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A Single-Type Ontology for Natural Language
Kristina Liefke
In (Montague 1970a), Richard Montague defines a formal theory of linguistic meaning which
interprets a small fragment of English through the use of two basic types of objects:
individuals and propositions. In (Partee 2006), Barbara Partee conjectures the possibility of
a comparable semantic theory, which only uses one basic type of object (hence, single-type
semantics). This paper supports Partee’s conjecture by identifying two suitable single-type
candidates. To single out the latter, we first introduce a set of semantic requirements on any
single basic type. The application of these requirements to the familiar types from (Montague
1973) reduces this set to two members. The paper closes with a comparison of Partee’s
preliminary single-type choice and our newly identified single basic types.
1. Introduction
Natural languages presuppose a rich semantic ontology. To provide an interpretation for, e.g.,
English, we require the existence of individuals (e.g. Bill), propositions (Bill walks), first- and
higher-order properties (walk, rapidly), relations (find), and many other kinds of objects.
Theories of formal linguistic semantics (paradigmatically (Montague 1970a; 1970b; 1973))
tame this ontological ‘zoo’ by casting its members into a type structure, and generating
objects of a more complex type from objects of a simpler type via a variant of Church’s typeforming
rule (Church 1940), (cf. Gallin 1975):
Type-forming rule (CT)
If α and β are the types for two (possibly different) kinds of objects, then is the
type for functions from objects of the type α to objects of the type β.
In this way, Montague (1970a) reduces the referents of the small subset of English from
(Montague 1973) to constructions out of two basic types of objects: individuals (or entities,
type e) and propositions (or functions from indices to truth-values, type ). Proper
names (e.g. Bill) and sentences (Bill walks) are then interpreted as entities, respectively
propositions, intransitive verbs (walk) as functions from entities to propositions (type ), and transitive verbs (find) as functions from entities to functions from entities to
propositions (type ).
Montague’s distinction between entities and propositions (or between entities, indices, and
truth-values) has today become standard in formal semantics. This is due to the resulting
semantics’ modelling power, and the attendant possibility of explaining a wide range of
syntactic and semantic phenomena. However, recent findings in language development
(Carstairs-McCarthy 1999; Cheney and Seyfarth 1990; Snedeker et al. 2007) suggest the
possibility of an even simpler semantic basis for natural language. The latter lies in a single
basic type (dubbed ‘o’) whose objects encode the semantic content of entities and
propositions. From them, objects of a more complex type are constructed via a variant of the
rule CT:
Single-type-forming rule (ST)
If o is the single basic type and α and β are single-type types, then is a singletype
type.