Expanded Notes on CPAs


You have learned how to spot a bad CPA, but how do you spot a good one?  Failing to find a counterexample is a start, but it doesn't really take you very far.  After all, you might search for hours and just be unlucky.  So it would help to have a positive strategy for determining the validity of a CPA.  This depends on the details of the arguments in question.  

Getting down to these details, CPAs generally consist of sets of sentences (viz., the reasons) that are connected by shared terms, phrases, or sentences (if at least one of them is complex).  Above, in argument (I), we see a CPA that involves three sentences which share the sentence pieces we have symbolized with "A" and "B".  The sharing of parts ensures that these sentences are about the same things.  Further, these terms or phrases are typically related in CPAs by terms that function to create a flow through the sentences.  These terms, or logical constants, include "if ... then", as in (I) above, but also "and", "or", "not", "just in case", as well as "all" and "some".  It is the "if ... then" that allows us to affirm "B" in our argument form, given "If A, then B" and "B".  These relate the pieces in the reason sentences so that when they are true, the conclusion must be true.  

For this to work, the conclusion must not go beyond the reasons.  If the conclusion is to follow from them with certainty, it must be guaranteed by them, and this won't be the case if it contains a word or phrase that isn't in the reasons or follows from the reasons.  In this case, the conclusion would depend on something that the reasons can't control, and so it could be false even when they are true.  Thus, if all of the substantive parts of the conclusion are contained in the reasons, then that is a good sign that the conclusion will follow from them, assuming (a) the argument flow set up by the reasons compels you to believe that the conclusion follows, and (b) you can't locate a counterexample.  

Still, this isn't foolproof.  Consider this argument:

  1. All humans are mortal.

  2. Chica is not human.

  3. Therefore, Chica is not mortal.

This looks like it might work, since the conclusion is fully connected to the reasons, and if you are impressed by that fact, you might not look very hard for a counterexample, leading you to conclude that the argument is good.  However, Chica is my cat and she is quite mortal, a fact that renders this argument quite bad.  It embodies what logicians call a fallacy, which is a pattern of reasoning that looks like it might work but generally leads you astray.  This particular fallacy is akin to the formal fallacy of denying the antecedent, which is what we do with (2).  However, (1) does not say that humans and only humans are mortal; many other things, including cats, are mortal as well.  For more on fallacies, see Chapter Seven of the UI Critical Thinking Handbook.

For still more detail on CPAs, also known as deductive arguments, see Chapter Six of the UI Critical Thinking Handbook, and in particular, Section III.2.