Nonfirstorderizability

From Academic Kids

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

The term was coined by George Boolos in "To Be is to be the Value of a Variable (or to be some Values of Some Variables)," reprinted in his Logic, Logic, and Logic. He 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 and also dicussed by Quine is:

(1) Some critics admire only one another.

Consider the sentence:

(2) Some of Fianchetto's men went into the warehouse unaccompanied by anyone else.

Quine argued that this could be captured in first-order logic as follows:

(3) (There was at least one man x such that)(x went into the warehouse and x was Fianchetto's man, and (For every man y)(If y accompanied x then y was one of Fianchetto's men)).

(3) <math>\exists x \mathrm{Man}(x) \wedge \mathrm{WentIntoWarehouse}(x) \wedge \mathrm{FianchettosMan}(x) \Rightarrow ((\forall y \mathrm{Man}(y) \wedge \mathrm{Accompanied}(y, x)) \Rightarrow \mathrm{FianchettosMan}(y)).<math>

Boolos, however, argues that "anyone else" should--at least, in some contexts--be read as referring not to "anyone who wasn't one of Fianchetto's men" but rather to "anyone who wasn't one of them", i.e. the "some men" referred to in the first half of the sentence. Since "these men" have not been given any distinguishing predicate (keep in mind that they may not have been the only ones to enter the warehouse, and presumably are not the only men of Fianchetto's), they can be referred to collectively and exclusively only by some sort of quantification. This can be done by speaking of (most commonly) the set of them, (more rarely) the mereological fusion of them, or (as Boolos urges, on the grounds that it better captues the intuitive meaning), by reading "some men" as a second-order quantification. Hence:

(4) (There were some men, X, (of whom there was at least one, x), such that)(For every man y)(if y was among X, then (y went into the warehouse and (for every man, z)(if z accompanied y then z was among the X)))

If somebody could put standard logical symbolism in here it'd help a lot.

Remove the less readable form of (3) ... or keep both?

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