Review of Type-Logical Semantics(4)

One of the many roles of linguistics is to address the semantics of natural languages, that is, the meaning of sentences in natural languages. An important part of the meaning of sentences can be characterized by stating the conditions that need to hold fo

Standard frames are subject to restrictions.For instance,the domain corresponding to boolean values must be a two-element domain,such as true false.Moreover,they must give afixed interpretation to the logical constants(i.e.,the conjunction operator should actually behave as a conjunction operator).Hence,we require:

false if true

true if false

true if true and true

false otherwise

true if

false otherwise

true if true for all

false otherwise

One can check that if we define as,it has the expected interpretation.Note that we will often use the abbreviations for,and for.

A formula of higher-order logic is a term of type.We say that a model satisfies a formula if true in the model.Two terms are said to be logically equivalent if they have the same interpretation in all models.One can check,for instance,that and are logically equivalent,as are and(that is,where every occurrence of is replaced by).

For example,consider the following simple three individual model,with constants and .

false

true

true

true

false

false

true

false

true

This model satisfies the term(Kala likes Tarzan)as

true.It also satisfies the term(There is someone Jane likes).It does not satisfy the term (Everyone likes himself/herself).

We will use higher-order logic to express our semantics.The idea is to associate with every sentence(or part of speech)a higher-order logic term.We can then use the semantics of higher-order logic to derive the truth value of the sentence.Consider the examples at the beginning of the section.We assume constants,and

of type,and a constant of type.We can translate the sentence Tarzan likes Jane as ,as infirst-order logic.But now we can also translate the part of speech like Jane independently as.

For a more interesting example,consider the treatment of noun phrases as given at the beginning of the section. The solution to the problem of losing the grammatical structure was solved by Russell by treating all noun phrases as though they were functions over their verb phrases.This is analogous to what is already happening with the definition of in higher-order logic,which has type.Such generalized quantifier takes a property of an individual(a property has type)and produces a truth value—in the case of, the truth value is true if every individual has the supplied property.A similar abstraction can be applied to a noun position.We define a generalized determiner as a function taking a property stating a restriction on the quantified individuals,and returning a generalized quantifier obeying that restriction.Hence,a generalized determiner has type

.Consider the following generalized determiners,used above:

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