[EM] Richard's frontrunners example

[LAYTON Craig]

Okay, this is a bit of a side track to the discussion, but I'll try to
clarify anyway;

>It's unavoidable that we speak English, or some other language that
>we all speak, in order to discuss these things at all. I don't know
>whether or not it's true that "if" has the different meaning in
>mathematics that you say it has. I've never run across that different
>definition. I notice that you didn't actually give us a definition
>of "if".
>
>Would I be right to say that you mean that in a situation where
>a statement isn't saying anything about proposition y, it's assumed
>to be saying something true about proposition y? When isn't saying anything

>at all about proposition y? If mathematicians say that's
>so, does that mean it's so, or could it be that the statement still
>isn't saying anything about proposition y?

I'm assuming that conditionals are the same in mathmatics as in logic (and
if they're not, they should be).  At any rate, here goes;

The reason that if...then statements are different to common usage, is that
we typically use them to describe causal relationships between two things.
If A then B, is commonly understood to mean that A causes B.  In mathmatics
this isn't the case.  An if...then statement is a truth functional
statement, saying that whenever A is true, B is true.  If we have
established the truth of 'if A then B' (or we are positing the truth of that
statement), the following is true;
- if A is true, B is true
- if B is false, A is false

Because it is a truth functional statement, if A is false, the statement A
then B will be true (irrespective of the truth of B).  This is the tricky
bit - if A is false, it is an impossible contradiction for A to be true.  As
a result; if A is true then anything is true.  The statement 'if A then...'
will always hold. eg. if water can talk then I'm a banana - a true &
logically valid statement.

The statement if A then B can only be false if A is true and B is false.

I hope that helps.