In an earlier note, I said that it's possible to teach first-order
logic in one hour. That led to an offline note with a skeptical
question: "How?" Here's the answer:
Modern textbooks on logic are written by mathematicians for students
who plan to become mathematicians. As a math major at MIT, I didn't
see a problem with that.
When I was working at IBM, I also taught courses on logic, for which
I used typical textbooks. But when I discovered Peirce's existential
graphs (EGs), I decided to add a lecture on EGs by showing the students
how to map formulas in predicate calculus to and from EGs.
But the students pointed out my mistake. They said that the graphs
were much, much simpler. They asked why I didn't start with them.
And they were right. I later learned that Don Roberts, who taught
philosophy and logic at the University of Waterloo, Canada, had made
a bet with one of his colleagues:
1. They were teaching two different sections of the introductory
logic course. And Don suggested an experiment.
2. Both of them would use the same traditional textbook, and they
would give the students exactly the same final exam. That exam
would not mention EGs, and the only notation for logic would be
the one in that book for predicate calculus.
3. The other teacher would teach logic "by the book" and never say
anything about EGs. But Don would start with EGs and show the
students how to do the exercises with EGs for representing
sentences in English *and* for deriving the solutions by using
Peirce's rules of inference.
4. During the course, students could read the book, if they wished.
But Don used it only for the problems and exercises, which were
stated in English. Don never mentioned the formulas for predicate
calculus until the last two weeks of the course.
5. In those two weeks, Don showed how to map EGs to and from the
formulas. He also showed how Peirce's rules of inference and
EG proofs could be adapted to the rules for predicate calculus.
6. Finally, the students in both sections took exactly the same final
exam in which predicate calculus was the only notation permitted.
7. Since Don told this story and I'm repeating it, you can be sure
that Don's students had a higher average score. But the most
significant point is that Don's students did the best on proving
theorems in predicate calculus, even though nearly all their
practice was in theorem proving with Peirce's rules for EGs.
Moral of the story: the syntax of the notation is a minor issue.
The most important point is learning how to think about logic, how
to relate language to logic, and how to think about proofs.
Before reading the rest of this note, please read slides 1 to 18
in http://jfsowa.com/talks/egintro.pdf .
Slide 4 shows the complete notation for first-order logic. It shows
the notation without using any words to describe its parts or how
they fit together. The diagrams are all you need to know. Any words
that describe them are useful only to point out the relevant features.
After you learn the diagrams, you can forget the words.
Then look at the examples in slides 5 to 18. They show all the ways
of putting the parts together to make syntactically correct EGs.
That is teaching by *showing*, not *describing*.
In one hour, I *show* the students these and other examples. It's
better to use a blackboard or whiteboard, since it's easier and
faster to draw and modify graphs (by inserting or erasing subgraphs).
After the first half hour, I ask the students to tell me what to
draw at the blackboard. I write a sentence at the top. Then I
draw whatever they say, and let other students verify or correct
what I draw. They do the thinking, and I just draw what they say.
At the end of the hour, they know how to map English to and from
first-order logic, as represented by existential graphs.
In two days -- 3 hours in the morning and 3 hours in the afternoon,
90-minute sessions with coffee breaks and lunch in between --
I cover all the material in those slides. At the end of the
first day, they know how to translate English to and from FOL
in *both* EGs and predicate calculus and how to prove theorems
in both notations. The second day goes into the fine points
and research issues.
With EGs, the diagrams are everything, and the words that describe
them are irrelevant. There is very little to remember (or forget!)
The students don't have to memorize syntax rules, precedence rules,
or scope of quantifiers. The examples that show how to draw the
diagrams take care of all those details. They don't have to learn
anything about commas, quotation marks, parentheses, angle brackets,
or options about what goes into the angle brackets.
After the students learn to think about logic with the diagrams,
they learn why the linear forms require extra rules to represent
what is "obvious" in the diagrams.
In fact, that's how creative mathematicians think: Diagrams are
fundamental. After they discover a proof (mentally or at the
blackboard), mapping it to a linear form is routine housekeeping.
See "Peirce, Polya, and Euclid": http://jfsowa.com/talks/ppe.pdf
John
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