In this video I'll talk about doing categorical translations ...
in relational logic.
We can combine categorical statements,
like statements about "some cat,"
with relations, like "loves."
So we get a statement like "Some cat loves Johnny."
But with relational logic, the category we're talking ...
about doesn't have to be at the beginning of the sentence.
It can be at the end, as in "Johnny loves all cats."
We can even relate two categories,
as in "Every person was born in some location";
"every person" and "some location"
are both categories that will wind up getting translated with ...
separate quantifiers.
All these translations will be built on the old categorical ...
forms that we used to translate sentences with "all,"
"some," "no,"
and so on in predicate logic,
things like the universal combined with the arrow for ...
"all," and the existential combined ...
with the and for "some."
But when we're doing relational categoricals,
we'll have to break these sentences down carefully step by step.
The categorical translations we'll do here will depend on ...
the four main frameworks that we used for doing categoricals ...
in predicate logic,
for all, some,
no, and not all or some are not.
They each have a quantifier and a symbol to join the ...
subject and predicate, and some have a negation.
(You should review these if you need to!)
But when we're doing relational categoricals,
the category may not be the subject of the sentence;
it may not be first! Still,
the category will go out in front of the form.
Here's an example with one categorical phrase.
Using this vocabulary
we translate "Some cat loves Johnny."
Since this is a categorical that begins with "Some,"
"some cat," we use the existential ...
categorical form
which starts with an existential quantifier and has an &
between the category and the rest of the sentence.
We translate the two parts of the sentence into half-English,
English but using the variable x.
So the category is "x is a cat," and what the rest of the ...
sentence is saying about that x is "x loves Johnny."
"x is a cat" translates to Cx
"x loves Johnny translates to Lxj
and we plug those back in to the existential categorical form.
When we're doing translations with a single category,
it's important to put the category at the beginning of ...
the translation, no matter where it appears in ...
the original sentence.
Let's look at "Johnny loves all cats."
Notice that the category, "all cats,"
is at the end of the sentence.
We use the "All" form for a categorical,
with a universal quantifier and an arrow.
But the category goes before the arrow.
The category will always go on the left,
even if it's at the end of the original sentence.
So we get "x is a cat" before the arrow,
and after the arrow we put the thing the rest of the sentence ...
says about x: That Johnny loves x.
Notice where the x goes there.
We put it into the sentence where the category "all cats" had been.
Now we can translate each piece.
"x is a cat" is Cx.
"Johnny loves x" is Ljx.
So we have universal x Cx arrow Ljx.
When we're doing relational logic we can also wind up ...
stacking categoricals one on top of each other.
So in "Some people love every cat,"
we have category terms in two different places in the ...
sentence, one on each side of the relation.
To do a sentence like this, we have to go step by step.
First we do the first categorical expression,
leaving the second half of the sentence in English.
Then we translate the second half of the sentence and plug it back in.
So our first categorical expression is "some people."
We use the "some" categorical form,
with an existential and an &. The category will go before ...
the &, and the rest of the sentence ...
will go after it.
The category is "x is a person."
To get the rest of the sentence,
we put the x in for "every person"
and leave everything else in English,
so we get "x loves every cat."
Now "x loves every cat" is itself a categorical,
so we have to translate that.
We use the "all" form.
Since we've already used x, we use y as the variable this ...
time. So that's a universal quantifier ...
at the beginning. Then we have the category ...
*before* the arrow,
y is a cat. After the arrow we have what ...
the rest of the sentence is saying about it, x loves y.
The category "y is a cat" has to come first,
before the arrow. And we put the y in the rest of ...
the sentence where "every cat" was.
Translating those two parts, we get that "x loves every cat"
is Ay(Cy -> Lxy).
Then we plug that back in with the existential and Px to get ...
our final answer.
Let's look at another example of doubled up categoricals.
Using this vocabulary, with the relation "born in"
we translate "Every person was born in some location."
Again we have two categoricals,
and we start with the first one, "Every person."
Since this is an "every" categorical,
we use a universal quantifier and an arrow.
The category, "x is a person,"
goes before the arrow. After the arrow is what the ...
rest of the sentence is saying about x:
x was born in some location.
Now we have to translate "x was born in some location,"
which has the categorical "some location."
We use the "some" form,
with an existential and an &. The existential is Ey because ...
we already used x. Before the &
goes the category we're working with,
y is a location. After the &
goes what the rest of the sentence is saying about y,
x was born in y.
Now we translate those elements into notation:
"y is a location" is Ly,
"x was born in y" is Bxy.
Plugging it back in, we get Ax(Px -> Ey(Ly &
Bxy)). Make sure that you have all ...
the parentheses. And it's a good idea to check ...
that the x's and y's line up in a way that makes sense.
x is the person, y is the location,
so it makes sense that we're saying x was born in y;
it'd be bad if we had it the other way around!
[no narration, just cat picture]