Nonfirstorderizability

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In formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in standard first-order logic. Nonfirstorderizable sentences are sometimes presented as evidence that first-order logic is not adequate to capture the nuances of meaning in natural language.

The term was coined by George Boolos in his well-known paper "To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)." Boolos argued that such sentences call for second-order symbolization, which can be interpreted as plural quantification over the same domain as first-order quantifiers use, without postulation of distinct "second-order objects" (properties, sets, etc.).

A standard example, known as the Geach-Kaplan sentence, is:

Some critics admire only one another.

If Axy is understood to mean "x admires y," and the universe of discourse is the set of all critics, then a reasonable translation of the sentence into second order logic is:

\exists X ( \exists x Xx \land \forall x \forall y (Xx \land Axy \rightarrow x \neq y \land Xy))

That this formula has no first-order equivalent can be seen as follows. Substitute the formula (x = 0 v x = y + 1) for Axy. The result,

\exists X ( \exists x Xx \land \forall x \forall y (Xx \land (x = 0 \lor x = y + 1) \rightarrow x \neq y \land Xy))

is true in all nonstandard models of arithmetic but false in the standard model. Since no first-order sentence has this property, the result follows.

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