Solve the formula for what?
And also, when you say formula, do you mean you have something like y=sqrt( (a-b)^2 + (c-d)^2 + (e-f)^2)? As it's just an expression rather then an equation unless you have an equals sign somewhere.
Are a,b,c,...,f known constants?
If you mean solve the equation y=sqrt( (a-b)^2 + (c-d)^2 + (e-f)^2) for y, then that is 'solved' in that form already, and can't be simplified any further unless some of a,b,c,d,e,f are related. If you have specific values for those variables, then you can sub them in as you did in your post to get a particular number (e.g. sqrt(38) if a=10, b=5, c=4, d=2, e=6, and f=3).
I will mention I'm not familiar with the idea of a Osgood semantic differential however, and wikipedia didn't help much, so perhaps theres something I'm missing about your question. (Also, help desk is usually for tech help, and this is more of a math help thing so far as I can tell. Again, perhaps I'm missing something that makes it more tech-y, but it looks like this might be more at home in math land.)