Nonfirstorderizability
Encyclopedia
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.).
A standard example, known as the Geach
–Kaplan
sentence, is:
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.
Formal logic
Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
, nonfirstorderizability is the inability of an expression to be adequately captured in particular theories in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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
George Boolos
George Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...
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
Second-order logic
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
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
Peter Geach
Peter Thomas Geach is a British philosopher. His areas of interest are the history of philosophy, philosophical logic, and the theory of identity.He was educated at Balliol College, Oxford...
–Kaplan
David Kaplan (philosopher)
David Benjamin Kaplan is an American philosopher and logician teaching at UCLA. His philosophical work focuses on logic, philosophical logic, modality, philosophy of language, metaphysics, and epistemology. He is best known for his work on demonstratives, on propositions, and on reference in...
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 (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.