Type logic

Definition
Type logic is a logical system based on Russell's theory of types. Every expression of a type-logical language belongs to a particular type indicating the set-theoretical denotation of that expression. There are two basic types, the type e (from entity) and the type t (from truth value). The formulas of predicate logic and propositional logic are expressions of type t in type logic, denoting truth values; the individual constants of predicate logic are expressions of type e in type logic, denoting individuals. All other expressions in type-logic are functional, i.e. they take an expression of type a as their argument and yield an expression of type b, which is indicated in their type as follows: &lt;a,b&gt;. The one-place predicates of predicate logic are of type &lt;e,t&gt; in type logic, denoting a function from entities to truth-values, which is another way to define a set. Two-place predicates are of type &lt;e,&lt;e,t&gt;&gt;. Type logic also allows functions of higher order. Noun modifiers can be treated as expressions of type &lt;&lt;e,t&gt;,&lt;e,t&gt;&gt;, mapping a set into a set. NPs are of type &lt;&lt;e,t&gt;,t&gt;, i.e. functions from sets to truth values, or equivalently, sets of sets. Determiners are relations between sets: &lt;&lt;e,t&gt;,&lt;&lt;e,t&gt;,t&gt;&gt;. In combination with lambda-abstraction, type logic is a very powerful logic for semantic representation. It has been fruitfully applied in Montague Grammar.

Links

 * Utrecht Lexicon of Linguistics