In some good arguments, the conclusion is said to follow from the premises. However, this commonsense notion of following from is hard to pin down precisely. The conclusion follows from the premises only when the content of the conclusion is related appropriately to the content of the premises, but which relations count as appropriate?
To avoid this difficult question, most logicians instead discuss whether an argument is valid. Calling something “valid” can mean a variety of things, but in this context validity is a technical notion. Here “valid” does not mean “good,” and “invalid” does not mean “bad.” This will be our definition of validity:
An argument is valid if and only if it is not possible that all of its premises are true and its conclusion false.
Alternatively, one could say that its conclusion must be true if its premises are all true (or, again, that at least one of its premises must be false if its con- clusion is false). The point is that a certain combination—true premises and a false conclusion—is ruled out as impossible.
The following argument passes this test for validity:
(1) All senators are paid. (2) Sam is a senator.
(3) Sam is paid. (from 1–2)
Clearly, if the two premises are both true, there is no way for the conclu- sion to fail to be true. To see this, just try to tell a coherent story in which every single senator is paid and Sam is a senator, but Sam is not paid. You can’t do it.
Contrast this example with a different argument:
(1) All senators are paid. (2) Sam is paid.
(3) Sam is a senator. (from 1–2)
Here the premises and the conclusion are all in fact true, let’s assume, but that is still not enough to make the argument valid, because validity
concerns what is possible or impossible, not what happens to be true. This conclusion could be false even when the premises are true, for Sam could leave the Senate but still be paid for some other job, such as lobbyist. That possibility shows that this argument is invalid.