Chapter 6, Tutorial 1
Predicate Logic Introduction
In this chapter, we extend our formal language beyond sentence letters and connectives. And even beyond predicates and names. Just one small wrinkle, adding two further connectives, allows our simple logic to become quite sophisticated. These two connectives are called "quantifiers" because they describe the quantities ALL and SOME.
Still, the price to pay in difficulty is not too bad. We will introduce the language of quantifiers in this tutorial. Then give it clearer meaning and better connection to natural language in later tutorials.
Let's begin with a review of our sentence logic but with names and predicates only; no quantifiers.
PL without Quantifiers
Now, let's review our ideas about symbolization. We use upper case letters as symbols for both whole sentences and predicates. This sounds confusing. But we will always be very clear in context. So, while 'A' once symbolized the whole sentence "Chris got an A", we sometimes have used this for "____ got an A". Whatever we use a letter to symbolize must be clearly specified.
...keep this sound argument in mind. It will be symbolized in a moment.
Halpin (whoever this guy is) is either a professor or a faker.
But (I'm telling the truth, really!) he is no faker.
So, Halpin is a professor
Here is such a specification, an "interpretation" that tells us exactly what is meant by each letter.
F_ : __ is a faker; P_ : __ is a professor
h: Halpin
Now, let's remember a bit of
"syntax" for our language with names and predicates.
To write
Halpin is a faker
we write:
Fh
This is a bit backward, but we're now used to it.
To write...
If Halpin is no faker, then he's a professor
you could write the hybrid form:
If ~Fh then Ph
and turn this into a correct answer,
remembering that a horseshoe goes in place of the word "then":
~Fh>Ph
again with names and predicates, we take simple, "atomic" sentences like
Fh
and
Ph
Then we put them together to form the likes of
~Fh
("Halpin is not a faker")
or
Fh v Ph
("Halpin is either a faker or a professor"; note that the outside parentheses have been dropped)
or even
~(~Fh>Ph)
This last is built up starting as we already have, constructing '~Fh>Ph', or really '(~Fh>Ph)' when we make the parentheses visible, then finally adding the tilde. So, tilde. is the main connective.
Let's check our sentence construction.
Each of the following expressions breaks the rules of correct grammar. Change the expression slightly (e.g., reorder, add a connective or parentheses) to make it fit the rules for sentence construction. Then press TAB. If an exclamation point is added, you're new expression is grammatically correct! If you get "???" added, then go back and do it again!
A '!' means your entry is correct. '???' means try again.
Trouble getting started?
Connective Tool:
Try to build sentences. Check them by entering in the box below. As you build, you should see what the main connective is: it's always the last one you put in. If you've entered a non-sentence, you'll get question marks and no main connective.
Start by "fixing" this incorrect sentence (hint):
(Example: Enter '~Fav(Kb>Rc)' and see what you get. Then enter some arbitrary sentences. Perhaps enter a couple of short sentences 'Fa=~Pa' and '~Fa&~Pa' maybe, then put them together with and '&'. This last '&' will be the main connective.)
Finally, we can symbolize the sound argument that began this page:
Halpin (whoever this guy is) is either a professor or a faker.
But (I'm telling the truth, really!) he is no faker.
So, Halpin is a professor
Ph v Fh
~Fh
Ph
Now, i t should be clear that this is a valid argument. Yes? (We can see it's valid by just one of our SD rules from 5.1. Which rule?