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Inverse View
It is not the case that Extensional sentences containing numerical singular terms, if true, are ontologically committed to numbers.
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Reasons For
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Reason for 1 of 2
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1.
Numerical singular terms function as nominalizations of predicates or quantifiers, not as genuine referring expressions that pick out objects.
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2.
Frege's context principle permits number-words to contribute to sentence truth conditions without requiring corresponding objects in the domain.
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3.
A sentence like '2+2=4' can be fully explained by its role in inferential practice without positing abstract objects as referents.
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Reason for 2 of 2
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1.
Paraphrase strategies (Field, Hellman) can restate all true arithmetic claims using second-order logic over physical regions, eliminating numerical singular terms entirely.
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2.
If true numerical sentences admit of nominalistically adequate paraphrases that preserve all empirical content, then ontological commitment to numbers is not mandatory but merely one interpretive choice.
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Reasons Against
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Reason against
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1.
Numerical singular terms are genuine singular terms.
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2.
Genuine singular terms, when occurring in true extensional sentences, refer to objects.
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3.
Numbers are the objects referred to by numerical singular terms in true extensional sentences.
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