I have discussed the importance of understanding logical form and how to create formal counterexamples. Understanding logic well is a lot easier when we know something about logical validity, and one way to better understand logical validity is to consider an argument that proves an argument to be valid. If we can know why an argument can be valid, then we can know more about logical validity in general. I will now produce a proof of logical validity here. It can take some time to understand the proof, so you might want to take your time to read it carefully.
Consider the following argument:
- If dogs are mammals, then they’re animals.
- Dogs are mammals.
- Therefore, dogs are animals.
This argument is logically valid. If the premises are true, then the conclusion must be true because it has a valid argument form. The argument form is the following:
- If A, then B.
- Therefore, B.
We can know that this argument is valid simply by knowing what “If A, then B” means. It means “If A is true, then B is true” or “B is true whenever A is true.” Since, A is true, B also has to be true because that’s what “If A, then B” means.
Nonetheless, we could construct a proof for this argument. Consider the following:
We can summarize the proof using the following words:
- We can assume the argument form is invalid. In that case the premises can be true and the conclusion can be false at the same time. Let’s assume the premises are true and the conclusion is false.
- If this assumption is impossible because it implies a contradiction, then we know the argument can’t be invalid (and must be valid).
- In that case “If A, then B” is true because it’s a premise, A is true because it’s a premise, and B is false because it’s a conclusion.
- In that case A must also be false because “if A (is true), then B (is true)” is assumed to be true, and B is assumed to be false. B is true whenever A is true, but B is false, so A must be false. (Consider the statement, “If dogs are mammals, then dogs are animals.” If we find out that dogs aren’t animals, then they can’t be mammals either. If the second part of a conditional statement is false, then the first part must be false.)
- Therefore, A is true and false.
- Therefore, the assumption that the argument form is invalid leads to a contradiction and must be false.
- Therefore, the argument must be valid.
We generally demand that our arguments are logically valid and we have an intuitive grasp about what it means for an argument to be logically valid. Validity isn’t sufficient to have a good argument, but it’s generally a very important element for constructing good arguments. Knowing more about logical validity beyond the intuitive level can help us achieve clarity and improve our thinking. Knowing why an argument is valid can help us achieve these goals.
Update (6/21/2011): I updated my proof that the argument form is valid because the other proof I gave was circular and therefore unpersuasive.