In this video, we're going to start learning ...
about relations and how to do translations with them.
To begin with, let's talk about what relations ...
are. Another name for the kind of ...
relation we'll talk about is two-place propositional ...
functions. (There can be relations with ...
more than two places, but we won't cover them.)
You might remember that "one-place propositional ...
function" was a fancy name for predicates.
So a sentence like "Pooh is a bear,"
which we would translate as Bp
is built around this structure "[blank] is a bear,"
which becomes a sentence if you fill in that one blank with a name.
This is called a one-place propositional function because ...
there's one place where you put a name in order to make a sentence.
Now let's look at a sentence like "Joanie loves Chachi."
This is built around the structure "[blank] loves ...
[blank]," which has two places that ...
both need names in order to make a sentence.
So it's called a two-place propositional function,
or a relation, because it expresses a ...
relation between Joanie and Chachi.
We would represent this with a capital L followed by two ...
lower-case letters. So Lxy stands for "x loves y."
Something about relations is that when you have the two ...
letters after the relation letter, their order matters.
So if we have Lxy for the relation x loves y,
and constants representing Chachi and Joanie,
Joanie loves Chachi
would be represented by Ljc. Joanie is the one who loves,
so j for Joanie comes first after the L.
On the other hand, Chachi loves Joanie
would be represented by Lcj. Now Chachi is the one who ...
loves, so c for Chachi is first after ...
the L; and Joanie is the one who is ...
loved, so j for Joanie is second.
And these two sentences aren't the same!
Sadly enough, when one person loves ...
another, the other one doesn't always ...
love them back.
We can also have the names in a different order.
If we say Chachi is loved by Joanie,
that's a passive voice way of saying Joanie loves Chachi.
Which is Ljc again. So we have to untwist our ...
original sentence before we translate it.
When we're doing relational translations like this,
we have to keep the letters together.
Let's look at this vocabulary.
If we want to say "Amy doesn't love Ted,"
we treat "Amy loves Ted" as a unit,
and then we put a negation on it.
So we get ~Lat. The Lat stays together and ...
the ~ goes out in front.
If we have "Bobby and Debra love Amy,"
then the "and" has to join two complete ...
sentences when we translate it.
We can't just stick it between two letters.
So we spell it out more fully: Bobby loves Amy and Debra loves Amy.
Then we translate each half. Bobby loves Amy is Lba,
Debra loves Amy is Lda, and we join it with the & sign.
Now let's talk about how to combine relations with quantifiers.
If we're saying Everybody loves Raymond,
that means that for all x, x loves Raymond.
For all x is Ax, and x loves Raymond is Lxr,
so we get AxLxr.
On the other hand, Raymond loves everybody
is still a statement about for all x.
It says that for all x, Raymond loves x.
Raymond loves x is Lrx, so now we get AxLrx.
Notice that the quantifier goes out front!
We can't separate the L, the r, and the x.
Notice also that it helps to translate our sentence ...
halfway, using the variable x but ...
keeping the rest of the sentence in English.
This half-English translation will become especially ...
important as we do more and more complicated translations.
Now let's talk about how quantifiers interact with ...
negations in relational logic.
So for the sentence "Nobody loves Chris,"
first we translate it into half-English,
turning "nobody" into the quantifier and x and ...
leaving the rest in English.
There are two ways to translate "nobody"
sentences, either with a universal and ...
then a negation or a negation and then an existential.
Whichever way we represent "nobody,"
we follow it up with what the sentence is saying is true of ...
no x: that x loves Chris.
"x loves Chris" is Lxc,
so we have Ax~Lxc or ~ExLxc,
depending on which way we chose to translate "nobody."
Now let's look at "Chris doesn't love everybody."
Here, because the word "not"
is in front of the quantifier "everybody,"
we translate it as ~ first and then Ax,
all in front of what the rest of the sentence is saying is not ...
true for all x: Chris loves x.
"Chris loves x" is Lcx,
so we get ~AxLcx.
You can see in both these cases,
when we translate into half-English the x went where ...
the quantifier word was. In the first case,
x went in where "nobody" was.
In the second case, "x"
went in where "everybody" was,
and we'd already taken care of the "not"
by putting a ~ in front of the Ax,
so "Chris loves x" is what was left.
There are some other ways of phrasing sentences with ...
relations and quantifiers.
Let's look at "There is someone who Raymond loves."
"There is someone" is existential.
Putting the x where "Someone"
was, we get basically "x is who ...
Raymond loves."
which is to say, Raymond loves x.
So this turns out as ExLrx.
This sentence shows that, when we turn a sentence into ...
half-English, we can't always put the ...
variable exactly where the quantifier word was.
Sometimes we have to untangle the sentence a little ...
so the variable can go somewhere that makes sense.
Finally, there are some sentences in ...
English that are ambiguous; they can have more than one meaning.
Take "Everybody doesn't love Ted."
Does this mean "For all x, x doesn't love Ted";
Ted is universally not loved?
That would be Ax~Lxt.
Or does it mean that it isn't the case that everybody loves ...
Ted? Not everybody loves Ted?
That would be ~AxLxt.
These are different meanings that original sentence can ...
have, at least in some ways of ...
speaking English. So it's ambiguous.
You have to tell which meaning is meant from the context.
I will try to avoid asking you to translate sentences that are ...
ambiguous in this way.