Validity and Soundness

Chapter Two
Philosophy 404
Summer 1999

I. Validity

1. Validity is a property that arguments either have or fail to have. If they have it, they are valid; if not, they are invalid. For us, an argument is a sequence of sentences where the last one---the conclusion---purportedly follows from the sentences that precede it---the assumptions (or premises or reasons). The presence of certain words---the warranting connectives---identifies a sequence of sentences as an argument. Validity is then defined in the following way: an argument is valid just in case the conclusion of the argument must be true given the truth of the assumptions.
   
   
2. Validity is hypothetical in character, which is to say that the premises in an argument must be related to the conclusion of an argument in a way that is independent of the actual truth values of the sentences involved. Thus, you can know that an argument is valid without knowing anything at all about the truth of the sentences that constitute it. All of the sentences involved can be true and the argument invalid (i.e., truth is not sufficient), and all of the sentences involved can be false and the argument valid (i.e., truth is not necessary).
   
   
3. Put another way, the validity of an argument depends on the structure of the sentences involved and not on their content. Thus, you can tell whether an argument is valid even if you have no idea what the terms involved are. For example:
   
   
   1. If the R/A distinction is semantic, then the proposition is singular.
   2. The proposition is not singular.

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   3. Therefore, the R/A distinction is not semantic.
      
      

   Unless you've been brushing up on your philosophy of language, you probably don't know what the R/A distinction is, or what it is for a proposition to be singular; nevertheless, you can tell that this argument is valid. We could make this even clearer by introducing symbols for the sentences involved here: (1) If A, then B; (2) Not B; (3) Therefore, not A. Here there is no question that we do not know the content of these sentences, since we do not know what "A" and "B" stand for, and therefore we are unable to determine the truth values of these sentences; even so, we are able to tell that (3) must be true here if (1) and (2) are. The truth of a sentence depends on its content, but the validity of an argument depends on the structure of the sentences it involves.
   
   

4. The Counterexample Test for Validity: while holding the premises true, try to devise a situtation that forces the conclusion to be false; if you can do this, the argument is invalid; if you can't, then that is a sign that the argument is valid. (It is not a proof, though---can you explain why?) Keep in mind that you are not to use anything you might know in advance about the terms used in the sentences---remember that validity does not depend on content. If it helps, you can use Venn Diagrams to perform this test. Below you will find two examples of this test in action:
   
   

   A
   
   

   1. (a) All Kansans are Jayhawks
   2. (b) All Jayhawks loathe Quantrill. .

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   3. (c) All Kansans loathe Quantrill
      
      

   B
   
   

   1. (a) All Republicans are fans of Rush.
   2. (b) All Republicans voted for Dole.

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   3. (c) All fans of Rush voted for Dole.
      
      

   We will assess these arguments verbally.
   
   

   1. Verbal Analysis: Argument (A): Assume (a) and (b) are true; from (a), we know that any Kansan we choose is a Jayhawk; and so from (b), we know that this person, being a Jayhawk, loathes Quantrill; thus, any Kansan we choose loathes Quantrill Thus, any person we choose who makes both (a) and (b) true will make (c) true---there is no way to hold (a) and (b) true without forcing (c) to be true as well. Argument (B): Assume (a) and (b) are true; from (a), we know that any Republican we choose will be a fan of Rush, and from (b) we know that this randomly selected Republican voted for Dole. However, this only tells us something about (c) if only Republicans are fans of Rush, but nothing about (a) and (b) force this on us. Indeed, all we have to do is find one fan of Rush who is not a Republican, and who did not vote for Dole, and we have a counterexample: all Republicans are fans of Rush, along with this guy, so (a) is true; all Republicans voted for Dole, so (b) is true; but not all fans of Rush voted for Dole since this guy didn't. As a result, we can accept (a) and (b) as true and reject (c) as false, and this is all it takes to show that (B) is invalid.
      
      
5. Consider the following analogy. A good canal is one that will channel water from point A to point B, which is to say that if the water is put into the canal at point A, it will come out at point B. We can make this determination without actually putting water into it. Likewise, a valid argument is one that channels truth from the premises to the conclusion, which is to say that if the premises are true, then the conclusion will also be true; furthermore, as we have seen, we can determine whether an argument is valid even though we do not know what the truth values of the sentences involved are. In both cases, our evaluation is based on an inspection of the structure and not the content.
   
   

II. Soundness

An argument can be good at the level of structure if it is valid, but that is usually not all that we want. Good arguments are going to be arguments that take us from true assumptions to a true conclusion. These are not only valid, but sound. An argument is sound just in case it is (a) valid and (b) all of its assumptions are true.

1. Determination of the validity of an argument is a matter of logic, since all we need to do is examine the structures of sentences; determination of soundness, on the other hand, is non-logical since it requires truth, and this depends on the correspondence between the content of the sentences and the world. This shouldn't come as a surprise, though, since successful arguments usually depend on what is said.
   
   
2. Remember: you don't need to worry about the actual truth values of the sentences in testing an argument for validity, but you do need to worry about these when you are testing it for soundness.