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:
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,
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.
[edit] See also
[edit] References
- George Boolos (1984). "To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)". Journal of Philosophy 81: 430-49. Reprinted in Boolos, George (1998). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press. ISBN 0-674-53767-X.
