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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.

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    – J D
    Jan 5 '20 at 20:39
  • Hartry Field explictly mentioned in SWN that he holds similar view as logical positivists' position on the analyticity of math including logic as mere tautological meaningless, so essentially he views any true statement other than scientific statements must come from math and logic alone. And you're right "if we have a mathematical theory, from this we cannot then draw conclusions about statements regarding non-mathematical objects". Non-mathematical statements can only be causally "entailed" by some body of nominalistically statable assertions... Oct 10 at 0:52
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See Principle C': Let A be any nominalistically statable assertion, and N any body of such assertions.Then A∗ isn’t a consequence of N∗ + S unless it is a consequence of N∗ alone.

If S is a mathematical theory, it "speaks about" mathematical objects while A∗ and N∗ do not.

Thus, IMO; we have to understand it as saying: in order to derive statements about the "nominalistic world" we have to use "nominalistic axioms" alone.

While this is equivalent to Principle C'': Let A be any nominalistically statable assertion. Then A∗ isn’t a consequence of S unless it is logically true ?

Because if we can prove it whitout nominalistic axioms at all, due to the fact that it cannot be consequence of the mathematical theory S, it must be "provable from logic alone".

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