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Math Circle: Logic Session #3: Noise, Normals, and Negators

Noise, Normals, and Negators

(January 29, 2013)  The kids bounded in, filled with curiosity about whether the new puppets, Wags and Rooney, were knights or liars.  Each child also wanted to hold a puppet.  We had 4 puppets and 8 kids.  Easy to manage, I thought:  each child could have a puppet for half the class.  Each child, though, wanted to make the puppets talk, and H, who hadn’t had a turn at all last week, quite reasonably wanted everyone to have an equal amount of time over the two-week span.    Uh oh; not so easy after all.  This concern about justice, tempered by excitement, infused the hour with a nervous energy.

Before using the puppets, we spent 20 minutes discussing Dodgson’s life.  Kids were fascinated by his early life, especially his struggle with academics versus leisure.  By the time Dodgson reached age 23 in the story, though, the kids were antsy for the puppets.

My plan had been for the kids to hold the puppets while I made them talk.  Since the talking was important to some of the kids, however, I decided to whisper lines into people’s ears.

But first, I clarified a requirement on the Island of Knights and Liars:  everything anyone ever says is a statement.  “What’s a statement?” asked someone.

“Something that’s not a question or an answer,” replied D.  While the class quickly agreed that a statement is definitely something that’s not a question, his reply led into an interesting debate about whether a statement could be an answer.  J and a few others provided some hypothetical examples that led to a consensus that statements are not questions but can be answers.

“But what is a statement?” demanded M, unsatisfied with the definition so far.   Ideas petered out, so I gave one definition:  a statement is a sentence that can be declared true or false.  The kids then attempted to identify which of a number of sentences,1 including some commands, were statements.  Things seemed clear until I announced that I was confused by what it means for something to be “true.”  M gave a definition that unfortunately I didn’t accept because it used the word itself in the definition.  Now there was silence.  I called on a few kids for ideas, and got the response “I don’t know.”  So I did that thing that’s so hard for us teachers: I let the silence hang there.

Finally, C bravely offered the conjecture that true means “right.”  “And what does right mean?” I countered.

“Correct,” said C.  A number of heads nodded emphatically at his response.  So I wrote “right/correct” on the board as our working definition of true.  Now it was finally time for the puppets to talk.

I introduced the puppy Wags, and asked each child to say any statement they can think of to him.  I whispered his answers into J’s ear, and J repeated them to the group.  The students’ statements for Wags flew out of their mouths so fast that my assistant R was not able to record them in her notebook quickly enough.  But here are a few:

  • “I am a cat,” said V.  (“You are not a cat,” replied Wags.)
  • “The blackboard is green,” said D.  (“The blackboard is not green,” replied Wags.)
  • “I don’t like flowers,” said J.  (“You do like flowers,” replied Wags.)
  • “Jack is taller than John and John is taller than Jack,” challenged D.  (“Jack is not taller than John, and John is not taller than Jack,” retorted Wags.)

Student comments now shifted into conjectures and questions:

  • “Wags is not from the island.” (L)
  • “Maybe he came from another island.” (C)
  • “Maybe he’s half knight and half liar.”  (J)
  • “Maybe his mother was a knight and his father was a liar.” (V)
  • “He was always saying the opposite of what we were saying.” (D)
  • “An Oppositer!” (M)

“What do you think Wags’ favorite word is?” I asked.

“Not!”  announced V, with E and the others chiming in.  Several students challenged Wags to reply without using the word “not.”  He did so successfully until D challenged him with a one-word sentence:  “Not.”  The kids laughed.  Wags cried.  I started to “cry” too until I realized that D’s sentence was invalid.

“Wait a minute,” I said.  “Isn’t the rule on this island that all utterances must be statements?”  M reminded the rest of the group of the definition of a statement, and the kids quickly realized the flaw in D’s challenge.  They were so disappointed that Wags hadn’t been tricked.  But they didn’t give up on him; they were determined to identify what type of person he was.

“We say something and then he says the opposite of it,” said V, thinking out loud.  Someone else added that sometimes Wags lies and sometimes tells the truth.

“He’s normal,” said L.

“He’s a Normal,” declared V.

“A contradictor,” said someone.

At this point I was giddy inside because I had not introduced Wags as a Normal, nor had I defined Normals.  Once again, I was reminded of the benefit of saying as little as possible.  I told the kids that yes, they were right, Wags is a Normal, a new type of person on the island.  Normals sometimes tell the truth and sometimes lie.  Wags was a particular type of liar:  a negator.

Now it was time to take turns with the puppets so that we could evaluate the statements of Rooney the raccoon.  There was some arguing and disappointment about puppet holding.  Once that was finally settled, Rooney spoke:  “Jack is taller than John, and John is taller than Jack.”

“Liar!”  announced the group.

“I love you and I don’t love you,” stated Rooney.

“Liar!”  It was obvious to everyone from the start that Rooney was a liar.  I asked if he was a particular type of liar.

“He’s a blender!” giggled M.  “No one likes those people,” she added, and the whole group laughed. I read the kids some sentences and asked whether Rooney, who always gives a statement coupled with its negation, could have said them.  One stumped them:

“John Lasagna will be a little late for the party.  He died yesterday.”2 This statement drove the kids into seeming chaos.  They were arguing, laughing, jumping, and shouting about it. What really happened, though, was that they paired off into lively independent conversations.  V said to C, “Garfield ate John Lasagne, and Garfield was a little late to the party, so that meant that John was a little late since Garfield ate him!”  Things were digressing into silliness, but still on-topic.   I couldn’t get a word in edgewise.  I did try.  I had planned to discuss this example with the class, but as we were out of time and they clearly didn’t need me, I decided to end class amidst the joyful noise.

After I finally got most of them out of the room, I asked H for advice on how to make the puppet action more just.  She proposed that she bring in her own puppet, designate its status (knight, liar, or normal), and prepare a statement for her puppet to recite to the class.  I agreed that this sounds fair.  C and another student who had lingered after class said, “Can I bring a puppet too?”

“Sure!” I said.  Since most of the kids had left by then, I’m telling you parents now that it’s fine for your child to bring one puppet if desired.  Prepare that puppet as per H’s proposal above, and we’ll have some more fun with puppets and logic next time.

Thanks to Randy Mayes for posting his excellent and entertaining logic notes online,2   and to Mary Eberlein for posting exceedingly clear logic definitions.1

Rodi

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1I used examples from Mary Eberlein’s course in “Logic, Sets, and Functions.”  She is in the Computer Science Department at the University of Texas.  http://www.cs.utexas.edu/~eberlein/cs313k/propLogic.pdf

2Randy Mayes is in the Philosophy Department at Sacramento State University.  http://www.csus.edu/indiv/m/mayesgr/phl4/Handouts/phl4contradiction.htm