2.4.2: Invalidity and Counter Examples

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Let's look at an example of an Invalid argument (an argument which is not valid):

AvB
A
B

Counterexample:

Case	A	B	AvB
1	t	t	t
2	t	f	t
3	f	t	t
4	f	f	f

I have set up a truth table which shows the argument to be invalid. First I use a '*' to mark each case in which the premises are all true. In one of these cases (the second) the conclusion is false. This is what can't happen in a valid argument. So the argument is invalid. I will use the term Counterexample for a case which in this way shows an argument to be invalid. A counterexample to an argument is a case in which the premises are true and the conclusion is false.

In fact, we can use this idea of a counterexample to reword the definition of validity. To say that an argument is valid is to say that any assignment of truth values to sentence letters which makes all of the premises true also makes the conclusion true. We reword this by saying: An argument is valid just in case there is no possible case, no assignment of truth values to sentence letters, in which all of the premises are true and the conclusion is false. To be valid is to rule out any such possibility. We can break up this way of explaining validity into two parts:

A Counterexample to a sentence logic argument is an assignment of truth values to sentence letters which makes all of the premises true and the conclusion false.

An argument is Valid just in case there are no counterexamples to it.

Now let us reexpress an of this using sentences of sentence logic and the idea of logical truth. Let us think of an argument in which X is the conjunction of an the premises and Y is the conclusion. X and Y might be very complicated sentences. The argument looks like this:

X
Y

I will express an argument such as this with the words "X. Therefore Y".
A counterexample to such an argument is a case in which X is true and Y is false, that is, a case in which X&~Y is true. So to say that there are no possible cases in which there is a counterexample is to say that in all possible cases X&-Y is false, or, in all possible cases ~(X&~Y) is true. But to say this is just to say that ~(X&~Y) is a logical truth. The grand conclusion is that

The argument "X. Therefore Y is valid just in case the sentence ~(X&~Y) is a logical truth.