Logic is defined simply as the art of reasoning well. Logic consists of arguments, and an argument is simply a set of two or more sentences, one of which is designated as the conclusion. An argument can be Valid, which is often defined as follows: “an argument is valid if and only if the conclusion follows logically and inescapably from the truth of the premises.” This is the definition I’ve heard from people like Dr. William Lane Craig, and in reflecting on it, I think it may be a very good definition of ‘validity’. However, this semester for the purposes of my Logic class, where we are looking at sentential logic, we have defined validity in such a way that there are many arguments which qualify as valid even when and where we would never consider countenancing them seriously. Validity, for the purposes of sentential Logic, is defined precisely as:
An argument is deductively valid if and only if it is not possible for all the premises to be true and the conclusion false
This definition is so broad that there are these special cases of valid arguments which are very curious. For instance, consider the following argument:
- The sky is brown
- Therefore, it is either raining in Boston or it is not raining in Boston
Here, since “it is not possible for all the premises to be true and the conclusion false” (namely because it is not possible for the conclusion to be false) the argument presented is valid.
Consider the following:
- Jane is taller than Anna
- Anna is taller than Cynthia
- Cynthia is taller than Jane
- Therefore, the Thesaurus is heavy
In this argument, “it is not possible for the premises to be true and the conclusion false” (namely because it is not possible for the premises to be true), and therefore the argument is valid.
Consider the following argument:
- Bod will not go to the office if Jane does
- Jane will not go to the office if Bob does
- Therefore, it is raining in Boston and it is not raining in Boston.
Since in this argument the conclusion is logically false, it it is possible for the premises to be true and the conclusion false. The argument is invalid, and it seems to be because the conclusion is not only false, but logically false, thus always false. Therefore it would seem as though, for any argument, if the conclusion is logically false we are dealing with an invalid argument… But we would be wrong.
Consider the following:
- It is raining in Boston
- It is not raining in Boston
- Jane will show up to the office
- Therefore it is both raining in Boston and it is not raining in Boston
In this argument we have a conclusion which is logically false, and yet it is not possible for the conclusion to be false and the premises true. Therefore, since the premises are inconsistent, the argument is always automatically valid. There are two ways for premises to be inconsistent: either two or more premises contradict each other, or else at least one of the premises is itself logically false (imagine, for instance, if we had the conclusion of the previous argument act as a premise).
This definition of validity is a curious one which I will have to ameliorate myself to over the course of the semester, but I think that I will likely want, by the end of the class, to retain the definition of validity proposed by Dr. Craig.