Is there any empirical evidence that doing math actually strengthens 
your logical skills?  It's the kind of claim you hear tossed around a 
lot but I'm wondering if there are studies showing that it's true. 

Of course I'm not talking about the fact that doing a particular kind 
of math problem over and over will make you better at solving that 
particular type of problem.  I know that if you do a lot of calculus, 
you'll get better at calculus.  But will it make you better at solving 
the type of logical/mathematical problem that you've never seen before? 

A completely rigorous study would be one in which: 
1) You have one group of test subjects practice on the kinds of math 
problems that you think will make them smarter.  The control group 
doesn't engage in any kind of extra practice. 
2) At the end of the practice period (which could be an hour, or 
several weeks, whatever), you give both groups a set of math problems 
that are completely different from the kind that you gave the test 
subjects during the practice period, and are also completely different 
from any type of math problem that you would normally encounter in 
school. 

Then you see if the test subjects who did the math practice, score 
better than the control group. 

Has a study like that ever actually been done?

Practicing math could help you solve various types of logical
problems, if, you apply the mathematical theories and concepts that
you learn to the problem. Math could also help strengthen your logical
skills, by helping you learn how to think in a logical way. For
example, if you take two people, and teach one person philosophy and
the other person math for one month, and then give each one of them a
logic problem, the person that studied philosophy will apply it to the
problem, while the other person would look for a mathematical, or
logical solution to the problem. The person who studied math, would
most likely get the correct answer.

Mathematics don't necessarily improve logical skills, but since you
are forced to use logic to solve math problems, you will start to
unconsciously look for logical solutions to problems.