The language sketched above is better referred to as first-order predicate calculus, as the language only quantifies over (first-order) individuals. A stronger language, second-order predicate calculus, can therefore be constructed by allowing quantification over predicates —that is, second-order sets of individuals. The language is the same as first-order predicate calculus, with the addition of the second order quantifiers ∃X (“there is at least one predicate X, such that …”) and ∃X (“for all predicates X, it is the case that …”). Even higher-order logics can similarly be constructed by allowing quantification over predicates of predicates, predicates of predicates of predicates, and so forth.
There are several examples of natural language expressions that are better expressed in a second-order language. The claim that there is something which a and b have in common would require a possibly infinite disjunction in a first-order language of the form (Fa & Fb) ∨ (Ga & Gb) ∨ (Ha & Hb) ∨ …, but can be succinctly rendered in a second-order language e.g. ∃X(Xa & Xb). There are also examples that cannot be expressed in a first-order language —for example, the statement “there are some critics who only admire each other” asserts the existence of a set of individuals with a certain property, but does not entail how many such individuals this includes.
The second-order predicate calculus is therefore more expressively powerful than firstorder predicate calculus. If it is a desideratum of our logical languages that they adequately capture our natural language arguments, then this motivates the adoption of higher-order logics. However, this expressive power comes at a cost; in particular, the standard semantics for second-order predicate calculus are incomplete. Thus, if it is a desideratum of our logical languages that they be both sound and complete (roughly that all theorems are provable and vice versa), then this is an argument against adopting higher-order logics. Finally, given that the extension of a predicate is a set of individuals, it has been argued that second-order predicate calculus is better considered as a branch of set-theory rather than logic, with the possible ontological commitments that entails.
