Validity

Valid Arguments

Validity is a term we use to identify a particular kind of logical structure in arguments.  Some reasons or premises can be said to support their conclusion, while others do not offer good support.  The notion of "support" here refers to the logical connection between the reasons and the conclusion.  But "support" is vague.  Some arguments, in virtue of their logical structure, have this special feature.  Consider these five examples, which all have the property of being deductively valid: 

 A.  1.  All people who are born in the United States are U.S. citizens. 

2.  McCormick was born in the United States. 

___________________________

3.  Therefore, McCormick is a U.S. citizen. Valid

B.  1.  If there is an accident on the highway, then Joe will be late getting home. 

2.  There is an accident on the highway.

______________________________

3.  Therefore, Joe will be late getting home. Valid

C.  1.  If the headlights are out, then a car is not legal to drive. 

2.  Denise's car is legal to drive.

_________________________________

3.  Therefore, the headlights are not out on Denise's car. Valid

D.  1.  All mammals have kidneys.

2.  Plants do not have kidneys.

__________________________

3.  Therefore, plants are not mammals. Valid

E.  1.  All spiders have 12 legs.

2.  All things with 12 legs are blue. 

___________________________

3.  Therefore, all spiders are blue.   Valid

What is the feature that all of these arguments have in common?  Is it that they all have true premises and true conclusions?  No.  E. has false premises and a false conclusion.  And we don't really know whether B.1. or C.2. are true.  Their truth values are unknown.  Is it that the premises support the conclusions?  Yes, but what does support mean here?  What exactly is the relationship between the premises and the conclusion?  

Validity defined:  An argument is valid when it has this property:  if the premises were true, then the conclusion would be guaranteed to be true.  In all of these cases, it is impossible for the premises to be true while the conclusion is false.  The support relationship between them means that if the premises are true, then they would guarantee the truth of the conclusion.  That is the definition of validity. 

A valid argument is one where the premises, if they were true, would guarantee the truth of the conclusion. 

Invalid  arguments are ones where there could be circumstances where the premises are true but the conclusion is false.  We can imagine situations where the premises are true but the conclusion is false.  Consider these:

.  1.  If Arnold is in Hollywood, then he is in California.

2.  Arnold is in California.

____________________________

3. Therefore, Arnold is in Hollywood. 

G.  1.  If Arnold is in Hollywood, then he is in California. 

2.  Arnold is not in Hollywood.

_________________________________

3.  Therefore, Arnold is not in California. 

H.  1.  All men are human. 

2.  Jessica is a human.

_______________________

3.  Therefore, Jessica a man. 

I.  1.  All men are human.

2.  Jessica is not a man. 

____________________________

3.  Therefore, Jessica is not human. 

J.  1.  No NFL quarterbacks are women. 

2.  Ben Roethlisberger is a not a woman. 

_____________________________

3.  Therefore, Ben Roethlisberger is an NFL quarterback. 

K.  1.  No dogs are reptiles.

2.  Tom Cruise is not a reptile.

______________________________

3.  Therefore, Tom Cruise is a dog. 

L.  1.  No movie stars are conservatives.

2.  No conservatives are liberals. 

_____________________________

3.  Therefore, movie stars are liberals. 

M.  1.  No reptiles are warm blooded.

2.  No warm blooded organisms are plants. 

_______________________________

3.  Therefore, reptiles are plants. 

In all of these cases (F-M), it is possible for the premises to be true while the conclusions are false.  In F, Arnold could be in Malibu, thus 1. and 2. would be true, but 3 would be false.  And in G.  Arnold could be in Sacramento, so 1. and 2. would be true, but 3 would be false. And in H and I, the premises are true but the conclusions are false.  K. shows why J. is invalid.  K.1. and K.2. are true, but K.3. is false.  So argument J, which has the same structure, is invalid as well.  And argument M. shows why argument L is invalid the same way.  M.1. and M.2. are true, but M.3. is false. 

The logical relationships of validity and invalidity hold no matter what we put in for the terms.  So argument F.  is the same as this one:

O.  1.  If something is stonky, then it is wonky. 

2.  Melb is wonky.

_______________________

3.  Therefore Melb is stonky. 

That is, O. is still invalid for the same reason F. is.  J.1. says that all things that are stonky are also wonky.  But it doesn't say that all wonky things are stonky.  J.1. leaves open the possibility that something could be wonky, but not stonky.  So we cannot infer that Melb is stonky from the claim that Melb is wonky. 

 Here are the patterns that the arguments we have been considering follow: 

A.  1.  All As are Bs.

2.  x is an A.

________________

3.  Therefore, x is a B.                                               VALID

B.  1.  If P then Q

2.  P

__________

3.  Therefore, Q                                                        VALID

C.  1.  If P then Q

2.  Not Q

____________________

3.  Therefore, not P                                                   VALID

D.  1.  All As are Bs.

2.  x is not a B.

______________

3.  Therefore, x is not an A.                                        VALID

E.  1.  All As are Bs.

2.  All Bs are Cs.

_________________

3.  Therefore, All As are Cs.                                       VALID

 F.  1.  If P then Q

2.  Q

_________________

3.  Therefore P                                                          INVALID

G.  1.  If P then Q

2.  Not P.

____________________

3.  Therefore, not Q                                                   INVALID

H.  1.  All As are Bs

2.  X is a B.

______________

3.  Therefore, X is an A                                              INVALID

I.  1.  All As are Bs.

2.  X is not an A.

________________

3.  Therefore, X is not a B.                                         INVALID

J.  1.  No As are Bs.  

2.  X is a not a B. 

_____________________________

3.  Therefore, X is an A.                                              INVALID

K.  same as J. 

L.  1.  No As are Bs. 

2.  No Bs are Cs. 

_____________________________

3.  Therefore, As are Cs                                               INVALID

M.  same as L.

So, once again, an argument is valid when the premises, if they were true, would guarantee the truth of the conclusion.  But a valid argument need not have true premises or a true conclusion.  In fact, the premises and the conclusion could be any combination of true and false except one:  there are no valid arguments that have true premises and false conclusion.  The definition of validity rules that out. 

 And an invalid argument is one where it is possible for the premises to be true and the conclusion false. 

A Test for Validity

Another way to apply this definition is to think of it as a test.  Consider this argument:

1.  All of the cars in the lot have CA plates.  

2. Vaughn's car does not have CA plates.

________________________________

3.  Therefore, Vaughn's car is not in the lot.  

Test it for validity:

A) Imagine or pretend that the premises are true.  

B) Ask this question: is it possible for the conclusion to be false in some situation while we are imagining that the premises are true? 

If you can imagine a situation where the premises are true but the conclusion is false, then it is invalid.  (Because in a valid argument, the premises, if they were true, would guarantee the truth of the conclusion.)

If you cannot imagine a situation where the premises are true and the conclusion is false, then the argument is valid.  

So in the case above, we assume or imagine that according to premise 1, every single car that is in the lot has CA plates.  There are no non-CA plates on any cars there.  

Then we imagine that Vaughn's car does not have CA plates.  

Now we ask the question, is it possible that Vaughn's car could be in the lot?  It cannot if all of the cars in the lot have CA plates.  So we cannot imagine a situation where the premises are true and the conclusion is false.  So this argument is valid.  The premises, if true, guarantee the truth of the conclusion.  

Now consider this argument:

1.  None of the cars in the lot have permits.  

2. Vaughn's car does not have a permit.

_______________________________

3.  Therefore, Vaughn's car is in the lot.  

Imagine that there is not a single car in the lot that has a permit.  And also assume that Vaughn's car does not have a permit.  Can we now imagine or picture a situation where Vaughn's car is not in the lot?  Can we coherently imagine that Vaughn's car is outside of the lot, and the two premises are true?  We can.  The premises don't assert or require anything to be true about cars outside of the lot; they could have permits or not.  So Vaughn's car could either be in the lot, or it could be out of the lot while the premises are true.  So, since we can imagine a situation where the premises are true and the conclusion is false, this argument is invalid.