second-order predicate calculus

Also known as: second-order logic

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July 13, 2026

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The second-order predicate calculus is an extension of the first-order predicate calculus that allows for quantification over predicate variables in addition to individual variables. This means it can express statements about properties and relations themselves, not just about individuals. For example, it can express the idea that "every property has at least one instance".12

In contrast, the first-order predicate calculus, also known as the lower predicate calculus, only permits quantification over individual variables. This distinction is fundamental in formal logic, with higher-order calculi being more expressive but also leading to incompleteness, as proven by Kurt Gödel.23

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formal logic
Higher-order predicate calculi
In particular, in the second-order predicate calculus, quantification is permitted over both individual and predicate variables; hence, wffs such as (∀ϕ)(∃x)ϕx can be formed. This last formula, since it contains no free variables of any kind, expresses a determinate proposition—namely, the proposition that every property has at least one…
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In particular, in the second-order predicate calculus, quantification is permitted over both individual and predicate variables; hence, wffs such as (∀ϕ)(∃x)ϕx can be formed. This last formula, since it contains no free variables of any kind, expresses a determinate proposition—namely, the proposition that every property has at least one…
Read More

metalogic
Nonelementary logic and future developments
…are also studies, such as second-order logic and infinitary logics, that develop the model theory of nonelementary logic. Second-order logic contains, in addition to variables that range over individual objects, a second kind of variable ranging over sets of objects so that the model of a second-order sentence or theory…
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…are also studies, such as second-order logic and infinitary logics, that develop the model theory of nonelementary logic. Second-order logic contains, in addition to variables that range over individual objects, a second kind of variable ranging over sets of objects so that the model of a second-order sentence or theory…
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