an x that's less than 3,

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because 1 is.

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And so that makes this Q true.

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All right, what about Q of 1?

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Well, again, there's an x that's
a non-negative integer, namely

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0, that's less than 1.

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And therefore, Q of 1 is true.

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On the other hand,
Q of 0 is false

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because there is no non-negative
integer that's less than 0.

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And so there's no
value that you can

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assign to x that's a
non-negative integer that

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will make it less than 0.

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OK, that one's not
so bad, I hope.

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Let's look at the same example
with the universal quantifier.

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This time, we'll say that R
of y means that for every x,

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x is less than y.

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Well, R of 1 is false.

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And the reason is that
5 is a counterexample.

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5 is not less than 1.

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And so it's not true that
every x is less than 1.

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R of 8 is false because
12 is not less than 8.

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And therefore, not
every x is less than 8.

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R of a googol, 10 to
the 100th, is false.

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Because if you
let x be a googol,

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it's not less than a googol.

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And so it's an example of the
fact that this doesn't hold

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for all x's.

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That part's obvious.

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OK.

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The thing that tends to
confuse people in the beginning

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is what happens when you
start mixing up quantifiers.

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So let's look at an
intuitive example

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first that might
help you remember

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what happens when you have
an A followed by an E.

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And then we'll look at
an E followed by an A.

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So suppose I look
at this statement.

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This time, I'm going
to tell you that v

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ranges over the possible
computer viruses, not

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biological viruses.

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d ranges over
antivirus software,

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defenses against viruses.

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And I want to look
at the predicate that

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says for every virus,
there is a defense such

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that d protects against
v. This defense is good

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against that virus.

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All right, so each virus,
I have a defense for.

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So an example would be-- these
are, by the way, dated viruses,

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but that's when the
slides were made.

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So against the Mydoom
virus, you could use

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Defender, Microsoft Defender.

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Against the ILOVEYOU virus,
you could use Norton.

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Against the Bablas virus,
you could use ZoneAlarm.

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Well, is that what we want?

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It's expensive.

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It means that for
every different virus,

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I need a different defense.

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I have to spend a
fortune on software.

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This is not what we want.

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So that's when for every
virus, there's a defense,

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but the quantifiers
are in the wrong order.

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Let's reverse them.

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Suppose I tell you that
there's one defense that's

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good for all viruses.

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There is a defense such
that for every virus,

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d protects against v. For
example, if d is MITviruscan,

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then it would be
wonderful if it was true

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that d protects
against all viruses,

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there's one defense good
against every attack.

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That's what we want
because it's a lot cheaper.

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All right, let's start looking
at a concrete mathematical

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example.

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And I hope that that
prelude with the viruses

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will help you decipher how
the quantifiers behave.

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So let's look at
this predicate now.

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Sorry, this is a proposition.

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It really doesn't depend
on the values of x and y.

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It's asking about all possible
x's and all possible y's,

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whether there is one.

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In order to figure out whether
a proposition like G is true,

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actually, I do need to know
what x and y are ranging over.

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Because as you'll see, whether
or not G comes out to be true

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will depend on that.

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So I'm going to look at
the domain of discourse

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that x and y range over.

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And we'll suppose
that the domain

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is the non-negative integers.

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So now, what G is saying
is that if you give me

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a non-negative integer x, there
is a y that's greater than x.

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In other words, I can find
another non-negative integer

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that's bigger than x.

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Well, that's certainly true.

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There's a simple
recipe for finding y.

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Give me an x.

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Let's choose y to be
x plus 1, or x plus 2.

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But I don't necessarily
need a recipe

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as long as somewhere
out there, there's

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a y that's bigger than x.

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This is true.

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So G is true when the
domain of discourse

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is the non-negative integers.

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On the other hand, when I
change the domain of discourse,

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different things can happen.

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So let's look at the negative
integers, the integers less

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than 0, and ask, is it
true that for every x,

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there's a y that's
greater than x?

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Well, for a lot
of them, there is.

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If x is minus 3, then
minus 2 is bigger than x.

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If x is minus 2, then
minus 1 is bigger than x.

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But then I'm in trouble.

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If x is minus 1, there's
no negative integer

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that's bigger than x.

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And so G is false when
the domain of discourse

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is the negative integers.

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Well, let's shift again,
the point being here

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both we're looking at
alternate quantifiers,

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and we're understanding
that the truth

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of a proposition with
quantifies depends crucially

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on the domain of discourse.

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If we let the
domain of discourse

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be the negative reals,
then what this is saying

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is that for every negative real,
there's a bigger negative real.

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And that, of course, is true.

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Because if you give me a
negative real r, then r

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over 2, because it's negative,
is actually bigger than r.

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And it will not be positive
if r isn't positive.

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So sure enough, G in
this case is true.

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All right, let's
reverse the qualifiers

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and see what happens.

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It's worth thinking about.

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So let's call H the assertion
that for every y-- sorry,

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that there exists a y such that
for every x, x is less than y.

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So intuitively,
what this is saying

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is there's a biggest element.

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Y is bigger than everything.

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Well, if the domain is
the non-negative integers,

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then H is false because
there's no biggest

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non-negative integer.

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If it's the negative
integers, it's

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false because there's no
biggest negative integer.

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If it's the negative
reals, it's false

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because there's no
biggest negative real.

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But the truth is
that this thing is

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going to be false in any
domain of discourse in which

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less than is
behaving as it should

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because all any y is not going
to be bigger than itself.

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So you can't possibly
find a biggest y.

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It would have to be
bigger than itself.

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This is going to be false in
all of the sensible domains

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where less than behaves
as we would expect.

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So let's make this
slightly more interesting

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and make that less
than or equal to.

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So now, it's actually
possible that there

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could be a biggest
element that is

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greater than or equal to
everything including itself.

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And if you look
at these domains,

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well, there this isn't any
greatest non-negative integer,

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because for any x,
plus 1 is bigger.

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There isn't any biggest
negative real-- same reasoning.

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For any r, r/2 is
going to be bigger.

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But for the negative
integers, there

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is a biggest y, namely minus 1.

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It's greater than or equal to
every other negative integer.