Quine argued that second-order logic is 'set theory in sheep's clothing,' meaning purportedly second-order quantification over relations reduces to first-order quantification over sets.

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Reasons For

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Reason for

1.Second-order quantifiers range over all subsets of a domain, which requires set-theoretic machinery to define properly.
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2.Without reducing second-order logic to set theory, we cannot formally specify what counts as a valid second-order interpretation.
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3.First-order logic with set variables can express everything second-order logic expresses, making the apparent distinction ontologically redundant.
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Reasons Against

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Reason against

1.Second-order quantification has a distinct logical character—quantifying over functions or relations directly—independent of set representations.
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2.First-order set theory itself requires second-order principles (like comprehension) to formalize, so reduction creates circularity rather than clarity.
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3.Second-order logic captures structural properties (like categoricity) that first-order logic cannot, suggesting genuine logical expressiveness beyond set theory.
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Key Terms

Quantification (in logic)(the statement is saying the analysis covers all relevant factors): A way of talking about how many things satisfy a condition—like 'all,' 'some,' or 'none.'
Quine(as a proper name referring to the philosopher whose theory is being discussed): Willard Van Orman Quine was a 20th-century American philosopher who wrote about how we know things and how language works. In this statement, we're discussing one of his specific ideas about observation.
Relations (in logic)(as what you supposedly quantify over in second-order logic): Connections or patterns between things—like 'taller than' is a relation between two people, or 'is between' is a relation among three numbers.
Second-order logic(as used in mathematical logic): A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.
Set theory(as used in mathematics): A branch of mathematics that studies collections of objects (called 'sets') and the rules for how they relate to each other.
Sheep's clothing (metaphor)(as Quine's way of saying second-order logic is really just set theory in disguise): A figure of speech meaning something is disguised or pretending to be something it's not—like a wolf hiding as a sheep to fool people.
first-order quantification(Cited as a motivation for using reification operators instead of higher-type Montagovian semantics): Quantification ranging only over individuals, as opposed to higher-order quantification which ranges over predicates or functions of higher types

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Truth & Knowledge1 linked Philosophy of Language1 linked

First-order logic with set variables can express everything second-order logic e...First-order set theory itself requires second-order principles (like comprehensi...

Details

Type: claim
Perspectives: 2 (1 for, 1 against)
Edits: 1 edit