Difference between revisions of "Type logic"
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'''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 constant]]s 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: <a,b>. The one-place predicates of predicate logic are of type <e,t> 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 <e,<e,t>>. Type logic also allows functions of higher order. Noun modifiers can be treated as expressions of type <<e,t>,<e,t>>, mapping a set into a set. NPs are of type <<e,t>,t>, i.e. functions from sets to truth values, or equivalently, sets of sets. Determiners are relations between sets: <<e,t>,<<e,t>,t>>. In combination with lambda-abstraction, type logic is a very powerful logic for semantic representation. It has been fruitfully applied in [[Montague Grammar]]. | '''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 constant]]s 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: <a,b>. The one-place predicates of predicate logic are of type <e,t> 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 <e,<e,t>>. Type logic also allows functions of higher order. Noun modifiers can be treated as expressions of type <<e,t>,<e,t>>, mapping a set into a set. NPs are of type <<e,t>,t>, i.e. functions from sets to truth values, or equivalently, sets of sets. Determiners are relations between sets: <<e,t>,<<e,t>,t>>. In combination with lambda-abstraction, type logic is a very powerful logic for semantic representation. It has been fruitfully applied in [[Montague Grammar]]. | ||
| − | + | == Links == | |
| − | + | *[http://www2.let.uu.nl/UiL-OTS/Lexicon/zoek.pl?lemma=Type+logic&lemmacode=200 Utrecht Lexicon of Linguistics] | |
| − | [http://www2.let.uu.nl/UiL-OTS/Lexicon/zoek.pl?lemma=Type+logic&lemmacode=200 Utrecht Lexicon of Linguistics] | ||
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| + | == References == | ||
* Gamut, L.T.F. 1991. ''Logic, language, and meaning,'' Univ. of Chicago Press, Chicago. | * Gamut, L.T.F. 1991. ''Logic, language, and meaning,'' Univ. of Chicago Press, Chicago. | ||
* Montague, R. 1974. ''Formal philosophy: selected papers of Richard Montague, edited and with an introduction by Richmond H. Thomason,'' Yale University Press, New Haven | * Montague, R. 1974. ''Formal philosophy: selected papers of Richard Montague, edited and with an introduction by Richmond H. Thomason,'' Yale University Press, New Haven | ||
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Latest revision as of 08:31, 30 August 2014
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: <a,b>. The one-place predicates of predicate logic are of type <e,t> 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 <e,<e,t>>. Type logic also allows functions of higher order. Noun modifiers can be treated as expressions of type <<e,t>,<e,t>>, mapping a set into a set. NPs are of type <<e,t>,t>, i.e. functions from sets to truth values, or equivalently, sets of sets. Determiners are relations between sets: <<e,t>,<<e,t>,t>>. In combination with lambda-abstraction, type logic is a very powerful logic for semantic representation. It has been fruitfully applied in Montague Grammar.
Links
References
- Gamut, L.T.F. 1991. Logic, language, and meaning, Univ. of Chicago Press, Chicago.
- Montague, R. 1974. Formal philosophy: selected papers of Richard Montague, edited and with an introduction by Richmond H. Thomason, Yale University Press, New Haven
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