In the last video we talked about relational logic and did ...
some translations with quantifiers.
In this video we'll see how relational logic can give us ...
translations with more than one quantifier.
Let's start by reviewing translations with single quantifiers.
Remember, a relation is a capital letter ...
followed by two lowercase letters;
here Lxy for x loves y. And we'll use r as a constant ...
to stand for Raymond.
To translate Everybody loves Raymond
we first put the sentence halfway into notation,
as Ax x loves Raymond. The x went where ...
"Everybody" was.
Since x loves Raymond is Lxr, we get AxLxr.
On the other hand, if we have Raymond loves ...
everybody, we put that halfway into notation
as Ax Raymond loves x; now the x is in the second ...
place, where the "everybody" was
and that's AxLrx.
So the variable can wind up in either place after the relation letter.
Now let's see how we can wind up with double quantifiers.
If we have "somebody loves somebody,"
first we put that partway into notation
and we do that step by step, using variables and leaving ...
the rest in English until the last step.
So first we turn the first "somebody"
into Ex, and put the x in to make it "x ...
loves somebody."
Now we have to figure out what "x loves somebody"
is. That has a quantifier that we ...
need to translate.
It'll be an existential, but it can't be Ex since we're ...
already using x. So we start with Ey.
THen we put y in where the "somebody"
was, and we get Ey x loves y.
So "x loves somebody" is EyLxy.
We put that after the Ex, and the final answer is ExEyLxy.
Now, the order in which quantifiers ...
appear can make a difference.
Let's say we have AxEyLxy.
That means that for every x, there is a y that that x loves.
Pick an x, and you can always find a y ...
for it to love.
But the x's could all love different y's.
We could represent this with this graph,
where the dots are people and the arrows are love ...
relationships. Notice that every dot has an ...
arrow going somewhere.
What if we switch the quantifiers so the Ey is first?
We have EyAxLxy.
This means that there is a y that every x loves.
Here it's the same y, beloved universally by every x.
That could be represented by this diagram,
where every dot has an arrow going to the upper right ...
corner. That's the one that everyone ...
loves. (Even that person loves ...
themself.) So these two sentences don't mean the ...
same thing, even though the only ...
difference is that we switched the order of the quantifiers.
So we need some way of telling which order the ...
quantifiers go in.
One thing is if we see the words "There is"
or "The same," that will get translated as an ...
existential, and it will go first no matter ...
where it was in the sentence.
So if we have "There is someone who loves ...
everyone," the "There is"
is an existential and we translate it first.
So we get Ex, x loves everyone.
And now we can translate "everyone."
That becomes Ay, and we put y in for ...
"everyone," so it's Ay x loves y.
That's Lxy, so we wind up with ExAyLxy.
Now if we have "everyone loves the same person,"
"the same" will be an existential,
and we have to translate it before "everyone,"
even though it's second in the sentence.
So we make it Ex, and we put the x where "the ...
same person" was,
so that part is "Everyone loves x."
Now we have to translate "everyone."
That'll be Ay, and we put y in where ...
"everyone" was,
so that's y loves x,
which becomes Lyx, so the final answer is ExAyLyx.
There are some more cues as to what order the quantifiers go in.
If you see "some or other," that's an existential that goes ...
last. The "or"
here isn't a wedge, it's just something that tells ...
you where to put the quantifier.
So if it's "someone or other loves everyone,"
we have to do the "someone" last,
and the "everyone" first.
The "everyone" is Ax,
and we put the x where "everyone"
was, so it's Ax someone loves x.
Now we can translate "someone"
as Ey, and put the y where ...
"someone" was,
so it's Ax Ey y loves x.
y loves x is Lyx--notice that y comes first!--so we get AxEyLyx.
If we have "everyone loves someone or other,"
we still translate "someone or other"
last, so we do "everyone"
first again, as Ax.
So we get Ax and put x where "everyone"
was, so it's Ax x loves someone.
Now we turn "someone" into Ey and put y where ...
"someone" was,
so it's Ax Ey x loves y.
Which becomes AxEyLxy.
Sometimes we have sentences with no explicit cue ...
that tells you which quantifier goes first.
The general idea is that if there isn't one of those cues,
you translate the quantifiers in the order they appear in the sentence.
So if we have "everybody loves somebody,"
we translate the "everybody" first.
That becomes Ax x loves somebody,
so now we translate the "somebody."
which becomes Ey x loves y.
And that's Lxy, so we get AxEyLxy.
But some of these sentences can be ambiguous.
Take "somebody loves everyone."
Does that mean "there is some one person who loves ...
everyone"? A universal source of love?
Or does it mean that somebody or other loves ...
everyone? So everyone has someone ...
who loves them, but they might all be different people?
We don't know. This sentence is ambiguous.
We need to rely on context to figure out which of these two ...
translations is accurate. So I'll try not to assign you ...
sentences that are ambiguous in this way.