Nonfirstorderizability

In formal logic, nonfirstorderizability is the inability of a natural-language statement to be adequately captured by a formula of first-order logic. Specifically, a statement is nonfirstorderizable if there is no formula of first-order logic which is true in a model 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 by George Boolos in his paper "To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)". Quine 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 is the Geach–Kaplan sentence: "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: That this formula has no first-order equivalent can be seen by turning it into a formula in the language of arithmetic . Substitute the formula (y = x + 1 v x = y + 1) for Axy. The result, states that there is a set X with these properties: There are at least two numbers in X There is a number that does not belong to X, i.e. X does not contain all numbers. If a number x belongs to X and y is x + 1 or x - 1, y also belongs to X. A model of a formal theory of arithmetic, such as first-order Peano arithmetic, is called standard if it only contains the familiar natural numbers 0, 1, 2, ... as elements. The model is called non-standard otherwise. Therefore, the formula given above is true only in non-standard models, because, in the standard model, the set X must contain all available numbers 0, 1, 2, .... In addition, there is a set X satisfying the formula in every non-standard model.