contraposition and 'most'

Is this odd, or am I just under-caffeinated at the moment?

The principle of contraposition (= the equivalence of 'If P then Q' and 'If not-Q then not-P') doesn't hold when the quantifier is 'most'. That is, 'Most As are Bs' is not equivalent to 'Most non-Bs are non-As'.

I take 'Most As are Bs' to mean: the number of things that are both A and B is greater than the number of things that are A but not B.

A minute or two of drawing (unless I've messed up somewhere) will get you a picture where
(1) the number of ABs > the number of A non-Bs
is true, but
(2) the number of non-A non-Bs > the number of A non-Bs
is false.

Again, maybe this point is as obvious as 2+3=5. But I am covering simple inductive arguments in my critical thinking class at the moment, trying to figure out which ones are good, and I had never thought about this case before, but it means the inductive analogue of quantified modus tollens (Most As are Bs, x is not B, Thus x is not A) is no good. And that surprised me, since the analogue of modus ponens is perfectly fine.


P.D. Magnus said...

I hadn't noticed that before, either. But part way through your post I was thinking of cases where there is only a small chance of some factor causing an outcome, but there is not other way to produce the outcome; as in:

Most syphilitics are paresis-free.
Al is not paresis-free.

Jonathan Livengood said...

Let the inference under consideration be the following:

If P(B|A) > 1/2, then P(~A|~B) > 1/2.

Under what conditions is the inference a good one? It looks to me like the following are sufficient:

1. If A is a subset of B, then the inference is valid.

2. If B is a subset of A and P(B) < 1/3, then the inference is valid.

3. If A and B overlap and 1 > 2P(A+B) - P(B), then the inference is valid.

Assuming I have these right (do I?), are these all of the cases in which the inference holds? How likely are these to hold and to be recognizable when they hold?