For Field, the following is 'perfectly obvious', but I would like confirmation that I understand it completely.
- Let A be a nominalistically statable assertion.
- Let A* be the assertion that results by restricting each quantifier of A with the formula not M(xi) (for the appropriate variable 'xi') (where M(x) is the predicate 'x is a mathematical entity').
- Let S be any mathematical theory.
Then Principle C'' is as follows: let A be any nominalistically statable assertion. Then A* isn't a consequence of S unless it is logically true.
My understanding of what this is saying is that if we have a mathematical theory, from this we cannot then draw conclusions about statements regarding non-mathematical objects. Please correct me if I am wrong on that. I am also still slightly confused about the wording "unless logically true" and the meaning behind it.