[EM] Richard's frontrunners example

[Richard Moore]

LAYTON Craig wrote:

> > > So you're saying that b is true when it isn't making any prediction
> > > about Y. When X is false, b says nothing true about whether Y is
> > > true, and says nothing false about whether Y is true. And you claim
> > > that b is true when it isn't taking a position about Y.
> >
> >That's the way "if" is used in mathematics.  I hope that we choose
> >mathematical usage over common usage because common usage is not
> >well-defined, and tends to be less convenient.
>
> At least in the field of logic, there is some doubt over whether
> conditionals (if statements) should actually be treated this way.  The
> consensus view, as I understand it, is that it is convenient, but not
> necessarily correct, especially if it is applied to an argument or statement
> outside pure logic or mathmatics.  eg from the statement 'the box is not
> red' we can deduce 'if the box is red then the box is green'.

"The box is not red" does imply that the conditional "if the box is red then
the box is green" is true. However we all know that the statement "if the box
is red then the box is green" cannot be *universally* true. It is only true for

those cases where the box is not red, and for those cases where the box is
green. Some people prefer to say that the statement is "consistent" rather
than "true".

Of course, a statement like "If the box is red then the box is green" is not
particularly useful if it cannot be applied to red boxes.

I know how counterintuitive it all sounds. But this convention can be
justified; here are a couple of ways to look at it.

1. Boolean logic -- If the conditional "If A then B", or "A implies B",
is true, then either B must be true or A cannot be true., So the conditional
is logically equivalent to "B or not A".

2. Venn diagrams -- Draw a big circle, and label it B. Draw a smaller
circle and label it A. "If A then B", or "A is a subset of B", is true if
and only if the A circle lies entirely within the B circle. That is, if you
are inside A you are also inside B. "If A then B" is true if B is true for
all points for which A is true.

Now, imagine that part (or even all) of A is outside of B. Then "If A
then B" is not universally true. However, draw a line dividing your
diagram in two, so that the part of A that lies outside B is entirely on
one side of that line; call that side of the line C. Now, for the special
case of "not C" (a reduced universe of discourse) you can again
assert "If A then B". And that is also the way the red/green box
paradox is resolved.

One last step is to apply probability to the statement "If A then B"
when the statement is not universally true. When we say the
statement has a 90% probability, we are saying that there is a 90%
chance of an outcome that lies within the region for which the
conditional statement is true (or again, for which it is consistent, if
you prefer). That means there is a 10% chance of an outcome
where A is true and B is false.

 -- Richard