Valid Arguments
Validity Some arguments, in virtue of their logical structure, have a special feature. Consider these four examples:
A. 1. All people who are born in the United States are U.S. citizens.
2. McCormick was born in the United States.
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3. Therefore, McCormick is a U.S. citizen.
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.
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3. Therefore, Joe will be late getting home.
C. 1. If the headlights are out, then a car is not legal to drive.
2. Denise's car is legal to drive.
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3. Therefore, the headlights are not out on Denise's car.
D. 1. All mammals have kidneys.
2. Plants do not have kidneys.
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3. Therefore, plants are not mammals.
E. 1. All spiders have 12 legs.
2. All things with 12 legs are blue.
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3. Therefore, all spiders are blue.
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? The relationship is that if the premises happened to be true, then the conclusion would have to be true too. 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. Consider these:
F. 1. If Arnold is in Hollywood, then he is in California.
2. Arnold is in California.
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3. Therefore, Arnold is in Hollywood.
G. 1. If Arnold is in Hollywood, then he is in Calfornia.
2. Arnold is not in Hollywood.
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3. Therefore, Arnold is not in California.
H. 1. All men are human.
2. Jessica Simpson is a human.
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3. Therefore, Jessica Simpson is a man.
I. 1. All men are human.
2. Jessica Simpson is not a man.
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3. Therefore, Jessica Simpson is not human.
J. 1. No NFL quarterbacks are women.
2. XX is a not a woman.
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3. Therefore, XX is an NFL quarterback.
K. 1. No dogs are reptiles.
2. Tom Cruise is not a reptile.
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3. Therefore, Tom Cruise is a dog.
L. 1. No movie stars are conservatives.
2. No conservatives are liberals.
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3. Therefore, movie stars are liberals.
M. 1. No reptiles are warm blooded.
2. No warm blooded organisms are plants.
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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 validty and invaldity 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.
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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.
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3. Therefore, x is a B. VALID
B. 1. If P then Q
2. P
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3. Therefore, Q
VALID
C. 1. If P then Q
2. Not Q
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3. Therefore, not P VALID
D. 1. All As are Bs.
2. x is not a B.
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3. Therefore, x is not an A. VALID
E. 1. All As are Bs.
2. All Bs are Cs.
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3. Therefore, All As are Cs. VALID
F. 1. If P then Q
2. Q
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3. Therefore P INVALID
G. 1. If P then Q
2. Not P.
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3. Therefore, not Q INVALID
H. 1. All As are Bs
2. X is a B.
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3. Therefore, X is an A INVALID
I. 1. All As are Bs.
2. X is not an A.
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3. Therefore, X is not a B. INVALID
J. 1. No As are Bs.
2. X is a not a B.
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3. Therefore, X is an A. INVALID
K. same as J.
L. 1. No As are Bs.
2. No Bs are Cs.
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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.