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Informal logic,nonfirstorderizability is the inability of a natural-language statement to be adequately captured by a formula offirst-order logic. Specifically, a statement isnonfirstorderizable if there is no formula of first-order logic which is true in amodel if and only if the statement holds in that model. Nonfirstorderizable statements 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 byGeorge Boolos in his paper "To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)".^[1]Quine argued that such sentences call forsecond-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 is theGeach–Kaplan sentence: "Some critics admire only one another."IfAxy is understood to mean "x admiresy," and theuniverse of discourse is the set of all critics, then a reasonabletranslation of the sentence into second order logic is:$${\displaystyle \mathrm{\exists }X((\mathrm{\exists }x\mathrm{\neg }Xx)\wedge \mathrm{\exists }x,y(Xx\wedge Xy\wedge Axy)\wedge \mathrm{\forall }x\mathrm{\forall }y(Xx\wedge Axy\to Xy))}$$In words, this states that there exists a collection of critics with the following properties: The collection forms a proper subclass of all the critics; it is inhabited (and thus non-empty) by a member that admires a critic that is also a member; and it is such that if any of its members admires anyone, then the latter is necessarily also a member.

That this formula has no first-order equivalent can be seen by turning it into a formula in the language of arithmetic. To this end, substitute the formula$(y=x+1\vee x=y+1)$ forAxy. This expresses that the two terms are successors of one another, in some way. The resulting proposition,$${\displaystyle \mathrm{\exists }X((\mathrm{\exists }x\mathrm{\neg }Xx)\wedge \mathrm{\exists }x,y(Xx\wedge Xy\wedge (y=x+1\vee x=y+1))\wedge \mathrm{\forall }x\mathrm{\forall }y(Xx\wedge (y=x+1\vee x=y+1)\to Xy))}$$states that there is a setX with the following three properties:

• There is a number that does not belong toX, i.e.X doesnot contain all numbers.
• The setX is inhabited, and here this indeed immediately means there are at least two numbers in it.
• If a numberx belongs toX and ify is eitherx + 1 orx - 1, theny also belongs toX.

Recall a model of a formal theory of arithmetic, such asfirst-order Peano arithmetic, is calledstandard if itonly contains the familiar natural numbers as elements (i.e.,0, 1, 2, ...). The model is callednon-standard otherwise. The formula above is true only in non-standard models: In the standard modelX would be a proper subset of all numbers that also would have to contain all available numbers (0, 1, 2, ...), and so it fails. And then on the other hand, in every non-standard model there is a subsetX satisfying the formula.

Let us now assume that there is a first-order rendering of the above formula calledE. If${\displaystyle \mathrm{\neg }E}$ were added to the Peano axioms, it would mean that there were no non-standard models of the augmented axioms. However, the usual argument for theexistence of non-standard models would still go through, proving that there are non-standard models after all. This is a contradiction, so we can conclude that no such formulaE exists in first-order logic.

There is no formulaA infirst-order logic with equality which is true of all and only models with finite domains. In other words, there is no first-order formula which can express "there is only a finite number of things".

This is implied by thecompactness theorem as follows.^[2] Suppose there is a formulaA which is true in all and only models with finite domains. We can express, for any positive integern, the sentence "there are at leastn elements in the domain". For a givenn, call the formula expressing that there are at leastn elementsB_n. For example, the formulaB₃ is:$${\displaystyle \mathrm{\exists }x\mathrm{\exists }y\mathrm{\exists }z(x\ne y\wedge x\ne z\wedge y\ne z)}$$which expresses that there are at least three distinct elements in the domain. Consider the infinite set of formulae$${\displaystyle A,B_2,B_3,B_4,\dots }$$Every finite subset of these formulae has a model: given a subset, find the greatestn for which the formulaB_n is in the subset. Then a model with a domain containingn elements will satisfyA (because the domain is finite) and all theB formulae in the subset. Applying the compactness theorem, the entire infinite set must also have a model. Because of what we assumed aboutA, the model must be finite. However, this model cannot be finite, because if the model has onlym elements, it does not satisfy the formulaB_m+1. This contradiction shows that there can be no formulaA with the property we assumed.

• The concept ofidentity cannot be defined in first-order languages, merely indiscernibility.^[3]
• TheArchimedean property that may be used to identify the real numbers among thereal closed fields.
• Thecompactness theorem implies thatgraph connectivity cannot be expressed in first-order logic.^{[clarification needed]}

• Definable set
• Branching quantifier
• Generalized quantifier
• Plural quantification
• Reification (linguistics)

• Printer-friendly CSS, and nonfirstorderisability by Terence Tao

• Logic

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