Logic Chapter Twenty Four

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Remember when I told you that "Ǝ" and "/\" were the last two new characters? Well, I screwed up, because there is one more character. That character is the deceptively simple "=." That's right, the "equals" sign. Perfectly innocent in mathematics, in symbolic logic it becomes a deadly weapon of . . . well, maybe not deadly, but dangerous. Grrr! Watch out!

Anyway, the "=" symbol (pronounced "iz," the same way you pronounce the word "is") means that something is identical with itself by another name. So "a=b" means that "a" is the same thing "b." 

This means that, under this scheme:

                                              Bx : x is a bear
                                              Cx : x is a cat
                                              a : King Arthur
                                              b : Bill Clinton

the formula "a=b" means "King Arthur is Bill Clinton." Now, it doesn't just mean that they're similar to each other, or that they're impossible to tell apart, it means that they are exactly the same thing. Weird, huh?

With those operators, and this scheme:
Ax:
Bx:
Cx:
d:
e:
f:
x is an aardvark
x is a bullfrog
x is a camel
Dick Cheney
Emo Phillips
Francis the talking mule

We can say things like:
  All things in the universe are bullfrogs
  At least one camel exists
  It's not true that everything in the universe is a bullfrog
  Not even one aardvark exists

A statement in which the universal operator is the main operator is a "universal statement." (Or just a "universal.")

A statement in which the existential operator is the main operator is a "existential statement." (Or just an "existential.")

And of course, a statement in which the negation operator is the main operator is a "negation."

So this means that one of the above statements is an existential statement, one is a universal statement, and two of them are negations.

Here are some statements you can practice translating. Don't worry if I've worded the translation differently from you. As long as the idea is the same, you're fine.
 
 

 

 

 

 
 

 

 

 
 

 

 

 


Write down what you think each sentence means before you click on it to see the answer. Be careful! Look at where the parentheses are before you decide what a statement means.

As you can probably guess, there are weirdnesses involved with these two operators. The existential weirdness doesn't really show itself until we get into the rules for creating existential statements, but the universal weirdness shows up right away.

Before we get weird, let's expand our world-construction system. We will represent a whole universe as a rectangle and empty space by cross hatching so that, for instance, the picture would represent a completely empty universe.


Now, on to the universal weirdness.

The classic universal statement is "all men are mortal." Well, that's boring, so I'll use the statement "all badgers are Corsican" because I like badgers, and Corsica is cool. Using one of our powerful new operators, we can symbolize the statement "all badgers are Corsican" as which more precisely says that "if something is a badger then it's Corsican." Right away, the weirdness starts, because this statement, "all badgers are Corsican" doesn't actually say that any badgers exist! It just says that if any badgers do happen to exist, then all of them are Corsican. So the statement is made true by all the following universes


Now, a clue to how to understand the universal weirdness is to understand that a universe will make true if and only if it makes it impossible for an object to turn up that makes false. And can only be false if there is some object that is a badger but not a Corsican. So if you rule out the possibility that an object can turn up inside the B circle but outside the C circle, you've made true, no matter what other properties that universe may have. Now lets look at these universes individually.

The first one is easy. Here is a universe that may or may not be empty but, if any badgers do happen to exist here they will perforce have to be Corsican because the B circle is completely inside the C circle. has to be true.

The second one is also easy. Here there is a part of the B circle that's outside the C circle, but it doesn't matter because that part of the B circle is crosshatched, which defines it as empty, so no non-empty part of the B circle is outside the C circle, so no non-Corsican badgers can exist, and is true.

The third one is a little trickier. You can't see the C circle here because it exactly follows the line of the edge of the universe. In this universe, everything is Corsican, so if any badgers do happen to exist, they're Corsican too. So is true here too.

The fourth universe is where the weirdness really kicks in. It is a universe bereft of badgers. No badgers here exist because the B circle is defined as empty. Since no badgers whatsoever exist, it is absolutely imposible for a badger to exist that is not a Corsican. Since no non-Corsican badger can exist, cannot be made false. If cannot be made false, then perforce it is true.

The fifth, totally empty universe has exactly the same property. Badgers cannot exist in it, so non-Corsican badgers cannot exist, so is true.

So it's important to remember that universal statements don't say anything about what does or does not exist. In a completely empty universe, the statement "all things are bullfrogs" is true even though no bullfrogs actually exist, because nothing else exists either. Thus the statement "all things are bullfrogs" is logically equivalent to "nothing exists that is not a bullfrog" or "if anything exists, it's a bullfrog." Existential statements on the other hand always assert that something exists or does not exist. Negations, as always, say that whatever follows the negation operator is not true.

Now, we can make some universal and existential statements true or a false with our regular old pictures. For instance:
A bullfrog exists
is made true by See the "k" (whatever "k" stands for) in the bullfrog space
All things are aarrdvarks
is made false by See the "b" (whatever "b" stands for) in the non-aardvark space

But to say more we need the new stuff.
A bullfrog exists
is made false by Notice the bullfrog space is empty.
All things are aardvarks
is made true by Notice the non-aardvark space is empty.

Here's it all put together with some more formulas.
This formula: Is made true by: And made false by:
 
 
 
 

 
 
 
 
 

 
 
 
 
 

 

Here are some more difficult ones. We'll do more with these next lesson.

This formula:

Is made true by:

And made false by:







This formula:

Is made true by:

And made false by:







This formula:

Is made true by:

And made false by:







This formula:

Is made true by:

And made false by:







For practice, cover the answers given above and, for each statement given below, draw a world that makes that statement true.

   
 
 
  

   
 

After checking your answers, cover them again, and for each statement, draw a world that makes that statement false.



 




Validity of Arguments involving Generalizations

Okay, we can show that some universal and existential arguments are invalid with our regular old pictures. For instance:

Keith is hairy
Mike is insane
Ossie is hairy
No-one is green


is proved invalid by
Jeff is green
Ossie is green
Everything is green


is proved invalid by

But there are others that need the new stuff.
Jeff is not a kangaroo
A kangaroo exists


is proved invalid by


Nigel is an elk
A non-elk exists


is proved invalid by



Remember these statements?
 
Now read the following little arguments. Are they valid? I hope so, because they're intended to be arguments in which the statement above the line says exactly the same thing as the statement below it! Did I do it right?



Now compare each verbal argument with the symbolic argument immediately below it. They're meant to have the same logical structure and in fact, with the right scheme, they say exactly the same things.

What's going on here? Well, the fact is that certain general statements are logically equivalent to other general statements involving the opposite quantifier. Take a look at the following diagrams.

Is x(Ax) true here?
Is ~x(~Ax) true here?
Is x(Ax) false here?
Is ~x(~Ax) false here?
Is x(Bx) true here?
Is ~x(~Bx) true here?
Is x(Bx) false here?
Is ~x(~Bx) false here?
Is ~x(Bx) true here?
Is x(~Bx) true here?
Is ~x(Bx) false here?
Is x~(Bx) false here?
Is ~x(Cx) true here?
Is x(~Cx) true here?
Is ~x(Cx) false here?
Is x(~Cx) false here?


Here's a rule that lets us switch between "" and "" and vice versa without making any invalid inferences,

Rule 17: Change Quantifier (CQ) Comes in eight little rules.
1. If "xP" is an available line, then "~x~P" may be written as a new line in the derivation, and
2. If "~xP" is an available line, then "x~P" may be written as a new line in the derivation, and
3. If "x~P" is an available line, then "~xP" may be written as a new line in the derivation, and
4. If "~x~P" is an available line, then "xP" may be written as a new line in the derivation, and
5. If "xP" is an available line, then "~x~P" may be written as a new line in the derivation, and
6. If "~xP" is an available line, then "x~P" may be written as a new line in the derivation, and
7. If "x~P" is an available line, then "~xP" may be written as a new line in the derivation, and
8. If "xP" is an available line, then "~x~P" may be written as a new line in the derivation.

Or, to put it another way



Whenever an instance of the sentence above the line appears as an available line, the sentence below the line can be written.

From left to right, these arguments might say things like:

If it's not true that everything is an ostritch, then something exists that isn't an ostritch.
If it's not true that a penguin exists, then everything is a non-penguin.
If something exists that isn't a quahog, then it's not true that everything is a quahog.
If everything is a non-robot, then it's not true that a robot exists.
If it's not true that everything isn't slithy, then something exists that is slithy.
If it's not true something exists that isn't a turnip, then everything is turnips.
If an ugly thing exists, then it's not true that everything is beautiful.
If everything is vile, then it's not true that something exists that isn't vile.

Arranging things this way reveals a helpful feature of CQ. Notice that in each "argument" there are always exactly two tildes. Never more, never less. These two tilde's are either both above the line, both below, or one up and one down. Notice also that no tilde ever appears in the same position both above and below the line. There's only two places to put a tilde, and if there's a tilde in that place above the line, there isn't below the line, and vice versa. Finally, of course, the quantifier always changes. "" becomes "" and "" becomes "" every time.

Now, what happens if you apply CQ to a the result of a previous application of CQ? Well, if you apply CQ to x~P you get ~xP, right? So what happens if you apply CQ to that ~xP? (Try it. I'll wait.)

Did you get x~P? (If you didn't, try again.) Try it with another formula. Is there any formula for which two applications of CQ doesn't bring you back to the same formula? Now think about it with "Fx" instead of "P." This symmetry gives us the chance to write the meanings of the CQ rules out as a set of four equivalences, (with Fx instead of P), like so.


You can check these by drawing pictures for each of the sides. What? You want me to draw the pictures! Why, the nerve of some people!

Okay, here's the pictures:

Now, one of these diagrams makes both and true, one makes both and true, one makes both
and true and one makes both and true. Which one is which?

On the other hand, one of these diagrams makes both and false, one makes both and false, one makes both
and false and one makes both and false. Which one is which for falsity?

Now, can you find a single picture in which and have different truth values? Can you find a picture in which and have different truth values? Is there one in which have and different truth values? Or can you find a picture in which and have different truth values?

Some of these arguments are valid. Some are invalid. You can't prove the valid ones yet, but you can show the others invalid by drawing pictures. Go for it. (Remember the differences between universals, existentials and negations.)


Homework 12. Use your own paper or the answer sheet at logic13homework.rtf.

For each of the following arguments do TWO things. First, translate it into English based on the following scheme.

Fx: x is fiendish
Gx: x is a gryphon
Hx: x is horrible
Ix: x is insane
Jx: x is a jumbuck
Kx: x is kinky
Lx: x is looney
Mx: x is mad
m: Merlin
o: Ozymandias
a: Ahuramazda
b: Bob

Second, circle it if it's valid, cross it out if it's invalid.

1. 2. 3. 4.
5. 6. 7. 8.







Copyright 2007 by Martin C. Young

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