If the condition specified by the definition does not hold, then the argument is invalid. A consequence of these definitions is that the following cases of valid arguments are all possible:

1 The premises are all (actually) true, and the conclusion is (actually) true.

2 The premises are all (actually) false, and the conclusion is (actually) false.

3 The premises are all (actually) false, and the conclusion is (actually) true.

4 Some of the premises are (actually) true, some (actually) false, and the conclusion is (actually) true.

5 Some of the premises are (actually) true, some (actually) false, and the conclusion is (actually) false.

The only case in which an argument cannot be valid is the case when the premises are all (actually) true, but the conclusion is (actually) false. For if that is so, then obviously there is a

2 This definition has the consequence that if any premise of an argument is a necessary falsehood, or if the conclusion is a necessary truth, then the argument is valid (a necessary falsehood is a proposition that could not possibly be true; a necessary truth is a proposition that could not possibly be false). In such cases the premises may be entirely irrelevant to the conclusion. For example, There is a married bachelor, therefore the moon is made of green cheese1 is valid, as is 'The moon is made of green cheese, therefore there is no married bachelor1. Our definition, then, is quite useless as a guide to reasoning, where necessary truths and necessary falsehoods are concerned. We believe this a reasonable price to pay, for the alternative - a definition of validity whereby an argument is valid by virtue of its form - is too difficult, for our purposes, to apply profitably to ordinary language. Further, it is really very seldom that necessary truths or falsehoods figure as the conclusions or premises of arguments encountered ordinarily.

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