------------------------------------------------------------------------
Subject: MORE ABOUT VALIDITY
------------------------------------------------------------------------
A student writes:
| Consider an argument where some premises may be true and some may be
| false, but the conclusion is false. From my understanding that would
| be a valid argument because some of the premises are false and the
| conclusion is false, therefore the argument is valid. Is this approach
| correct?
An argument is valid if and only if it is *necessarily* "truth-preserving",
i.e., if & only if it's *impossible* for all premises to be true but the
conclusion false.
Note that this has nothing at all to do with whether any of the premises
actually *are* true or false; it's a "what if" kind of situation.
So you can have an argument with false premises and a false conclusion
that's valid, and you can have one like that that's invalid.
Here's a valid one:
All cats are fish.
All fish can fly.
.'. All cats can fly.
Here, everything's false, but the argument is valid.
It's valid because it has the form:
All Ps are Qs.
All Qs are Rs.
.'. All Ps are Rs.
and there's no way for a P to be a Q, and a Q to be an R, without having
the P be an R. I.e., it's impossible for the premises to be true and
the conclusion to be false.
Here's an invalid one:
All cats are fish.
All cats can fly.
.'. All fish can fly.
Again, everything's false; moreover, the argument is invalid.
It's invalid, because it has the form:
All Ps are Qs.
All Ps are Rs.
.'. All Qs are Rs.
and an example of an argument with this form that has true premises and
a false conclusion is this one:
All cats are mammals.
All cats purr.
.'. All mammals purr.
So, it's *possible* for this argument (form) to have true premises and a
false conclusion; hence, it's not valid.
Missing premises can *sometimes* make an invalid argument valid,
as we saw in PP1. In other cases of invalid arguments, no missing
premises will help; for an example, look at the invalid argument above.
One more point: An argument with *inconsistent* premises
(i.e., premises that contradict each other) is always valid(!),
because it's impossible for it to have all true premises with a
false conclusion, and that's because it's impossible for it to have
all true premises, period. Of course, such an argument cannot be
sound.