Classical Logic II: Higher‐Order Logic
Stewart Shapiro · 2017
A typical interpreted formal language has (first-order) variables that range over a collection of objects, sometimes called a domain-of-discourse. The domain is what the formal language is about. A language may also contain second-order variables that range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second-order variable ranging over qualities. Similarly, the sentence ‘there is a property that holds of all and only the prime numbers’ has a variable ranging over properties of natural numbers. Third-order variables range over properties of properties, sets of sets, functions from properties to sets, etc. For example, according to some logicist accounts, the number 4 is the property shared by all properties that apply to exactly four objects in the domain. Accordingly, the number 4 is a third-order item. Fourth-order variables, and beyond, are characterized similarly. The phrase ‘higher-order variable’ refers to the variables beyond first-order.