Traditonally textbooks present material on what is called
predicate and quantifier logic well after a student has been working
with derivations involving propositional logic--the P's and Q's you
have worked with in the last two sections. When Russell and
Whitehead first presented their work on symbolic logic a century ago
they started right off with symbolization both for full statements (the
P's and Q's) and for predicate statements, the way in which we talk
about individuals having certain characteristics (predicates). I
think they had the right approach, and that is why I will ask you to
get started early on this other type of symbolization.
First off, why do we need it? Well, think about those syllogisms
we saw a while back, especially my example of salt being black and
everything black being sweet so that we would then say salt is
sweet. The type of symbolization we have used so far only allows
us to connect complete statements so we could not then show why any
syllogism--which relies on connections among the terms used--would
work. What we are going to learn in this and the following
section is how in fact we can handle any of the statements about
individual characteristics.
Again we have atomic statements, the basic building blocks. What
we do first off is state something about an individual. For
instance, we want to note that Alice is a student and maybe also that
she is ambitious. We'll use the first letter of her name in lower
case and let what we are saying about her ("predicating" is the more
formal term) be expressed through a capital letter: Sa would express "Alice is a
student." Aa would
express "Alice is ambitious."
yes, we are using
small letters for the names and capitals for the characteristics--not
what you would have expected, right?
If we
want to talk about someone else we'll use a different letter for
the name. St would express "Ted is a
student" while At would
express "Ted is ambitious."
If we want to make the point that both Alice and Ted are students but
Alice is ambitious while Ted is not we would have (Sa & St) & (Aa & ~At).
Note how we need to use parentheses in order to present ideas as
packages.
There are some things you need to remember. One is that an
expression such
as Sa or At cannot be broken apart.
We can deny that someone does have a particular characteristic, as we
do with ~At, but we never
have something like ~t
(there is no such thing as a non-Ted).
These are all examples of one-place
(monadic) predicates. Later we will talk about something
more elaborate that lets us express relationships, as when we might say
"Alice is Ted's friend."
Now for
a practice exercise: On a
piece
of paper write how you would symbolize the following sentences.
When
you think you have your answers click on to
compare
with what I would have written.
Al is
young but Claudio is not.
Barbara is not
a student but Donna is.
Although both
Al and Claudio are students,
they are not both young.
Al and Claudio
are not both young but they
both work hard.
If Barbara is
not ambitious then she will
not work hard.
If either
Barbara or Donna is ambitious then
they will both work hard.
Claudio will
work hard unless he is not
ambitious,
but Donna will not work hard unless she is ambitious.
If Al and
Claudio are not both ambitious then
one of them will not be working hard.
Keep
in mind the easiest mistake to make at the beginning is to think the
capital letter stands for someone's name and should come first in a
term.
yes, we are using
small letters for the names and capitals for the characteristics--not
what you would have expected, right?