Shantila's Inside Logic #21

All

*

You are still at the airport. It is pretty boring here. You are sleepy. You begin to doze. You find yourself eavesdropping again on the nice couple who were kissing each other and who are going to get married. The woman is the one who saved you and Joe from the guy who tried to steal the lottery ticket.

"The plane to Atlanta will be two hours late," says the man.

"Yes, and then it will take us a good half-hour to get from the airport downtown," the woman says.

"The Peachtree Plaza is all the way downtown?" he asks.

"Yes," she says. "I'm afraid so. And the dinner is at 6 o'clock. We are going to miss the whole thing again."

"That is really sad," he says. "Our comrades will be disappointed."

"Yes, we should never treat a comrade so poorly. I just feel terrible," she says. "I wonder how we could make it up to her?"

"That's a great idea!" she exclaims.

The Peachtree Plaza? A birthday dinner? At 6 o'clock tonight? -- Your sister's dinner is at Atlanta's Peachtree Plaza at 6 tonight! These people must be going to your sister's birthday party!

Joe had been sleeping but he wakes up. He rubs his eyes. "Do you have the lottery ticket?" he asks you suddenly.

"No, I gave it back to you" you say.

"No you didn't," he says.

"Yes, I did, remember?" you insist.

"You did?" Joe asks anxiously.

"Yes, I did, you idiot! When that guy who was going to confess pulled out the gun!" you say.

"I don't think so," Joe says.

"Yes, I did, you idiot!" you scream.

Now the lottery ticket is missing!

The whole situation is screwed up, but who cares? It isn't your problem.

And then you hear the woman who knows your sister and who is going to marry the man who kissed her nicely whisper something awful to the man.

"If you kiss me nicely, I will give you half of the eight million dollars!" she laughs quietly.

What?!

She stole the ticket! She must have done it when she jumped up and knocked down the man with the gun.

You stand up and turn around.

"Give the ticket back to Joe," you demand.

But she won't give it back.

"I bought the ticket last night at a 7-11 along with a bag of peanuts and a strawberry Big Fizz," she explains.

"Oh, boy, that is a huge lie!" you say. "Joe, just go get the ticket."

Joe goes over and tries to grab the ticket from her, but she won't let him. They are arguing about it. Joe says that he never heard of a strawberry Big Fizz. As the woman yells back that she drinks Big Fizzes all the time, you notice long upper teeth pointing down at the sides of her mouth!

Joe grabs for the ticket but misses. Then she grabs Joe! And then she begins to BITE his NECK! You see her long incisors going towards his neck--

SHE IS A VAMPIRE!!!

Fortunately at that very moment Buffy, the Vampire Slayer, arrives on a plane from Los Angeles and she and the vampire woman have a big fight, breaking windows all over the airport. Finally Buffy beats her up and drives a stake through her heart and she just disappears. Then Buffy comes back and talks with you awhile.

Things are turning out better now! Also, the plane is going to leave soon.

But wait!

Maybe that man also is secretly a vampire! -- After all, they were comrades! What did that mean? -- WAIT! They said your sister also is a COMRADE!! --

YOUR SISTER is a VAMPIRE TOO!

This is terrible news. You never even suspected it. And now she is going with them to Transylvania!

Somebody is shaking your shoulders.

You wake up.

It is the very woman who is a VAMPIRE!

SHE CAME BACK TO LIFE AGAIN!

"Help!" you scream. "Help! Where's Buffy?"

The woman backs away, startled. The man who kissed her nicely is standing beside her.

Joe comes running over.

"What's wrong?" he asks.

"My sister is a vampire!" you say.

"Oh no! That is terrible" Joe says.

"And so is she!" you scream, pointing at the woman.

"I don't think so," Joe says. "Not her. She helped us."

"But she stole the ticket and then she tried to bite you!"

"No. She saved our lives," Joe says. "She jumped over three rows of seats and knocked that guy's gun away."

"And then Buffy killed her!" you scream. "Where's Buffy?"

"I don't think so," says the woman who was a vampire, her voice a bit cool. "I think you just had a nightmare. You were screaming something about your sister."

"Yes," Joe says. "And you slobbered all over your t-shirt."

"Oh," you say.

You look at your t-shirt.

"Look. I have the ticket right here," he says.

He pulls out the ticket out of his pants pocket and shows it to you.

"Oh," you say. "Sorry. I guess I was sleeping. I was dreaming. Sorry."

"It's ok," the woman says. "It happens to me too sometimes."

"Where did you meet my sister?" you ask.

"Your sister? Who is your sister?" she asks. "I don't think I ever--"

"Oh," you realize. "That also was part of my nightmare."

At that moment your converation is interruped by Jessica Simpson who walks into the corridor, sees Joe, and immediately falls in love with him.

"Excuse me. What is your name?" she asks Joe.

"Joe," Joe says.

"Where are you going?" she asks, giving him a little kiss.

"Atlanta," he says.

"Me too!" she says.

"Guess what happened! I just won the lottery--" he says.

"Wanna' ride with me to Atlanta, Joe?" she asks fetchingly. "I am flying my own private jet."

"Sure," Joe says. "But, uh --"

"What?" Jessica asks.

"I should tell you something first," Joe says.

"What?" Jessica asks.

"I have a logic block," Joe says.

"Oh, don't be silly," says Jessica.

"But its only partial," Joe says.

"If you have a logic block, how did get dressed? How did you get to the airport?" asks Jessica.

"Ok. Fine," Joe says. "Could we take my friends along with us too?"

"Sure, no problem about that," she says. "There's plenty of room, we just have to wait about ten minutes while it is re-fueling."

*

Ok, great. Things are looking up for you and you finally are going to make it out of the Detroit airport despite the unfortunate scene you created with your nightmare. By the way, even when we are having a nightmare or daydreaming we can reason validly. For example, consider this reasoning:

All vampires are scary.

My sister is a vampire.

Therefore, my sister is scary.

Notice three things. First, this reasoning is valid -- if the two premises are true, the conclusion must also be true. There is no way both premises can be true without guaranteeing the truth of the conclusion.

Secondly, there probably is no reason to think that the reasoning is sound (even if your sister is a bit odd in some ways).

Thirdly, even though the argument is valid, we cannot interpret it using the special symbolic language and system that we have developed so far. We cannot use the language and system so as to prove that it is valid. This is a shortcoming of the language we have built up so far. To see this, let us consider how we might try to represent this argument. We can represent the first sentence simply as V, the second as S, and the third as C. But obviously we do not have any way to prove the following sequent (and we wouldn't want to be able to prove it, because it is not a valid form -- it is easy to find counter-examples to it where the premises are true but the conclusion false):

V, S } C

The problem with our representation of the argument is that the first sentence "All vampires are scary" has an internal structure that we cannot represent using the special language we've developed so far. The key idea in that first premise is all. Our language so far is designed to be useful for exploring the basic logical notions: and, not, or, if. And indeed we have developed a system of logic that captures important relationships between these basic notions -- and it is "complete" in that sense that all the valid sequents we can express in our language can be proved valid.

But now we see that there are other basic logical relationships that cannot even be expressed in the language we have developed so far.

For this reason we are going to make further developments in the language and proof system so as to be able to represent additional valid forms of reasoning.

There many simple examples of reasoning involving the concept all that we can make in ordinary thought and action. For instance:

All lottery winners are happy. Joe is a lottery winner. So Joe is happy.

All residents of Boston are cold today. Bill is a resident of Boston. Therefore, Bill is cold today.

All tigers are ferocious. Tinker is a tiger. So Tinker is ferocious.

All squirrels are shy. Frisky is a squirrel. So Frisky is shy.

All her boyfriends are losers. Sam is her boyfriend. So Sam is a loser.

All of these examples involve valid reasoning, as you can verify for yourself simply by thinking about them. In each case, the conclusion MUST be true if the two premises are true. But given the simple language we have so far, we cannot represent them to show they are valid. As with the "sister / vampire" example, in each case we can use sentence letters, say, P and Q for the two premises, and some different letter R for the conclusion. That is the best we can do, but of course the sequent P,Q } R is not a valid sequent.

In order to represent inferences or arguments like these, we now are going to add to our language a new way to represent all statements.

Here is how we will do it. So far we have used capital letters to stand for simple statements; for example, letting S stand for "Sally lives in Cleveland." We will continue to do that. But we also will use capital letters in a new way as well. And beginning now, we also will use small letters too: we will use small letters (a,b,c, ...) as names. The new way of using capital letters is as "predicates." We now can put a capital letter and a small letter together, for example, to let Lj say "Joe is a lottery winner." In this example, Lx stands for "x is a lottery winner." Here we use x as a "variable" ranging over people and things that can be named; the variable x doesn't name any particular thing, but simply is a place-holder where a genuine name can appear. Here are some examples of sentences in this language.

Bb: Bill is a resident of Boston

Cb: Bill is cold today

Hj: Joe is happy

Hs: Sally is happy

In these examples, we are using the capital letters as predicate letters and we can specify the meaning of a predicate letter by using a variable, as in these examples.

Bx: x is a resident of Boston

Hx: x is happy

Cx: x is cold today

We also indicate how we are using the small letters as names:

b: Bill

j: Joe

s: Sally

Given these new developments in our language, we now can go on to devise a simple way to express "all" statements. We simply will use parentheses around a variable

(x)

to express the idea of "all". So, for example, (x) preceding Cx -- that is, (x)Cx -- says All things are cold. (Expressed in a somewhat stilted way, we can say: "All things x are such that x is cold.")

So this gives us another new type of sentence.

Now let us consider some reasoning in this new language. Consider the following sequent

(x)Cx } Cb

Is this valid? Yes it is! If everything is cold, (x)Cx, then surely Bill is cold, Cb. Basically, this is one of the core ideas involved in reasoning with all. It is a simple yet signficant idea. If some predicate is true of everything (or at least, everything that is being talked about in a particular context), then no matter who or what you specify, the predicate is true of the specified person or thing. This gives us a Rule for a system of reasoning based on the more complex language.

More generally, the UI rule says that given a sentence of the form (x)P, where P contains at least one occurrence of the variable x, we can write the sentence that results by replacing x with c wherever x appears in P. So, for example, given (x)(Fx&Gx) we can use UI to write Fc&Gc. --We call this rule Universal Instantiation because (x) is called a Universal Quantifier, and we "instantiate" the name c when we move from the universally quantified sentence (x)Fx to its instance Fc.

The new rule can be used straightforwardly as follows to prove

(x)Fx } Fb

Now here is how we can we prove the argument about Joe being happy by virtue of winning the lottery.

All lottery winners are happy.

Joe is a lottery winner.

So Joe is happy.

First we need to figure out how to use our new symbols to interpret the first premise, All lottery winners are happy. The key is to combine the use of "all" with "if". The sentence tells us that all things x are such that if x is a lottery winner, then x is happy. We can express this as follows using our new way of representing "all" and using our familiar > for "if":

(x)(Lx>Hx)

This is how we represent All lottery winners are happy.

Given this interpretation of the first premise, our new Rule UI can be used to construct the following proof (where the representation of the second premise is Lj and the conclusion is Hj):

(x)(Lx>Hx), Lj } Hj

Notice that in English, the words "every," "any," "each," and "a" can be used to say things that the same meaning as "all". That is,

All lottery winners are happy

means just the same thing as each of the following sentences.

Each of these sentences can be represented as (x)(Lx>Hx). Notice that in the final sentence "a lottery winner" is used in a way so that it should be interpreted as meaning "all". But, of course, the indefinite article "a" is not always used to mean all -- consider, for example, "a vampire is trying to bite Joe" which does not mean that all vampires are trying to do it! So, as we have seen in other cases, careful interpretation is required in going from English statements into our formal language.

Even more complex statements can be represented as well. We might want to talk about lottery winners from Texas, and we might want to say that all Texan lottery winners own a big ranch. Letting Tx say "x is a Texan" and Lx say "x is a lottery winner" and Rx say "x owns a big ranch," then we express this idea by using a conjunction in the antecedent of the conditional that is in the scope of the universal quantifier, as follows:

(x)((Tx & Lx) > Rx)

This says that all x are such that if Tx and Lx, then Rx -- which is what we wanted to express in saying all Texan lottery winners own a big ranch.

*

What now about no, nobody and nothing? These words often are used in English sentences so as to be properly interpreted using the universal quantifer. Notice that Everything is NOT cold,

(x)~Cx

says the same thing as

Nothing is cold.

Now consider

Nothing cold is happy.

This can be represented as

(x)(Cx>~Hx).

Likewise No tigers are friendly can be represented as

(x)(Tx > ~Fx).

Likewise No Michigan lottery winners are happy can be represented as

(x)((Mx & Lx) > ~Hx)

Another type of sentence we can represent is those like Only tigers are friendly: this can be represented as (x)(~Tx > ~Fx). That is, for all x, x is friendly only if x is a tiger.