How to Prove an Argument Invalid by a Counterexample

How to Prove an Argument Invalid by a Counterexample An argument is invalid if the conclusion doesnt follow necessarily from the premises.  Whether or not the premises are actually true is irrelevant.  So is whether or not the conclusion is true.  The only question that matters is this: Is it  possible  for the premises to be true and the conclusion false?  If this is possible, then the argument is invalid. Proving Invalidity: a Two-step Process The counterexample method is a powerful way of exposing what is wrong with an argument that is invalid.  If we want to proceed methodically, there are two steps: 1) Isolate the argument form; 2) Construct  an argument with the same form that is obviously invalid. This is the counterexample. Lets take an example of a bad argument. Some New Yorkers are rude.Some New Yorkers are artists.Therefore Some artists are rude. Step 1: Isolate the Argument Form This simply means replacing the key terms with  letters, making sure that we do this in a consistent way.  If we do this we get: Some N  are RSome N are ATherefore some A are R Step 2: Create the counterexample For instance: Some animals are fish.Some animals are birds.Therefore some fish are birds This is what is called a substitution instance of the argument form laid out in Step 1.  There is an infinite number of these that one could dream up.  Every one of them will be invalid since the argument form is invalid.  But for a counterexample to be effective, the invalidity must shine forth.  That is, the truth of the premises and the falsity of the conclusion must be beyond question. Consider this substitution instance: Some men are politiciansSome men are Olympic championsTherefore some politicians  are Olympic champions. The weakness of this attempted counterexample is that the conclusion isnt obviously false.  It may be false right now, but one can easily imagine an Olympic champion going into politics. Isolating the argument form is like boiling an argument down to its bare bonesits logical form.  When we did this above, we replaced specific terms like New Yorker with letters.  Sometimes, though, the argument for is revealed by using letters to replace whole sentences or sentence-like phrases. Consider this argument, for instance: If it rains on election day the Democrats will win.It wont rain on election day.Therefore the Democrats wont win. This is a perfect example of a fallacy known as affirming the antecedent.  Reducing the argument  to its argument form, we get: If R then DNot RTherefore not D Here, the letters dont stand for descriptive words like rude or artist. Instead, they stand for an expression like, the Democrats will win and it will rain on election day.  These expressions can themselves be either true or false.  But the basic method is the same. We show the argument s invalid by coming up with a substitution instance where the premises are obviously true and the conclusion is obviously false.  For instance: If Obama is older than  90, then hes older than 9.Obama is not older than 90.Therefore Obama is not older than 9. The counterexample method is effective at exposing the invalidity of deductive arguments.  It doesnt really work on inductive arguments since, strictly speaking, these are always invalid.