[EM] Anthony & reductio ad absurdum issue

MIKE OSSIPOFF nkklrp at hotmail.com
Wed Mar 20 16:45:05 PST 2002

It's true that I don't have time for this. There are several voting-
system postings that I should be replying to instead. But I reserve
the right to reply to anything about me, no matter what I promised.
And Anthony still needs some help with reductio ad absurdum. So
Anthony will just have to retract all that praise that he heaped on
me for not replying.

Anthony said:

Well, this is amazing -- a message running over two hundred
lines (quoted in full below), and ending with "I don't have

I reply:

Not really amazing at all. If my reply had only been one brief paragraph, I 
wouldn't have had much reason to say that I didn't have
time for more such replies. The reason why I don't have time for
replies like that last one, Anthony, is that your messages that
I've been replying to are very long, and so the replies are time-
consuming. I don't have time for this reply either, but I'm going
to reply anyway.

Anthony continued:

There is no way a court is going to accept that "equal
protection" prohibits IRV. Courts don't take kindly to
manipulation of verbiage at the expense of intent.

I reply:

I guess I have to repeat that I make no claim to knowing what the
courts would say about that. One thing for sure is that when IRV
ignores some preferences, for no good reason (IRVists' "reasons"
could be debated separately), that can't be called equal fairness
to voters.

Suppose the government decided to reduce population by executing
a randomly chosen 10% of its population. That meets your standard
of equal protection, but executing some but not all people, with no
good reason why those people should be executed, is not going to
seem like equal fairness to those people.

Does the actual equal protection clause say that its intent is
only about uniform application of laws, rather than about arabitrarily
laws uniformly applied? I realize that, even in the 18th century,
equal opportunity and fairness wasn't really on the table for citizens
in general, but that doesn't mean that it wouldn't be quite feasible
for voters, then as well as now.

Again, I'm not claiming that the courts will do other than what
you claim, ok?

Anthony continues:

I mentioned parody and reductio ad absurdum, and used the latter
by applying your reasoning to plurality, showing that it
leads to an absurd conclusion (that plurality is
unconstitutional) and is therefore not valid reasoning.

I reply:

But below, you say that reductio ad absurdum should be worded in
terms of a contradiction, not an absurd or false conclusion. Which
is it?

Right. You said that "all voted pairwise preferences should be counted"
implies "everyone's favorite candidate should win". If I refer to that
again, I call it "Anthony's claim". You thought that Anthony's claim
could be used to make a valid reductio ad absurdum. That's why I said that 
you don't know what reductio ad absurdum means, and that is still
a reasonable conclusion.

Plurality and Approval & Condorcet reliably count every
voted preference, but they don't make everyone's favorite win. So
much for Anthony's claim that one implies the other.

Anthony continues:

suppose I should have explained that reductio is also applied
to derived inference rules, instead of assuming you knew
that. Anyway, you know now.

I reply:

What I said was that Anthony's claim can't be used in a valid
reductio ad absurdum. If you have some sort of inference rule that
establishes Anthony's claim, then I must have missed it when you
posted it.

Anthony continues:

Incidentally, your formulation of reductio is a bit off.
Formally, it's generally of the form:

If assuming -P allows us to derive a contradiction, then
P must be true.

I reply:

We find it worded both ways. Sometimes it's described as a way
of dispoving P. Sometimes as a way of disproving -P, to establish P.
Is that really an important difference?

If we speak of it as a way of disproving P, then of course any propositioin 
could be used as that P, including (-Q), where Q is
some proposition that we'd like to prove. I'm glad that I could be
helpful by explaining that to you.

Sometimes it's defined in
terms of P leading to a contradiction. Sometimes it's defined in
terms of P implying a proposition known to be false. Anthony, those
aren't different meanings for reductio ad absurdum. They're merely different 
wordings for saying it.

Better authorities than Anthony state it the way I stated it:

P is shown to be false if P implies a proposition known to be false.

I'd believe them before I'd believe Anthony that they're wrong,
because, for one thing, they can spell "reductio ad absurdum".

I'll go around gathering references if you really want me to, but
you know this is off-topic.

Anthony continues:

Often, Q is P, giving a simplified rule: If assuming -P
allows us to derive P, then P is true.

Note that in the latter case we derive P, _not_ known in
advance to be false (or true, for that matter), so your
formulation of reductio is clearly not quite right.

I reply:

If -P implies P, then -P implies (P& -P), a proposition known to
be false even if we have no idea whether or not P is true or false.

So saying that -P is false because it implies (P& -P) is a special
case of saying that -P is false because it implies something known to be 

I've sometimes seen that in a slightly more general form, where
P is shown to be false because it implies (Q& -Q).

Anthony continues:

could, of course, make the rule that says that the derivation
is completed when the constant "False" appears, and define
reductio that way, but it's a bit roundabout, since this is
always done by way of exhibiting a contradiction.) Thus the
standard statement of reductio is that if assuming -P leads
to a contradiction, then P is true.

I reply:

As I said, better better authorities than Anthony have worded it
in terms of a known false proposition instead of a contradiction.

But it's the same thing. If a contradiction is some statement that
Q is true and -Q is true, then it's a statement of the proposition
(Q& -Q), a proposition known to be false. And if, as you said,
falsity is always demonstrated by a contradiction, then, from what
I've said in this paragraph, both wordings mean the same thing.
But if falsity sometimes weren't assocated with a contradiction, then of
course we'd have to choose between the contradition wording and the
false proposition wording. The false proposition wording then would probably 
then be considered more useful.

Maybe an example would help:

If the proposition "Anthony is a mathematician" implies the
proposition "Anthony can spell "reductio ad absurdum without coaching
from Mike, and Anthony hasn't demonstrated ignorance of the meaning
of reductio ad absurdum", then the established falsity of the 2nd
proposition establishes the falsity of the 1st proposition.

Mike Ossipoff

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