Els said:
.... snip
I guess I was puzzled by you saying that 2 was false when you
were tackling the argument - because it is irrelevant to the
argument that it is false.
Anyway, there are interesting and tricky notions surrounding
these types of considerations. Validity is only one type of
goodness in arguments (just as it it is in website pages) but it
is what arguments are best at. Truth is something else
altogether.
Usually, a person will usefully (educatively) advance an argument
if he is confident that his audience is likely to assent to the
premises. The truth of the premises in any final argument from
one person to another is taken for granted, The point is to lead
the audience to see that the conclusion results from what they
already accept.
There are many exceptions to this description. For example, in
the teaching of argument to people, an example is often given,
there being no assumption of truth in either the premises or the
conclusion. Nor in someone who is considering various arguments
so as to work out what might be worth exploring further. And,
curiously enough, in the argument by reductio, this is not so
either. More on this in a moment.
While it is true that everyone has a good enough grasp of the
ideas of truth and falsity, it is not the case that these, when
combined with the notion of argument, have commonsensical
relations. It is therefore not surprising that there can be some
confusion about them.
A valid argument, to use a concept in perhaps a stricter than
commonsensical way, is an argument whose premise(s) entail the
conclusion(s). It is not one where the conclusion is true, or one
where the premises are true. It is one where, if the premises are
true, the conclusion must be true. (There are some equally strict
but weaker types of validity and implication in formal logical
systems but I am not talking about these).
Validity is purely a relation between statements and there are
many assertions (including the conclusion) that can be false
while yet the argument remains valid. (Just as a website can pass
a validation test and yet have every other kind of fault)
There is only one particular arrangement of truth and falsity in
a valid argument that is ruled out, namely the combination of
true premises with false conclusion. Any argument that is known
to have true premises with a false conclusion is thereby known to
to be invalid. It is a bad argument by being invalid. The
conclusion does not follow from the premises.
The condition of an argument being valid is simply this: if the
premises are true, the conclusion must be. This says nothing
about the truth of either the premises or the conclusion. How to
know the truth of the premises is entirely out of the scope of
any one argument being assessed.
A reductio is a rather special sort of argument, it is a sort of
meta argument in which people are invited to look at how more
normal arguments within play themselves out.
If I thought anyone in the whole world was still awake I would go
on... but even I have limits when I hear snoring... said:
As for the term 'reductio ad absurdum', I didn't know it, but it
sounds like a good term for the concept. Would that be the same thing
as that mysterious mathematical calculation that (by sneaking a
non-truth into the equation) seems to prove that 1 = 0?
One can show the most absurd conclusions by including what is not
true in the premises. A reductio is more strictly an exhibition
of how an assumption or set of assumptions leads to a
contradiction.
