Blackwell Reference Online

Extract

L ogic, philosophy of mathematics If in a logical language the quantifiers only contain variables ranging over individuals , this language is called a first-order language, and these variables are called first-order variables. The study of the rules of inference in a first-order language is called first-order logic. In this logic, individuals are the only arguments of predicates . If the variables range over properties, relations, functions , and classes of the individuals, they are called second-order variables. A language containing second-order variables is a second-order language, and the logic of this language is second-order logic. The domain of second-order logic is determined by the first-order logic. If the variables range over the domain of properties or the relations of properties, then we have third-order variables, language, and logic. This construction can go on to even higher orders. Any logic that is at least a second-order logic is called higher-order logic. Strictly speaking, first-order logic emerged with Hilbert in 1917. For most mathematicians, it is the proper and natural framework for mathematics. “I have distinguished between a logician's use of first-order logic (where quantifiers range only over individuals), second-order logic (where quantifiers can also range over sets or relations), w-order logic (essentially the simple theory of types), ... log in or subscribe to read full text