Nonfirstorderizability

In formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in particular theories in 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.).

Examples
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 as follows. Substitute the formula (y = x + 1 v x = y + 1) for Axy. The result,
states that there is a nonempty set which is closed under the predecessor and successor operations and yet does not contain all numbers. Thus, it 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.
gollark: I only write malware ironically, and it's not very dangerous.
gollark: <@!356107472269869058> Writing malware is bees.
gollark: Shut what down?
gollark: Their free API lets me do 1000 calls a day and it has nice stuff like forecasts and maps.
gollark: OpenWeatherMap is pretty good!

See also

References
  1. Noonan, Harold; Curtis, Ben (2014-04-25). "Identity". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.


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