There is a tendency for people to think that invalid arguments are false arguments. After all, we know that when a valid argument has true premises, it must have a true conclusion. As we can see, the notion of truth plays a role in the definition for validity. People then infer, incorrectly, that invalid arguments must have false premises and a false conclusion. We will see why this is incorrect when we discuss the following argument:
Sample paper
There is a tendency for people to think that invalid arguments are false arguments. After all, we know that when a valid argument has true premises, it must have a true conclusion. As we can see, the notion of truth plays a role in the definition for validity. People then infer, incorrectly, that invalid arguments must have false premises and a false conclusion. We will see why this is incorrect when we discuss the following argument:
All invalid arguments are non-arguments with with mood and figure AAA-1
Some arguments with mood and figure EEE-1 are not valid .
No arguments with mood and figure EEE-1 are arguments with mood and figure
AAA-1
We will start our paper by putting this argument in standard logical form. This argument has the following instance:
All arguments with mood and figure AAA-1 are valid (contraposition)
Some arguments with mood and figure EEE-1 are not valid .
No arguments with mood and figure EEE-1 are arguments with mood and figure
AAA-1
When we put this argument in standard logical form, it will have this form:
All P are M
Some S are not M
No S are P
This form is invalid. We can prove it is invalid by Salmon’s rules.
- M-1 ok
- S-1 xxx, P-2 ok
- 1=1 ok
We can also prove it is invalid by a Venn diagram. Shade regions 2 and 3. Place an x in region 1. We can prove that it is invalid by means of the following counterexample:
All poodles are dogs
Some animals are not dogs .
No animals are poodles.
A valid argument cannot have even one instance with true premises and a false conclusion. If we can find such an instance, then we know the form is invalid. But interestingly enough, this argument expresses no false statement. Each statement is true.
Consider the first premise: All arguments with mood and figure AAA-1 are valid. This premise is true. AAA-1 has this form:
All M are P
All S are M
All S are P
We can prove that this form is valid by both Salmon’s rules and a Venn diagram. By applying Salmon’s rules, we have the following:
- M-1
- S-2, P-0
- 0=0
Since all three of Salmon’s rules are met, this argument is valid. We can also prove the argument is valid by means of a Venn diagram. In our Venn diagram, the regions 1,2,4, and 7 would all be shaded in. For the conclusion to be true, regions 1 and 4 would be shaded, which they are.
Now consider the second premise: Some arguments with mood and figure EEE-1 are not valid. This is true. In fact, no argument with mood and figure EEE-1 is valid. EEE-1 has this form:
No M are P
No S are M
No S are P
We can prove that this argument is invalid by means of a Venn diagram and a counterexample. If we constructed a Venn diagram, regions 4,5,6 would be shaded in. For the conclusion to be true, regions 2 and 5 would have to be shaded. Since region 2 is not shaded, the argument is invalid. We can also prove this form is invalid by a counterexample.
No cats are poodles
No dogs are cats .
No dogs are poodles.
Since a valid argument will not have a single instance with true premises and a false conclusion, we know that EEE-1 is invalid.
Now consider the conclusion: No arguments with mood and figure EEE-1 are arguments with mood and figure AAA-1. This is also true. Take a look at the two forms side by side:
No M are P All M are P
No S are M All S are M
No S are P All S are P
While these forms have the same figure, they have different statements and are thus different.
Our original argument was invalid. Yet it did not have false premises and a false conclusion. Invalid arguments need not have false premises or a false conclusion. Thus, validity is not equivalent to truth. And invalidity is not equivalent to falsity.
© September, 2008
C. Bolton
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