Consider the following argument:
AvB Adam loves Eve or Adam loves Bertha.
~A Adam does not love Eve.
B Adam loves Bertha.
If you know, first of all, that either 'A' or 'B' is true, and in addition you know that 'A' itself is false; then clearly, 'B' has to be true. So from 'AvB' and '~A' we can conclude 'B'. We say that this argument is Valid, by which we mean that, without fail, if the premises are true, then the conclusion is going to turn out to be true also.
Can we make this idea of validity more precise? Yes, by using some of the ideas we have developed in the last three chapters. (Indeed one of the main reasons these ideas are important is that they will help us in making the notion of validity very precise.) Let us write out a truth table for all the sentences appearing in our argument:
We know that cases 1 through 4 constitute all the ways in which any of the sentences in the argument may turn out to be true or false. This enables us to explain very exactly what we mean by saying that, without fail, if the premises are true, then the conclusion is going to turn out to be true also. We interpret this to mean that in each possible case (in each of the cases 1 through 4), if the premises are true in that qe, then the conclusion is true in that case. In other words, in all cases in which the premises are true, the conclusion is also true. In yet other words:
To say that an argument (expressed with sentences of sentence logic) 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.