Logic Chapter Three
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VALIDITY

An argument is valid if and only if it is impossible to have a situation in which the premises are true and the conclusion is false. Otherwise, it is invalid. (A valid argument will onlyprove something if it is also sound.)

An argument is sound if and only if it is valid and all its premises are true. If an argument is sound, then its conclusion is true. Thus, a deductive argument will have persuasive force to the extent that we think that it is sound. Just being valid isn't enough. Neither is just having true premises. It's gotta have both. If we are convinced that the argument is sound, then we should be convinced that the conclusion is true. To put it another way, a sound argument proves its conclusion absolutely.

Now go back and read the definition of validity again. Isn't it weird? I mean, validity isn't really about truth at all. It's about possibility. If a certain kind of situation is possible for an argument, that argument will be invalid, even if the conclusion is true!

For instance, the following arguments are all completely invalid.
 
Paris is in France
Berlin is in Germany
Compton is in America
Cats are mammals
Dogs are mammals
Ferrets are mammals
People have two legs
Mammals have four legs
Insects have six legs
Dumbledore is a wizard
Gandalf is a wizard
Merlin is a wizard

Remember, the actual truth or falsity of the premises is irrelevant, completely irrelevant, to the validity of the argument.  Validity is just about the logical relationship between the parts of the argument, nothing else.

Don't worry about the fact that none of these things is true. Worry about the fact that it's possible for the conclusion to be false even if the premises are true.

(Test yourself: An argument where the conclusion could be false even if the premises are true is VALID INVALID)

We can test for validity by trying to draw pictures. Actually, we can test for invalidity by trying to draw pictures. For arguments with the type of premises we can draw pictures for, an argument is valid if and only if it is impossible to draw a picture in which the premises are true and the conclusion is false. Otherwise, it is invalid.

Read that again carefully. Now test yourself. Which of the following statements (A, B, C & D) is true?

A: If you can draw a picture that makes the premises true and the conclusion true, the argument is VALID

B: If you can draw a picture that makes the premises true and the conclusion false, the argument is VALID

C: If you can't draw a picture that makes the premises true and the conclusion true, the argument is VALID

D: If you can't draw a picture that makes the premises true and the conclusion false, the argument is VALID

Did you get that? It means that to test an argument, we try to draw a picture in which the premises are true and the conclusion is false. If we can, the argument is invalid. If we can't, it's valid.

Here's a scheme.

                                              Fx : x discovered France
                                              Px : x eats cheese pizza
                                              Mx : x is a Martian
                                              a : Albert Einstein
                                              w : my wolverine
                                              l : Laura Schlessinger

So"a" stands for Albert Einstein, "w" for my wolverine and "l" for Laura Schlessinger. Anything inside the "F" circle discovered France, inside the "P" circle eats cheese pizza and anything inside the "M" circle is a Martian

Given that scheme, is this argument valid or invalid?
 
  Albert Einstein discovered France
My wolverine eats cheese pizza
Laura Schlessinger is a Martian
Fa
Pw
Ml

Now, it's true we can draw a picture in which the premises and conclusion are all true. Here's a simple one:


But it doesn't prove anything. Being able to make everything true doesn't matter. We need to know if its possible to make all the premises true at the same time that the conclusion is false. The following picture does this, so the argument is invalid.


This picture proves that it's possible for Albert Einstein to have discovered France and for my wolverine to eat cheese pizza even if Laura Schlessinger is not a Martian. If that's possible, then the argument is not valid.

An argument with conclusion and premises that are true still isn't neccesarily valid.
 
  Elvis is dead. (Accept it.)
The X-Files was a popular TV show
The Eiffel Tower is in France
De
Pf
Ft

This time, don't worry about the fact that all of these things are true. Worry about the fact that it's possible for the conclusion to be false even if the premises are true. Again, the following picture does not prove the argument valid.


But this next picture does prove the argument invalid.


This picture proves that it's possible for Elvis to be dead and for the X-Files to have been a popular TV show even if the Eiffel Tower is not in France. If that's possible, then the argument is not valid.

So if you're trying to check the validity of an argument, and you figure out a way that the premises and conclusion can all be true, then you haven't checked the validity of that argument. You gotta try to figure a way to make the premises true and the conclusion false. If that can't be done, the argument is valid. If it can be done, then it's invalid.

You think that's weird? Well check this out.

An argument with premises that can't all be true IS necessarily valid.

Read that again. It says that if the premises can't all be true, then the argument is valid. It doesn't even mention the conclusion, which means that an argument with a false, stupid or impossible conclusion can be perfectly valid, provided it has premises that somehow contradict each other.
Test yourself. Which of the following two sentences says the same thing as the sentence underlined above?

A. An argument with premises that can't all be true is necessarily valid.

B. An argument with premises that can't all be true is necessarily invalid.

If you said "A," you're right!
If you said "B," you're wrong.

Here's an example of an argument that's valid because of contradictory premises.
 
Elvis is dead.
Elvis is alive.
Laura Schlessinger is a woolly mammoth.

De
~De
Wl                  VALID!!

Think about it. Is it possible to have a situation in which the premises are true and the conclusion is false? Sure, it's possible to have a situation in which the conclusion is false, but for the argument to be invalid, it has to be possible for the premises to all be true at the same time the conclusion is false. So if the premises can't all be true, the argument is valid. (If you still think the argument is invalid, draw a picture in which the premises are all true and the conclusion is false. Remember, there's only one Elvis, and you can't be both dead and alive.)

Is this a startling concept? Well, remember that logic is startlingly different from the way people usually think, and from the way they expect you to think.

Test yourself: Does the fact that we can make a valid argument for absolutely any conclusion mean that logic can prove absolutely anything? YES NO

To put it another way, can you construct a sound argument for a false conclusion? YES NO

Here are some arguments for you to work on by yourself. Some are harder, so just do the ones you can.

Practice: For each of the following arguments, try to draw a world-picture in which the premises are true and the conclusion is false.

Nu
~Nu        
Ga ^ Ki
Fa 
Me        
Fa ^ Me
Gi
Lo        
Lo v Gi

~Jo       
Jo
Ke
Ko          
Ku
Hu
Je        
Mo v Je
Fo
Fi              
Fe v Fu
Ma
Ji              
La ^ Ji
Na v Lu
~Na            
Lu
Ka ^ Mi
~Ka            
Mi
Ne v Ha
Go v Li

Go
Ho v Ju
Le ^ Ii

Le
Ie ^ Hi
No v Gu

No
Mu ^ Ia
He ^ Ni

He
Iu     
Ge ^ Io

Ja


practice 3. Use your own paper or the answer sheet at don't turn this in. For each of the following arguments, say whether it is valid or invalid.
  My monkey can fly.
My monkey can fly. 

1. ____________ 
 
  My monkey can fly.
My monkey can not fly. 

2. ____________ 
 
  My monkey can not fly.
My monkey can fly.
My monkey can fly.
3. ____________ 
  My cat can not fly.
My dog can not fly.
My snake can not fly.
My rabbit can not fly.
My monkey can not fly.
4. ____________

If you have trouble, I suggest trying to draw pictures based on the following scheme

                                              Fx : x can fly
                                              m : my monkey
                                              c : my cat
                                              d : my dog
                                              s : my snake
                                              r : my rabbit

Question 5. Write out the correct definition of validity in your own words as completely as you can. Don't worry about making it elegant, just worry about getting it right.

Copyright 2005 by Martin C. Young

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