Why the term “Valid” is Misleading

     In logic, the concept of validity is an interesting one. What makes this concept especially hard to grasp is the idea that for an argument to be valid, it does not need to have true premises. 

An argument usually takes the form of several premises, and a conclusion: 
     Premise 1
     Premise 2
     Premise 3
     …
     Conclusion

If an argument is valid, it is impossible for its premises to be true and it’s conclusion to be false at the same time. In other words, the truth of the premises guarantees the truth of the conclusion. 

Argument A:
Premise 1: If I exercise every day, I will eventually get stronger.
Premise 2: I exercise every day.
Conclusion: I will eventually get stronger. 

Notice that the above argument is valid because if the premises are all true, then it is impossible for the conclusion to be false. 

There is also a second way to understand the definition of validity, and it is this: an argument is valid if the set consisting of an argument’s premises and the negation of its conclusion (also known as the counterexample set) is inconsistent. A set of sentences is inconsistent if they cannot all be true together in some possible world or circumstance.

Counterexample set to Argument A: (This is the counterexample set to example A)
Premise 1: If I exercise every day, I will eventually get stronger.
Premise 2: I exercise every day.
Conclusion: It is not the case that I will eventually get stronger. 

Notice that these sentences cannot all be true together (they are inconsistent). Since the counterexample set to Argument A is inconsistent, Argument A is valid. 

With me so far? Here’s where things start to get tricky.

An argument is always valid if it has inconsistent premises. Since it is impossible for all the premises to be true together, it is impossible for all the premises to be true while the conclusion is false simultaneously. This means that if you have a bunch of claims (premises) that cannot possibly all be true together, you can assert anything you like in the conclusion, and the argument will always be valid. 

Argument B:
Premise 1: The sky is blue.
Premise 2: The sky is not orange.
Conclusion: The moon is made out of blue cheese!

Argument B is valid because it has inconsistent premises. 

The second way that an argument can always be valid is if it is impossible for the conclusion to be false. This is because there is no possible scenario where the premises are true and the conclusion to be false at the same time (because the conclusion is guaranteed to always be true!). Common examples of logical truths include “a=a” and “P or not P.” Thus, you can pretty much assert anything you want in the premises, and as along as the conclusion is a logical truth, the argument is always valid.

Argument C:
Premise 1: The fish kingdom is going through recession.
Premise 2: Unicorn blood is black.
Premise 3: Harry Potter is real.
Conclusion: a=a

Argument C is a valid argument. 

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