Second-order predicate
Encyclopedia
In mathematical logic
, a second-order predicate is a predicate that takes a first-order predicate
as an argument. Compare higher-order predicate.
The idea of second order predication was introduced by the German
mathematician and philosopher Frege
. It is based on his idea that a predicate such as "is a philosopher" designates a concept, rather than an object
. Sometimes a concept can itself be the subject of a proposition, such as in "There are no Albania
n philosophers". In this case, we are not saying anything of any Albanian philosophers, but of the concept "is an Albanian philosopher" that it is not satisfied. Thus the predicate "is not satisfied" attributes something to the concept "is an Albanian philosopher", and is thus a second-level predicate.
This idea is the basis of Frege's theory of number
.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, a second-order predicate is a predicate that takes a first-order predicate
First-order predicate
A first-order predicate is a predicate that takes only individual constants or variables as argument. Compare second-order predicate and higher-order predicate.-See also:*First-order predicate calculus...
as an argument. Compare higher-order predicate.
The idea of second order predication was introduced by the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician and philosopher Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
. It is based on his idea that a predicate such as "is a philosopher" designates a concept, rather than an object
Concept and object
In the philosophy of language, the distinction between concept and object is attributable to the German philosopher Gottlob Frege.According to Frege, any sentence that expresses a singular thought consists of an expression that signifies an Object together with a predicate In the philosophy of...
. Sometimes a concept can itself be the subject of a proposition, such as in "There are no Albania
Albania
Albania , officially known as the Republic of Albania , is a country in Southeastern Europe, in the Balkans region. It is bordered by Montenegro to the northwest, Kosovo to the northeast, the Republic of Macedonia to the east and Greece to the south and southeast. It has a coast on the Adriatic Sea...
n philosophers". In this case, we are not saying anything of any Albanian philosophers, but of the concept "is an Albanian philosopher" that it is not satisfied. Thus the predicate "is not satisfied" attributes something to the concept "is an Albanian philosopher", and is thus a second-level predicate.
This idea is the basis of Frege's theory of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
.