Truth Tables and Counter-examples

On the first exam, the portion that seemed to give a lot of you the most difficulty was the counter-example section. That is not very surprising. To construct a good counter-example can be quite difficult, and often takes quite a bit of time and ingenuity. 

The truth table method of determining validity and invalidity eliminates the need for ingenuity to determine validity or invalidity. The process is algorithmic. All you have to do is plug in the truth-values consistently and exhaustively, determine the main connectives, and check the rows accurately. If there is a row where each premise is true and the conclusion is false, then the argument is invalid; otherwise, it is valid. 

The indirect truth-table method does require a bit more ingenuity, but not much. All you really have to do is get very familiar with the nature of the logical connectives. Once you have mastered them, the indirect method is usually fairly simple. In fact, the indirect method will require at most three rows to determine validity or invalidity no matter how many statement letters are in the original argument.  

But, I want you to see that the truth table methods you are currently learning are not as disconnected from the counter-example method you learned earlier. Think of it this way: if an argument is valid, that means that there is no counter-example that you (or anyone) can construct that will have true premises and a false conclusion. But if an argument is invalid, then there is a counter-example that someone (perhaps only God or an angel) can construct with true premises and a false conclusion. In other words, when you determine by the truth-table method that an argument is invalid you have determined that there are ways to make all of the premises true and the conclusion false. Put differently, you have shown that there are ways to consistently interpret the premises and the conclusion such that the premises simply do not provide one with the best of reasons to accept the conclusion. If I were really mean, I'd make you attempt to construct counter-examples for each invalid argument. But I'm quite sweet.

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