MIKE OSSIPOFF wrote:
> Depending on whether we accept Blake's claim, and Richard's claim
> that's based on it, Richard is saying one of the 2 following things:
That's true...
> A. He agrees that if i & j are frontrunners, then if there's a tie
> it's between them, but he denies that if it's true that i & j are
> frontrunners then it's true that if there's a tie it's between them.
No, not that one.
> B. He agrees that if i & j are not frontrunners, then if there's a tie
> it's not between them, but he denies that if it's not true that i & j
> are frontrunners, then it's not true that if there's a tie then it's
> between them.
Yes, that is what I'm saying. The paragraph can be sorted out
symbolically (f means i & j are front-runners, t means a tie, tij
means a tie between i & j, -> is logical implication, ! means not,
+ means logical or, * means logical and):
Agree: !f -> (t -> !tij)
Deny: !f -> !(t -> tij)
!f -> !(!t + tij) implication equivalent
!f -> (t * !tij) deMorgan equivalent
f + (t * !tij) implication equivalent
The last equivalent of the denied part says that either i & j are
front-runners, or we have a tie and it is not between i & j. This is
false whenever i & j are not front-runners and there is no tie.
> Is that self-contradiction, or what?
It is merely a self-consistent statement.
> That's because he claims that if a statement says nothing about a
> proposition, then it's saying something true about it. Does that sound
> convincing enough to accept the absurd self-contradiction in B above?
I don't claim that at all. The statement in question (b in Mike's previous
post) says that everywhere p is true, q is true. So it is definitely saying
something about proposition q.
> Still accepting Richard's notion of how Bart's Pij definition is worded
> (I'll get to that later), I took it to have a different meaning.
> To me it meant "In the event that there should be a tie, it would be
> between i & j."
>
> In other words, I took it as a general statement, one that doesn't
> become paradoxical and meaningless nonsense if there's no tie.
>
> That's why, yesterday I worded it: "If there is/were a tie, it is/would
> be between i & j. (use the verb before the "/" if there is a tie, and
> the verb after the "/" if there isn't a tie).
The subjunctive form doesn't help here. That form is used to describe
things outside the universe of discourse. So you can't really apply
logical analysis to a subjunctive statement.
>But, even if we accept Richard's apparent notion about Bart's wording,
>and the conclusions that Richard draws from that notion, the only
>difference between my Pij definition and Bart's is when there's no
>tie--a situation of no interest for strategy. Bart's Pij and mine
>would result in the same Approval strategy.
Conditional probability isn't affected by affirming the antecedent.
If Pij is the conditional probability that any tie is between i and j, then
if you affirm that there is a tie, Pij is still the probability that any tie is
between i and j.
>But now let's take a closer look at what Bart actually said:
>"Pij is the probability, given that there's a tie, that the tie is
>between i & j."
>Richard seems to believe that Bart said:
>"Pij is the probability that, given a tie, it's between i & j."
Oddly enough, I'm really not interested in any Pij as defined
by the second statement, if we understand the "given" phrase
to belong to the second half. Because then it's the probability
of a conditional, rather than a conditional probability. But if we
take the commas to indicate that "given" is subordinate to the
whole clause about Pij, then it's the same as the first one.
>But that isn't the same thing. What follows "that" is what the
>probability is about. The "given..." clause is not in the "that..."
>clause. The "given..." clause doesn't refer to what the probability
>is the probability of. The "given..." clause is part of the sentence's
>main clause. It modifies the statement about Pij being the probability
>that...
If you are trying to say that the probability of the conditional is
not equal to the conditional probability, you'll get no argument
from me. But the probability of the conditional is irrelevant to
strategy, as far as I can tell.
>In other words, Bart's wording means this:
>If there's a tie, then Pij is the probability that it's between i & j."
>Certainly Richard's misreading of Bart's wording is understandable.
Sounds more like you misread what I'm saying. I've been saying
all along that we should use the conditional probability.
>The application of ideas about conditional probability has been
>controversial. I didn't expect that I'd enounter that here, but I
>guess it was inevitable.
I'm unaware of any general controversy over conditional
probability. It's used successfully in a lot of fields. And it's
a natural tool for discussing strategy based on statistical
information.
I think it's time to move off this lengthy digression. The original topic
of strategy is what we should be discussing. For instance:
1. We know what the ZI and non-ZI strategy matrices look like for
approval. The strategy matrix for plurality is the same, but the
application is different. What conclusions can we draw from those?
2. What does a strategy matrix look like for Condorcet? For Borda?
For IRV? What conclusions can we draw from those?
-- Richard
