Eliminating the Universe

Eliminating the Universe

Logical Properties of Natural Language

Edward L Keenan


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This book synthesizes the author's work (1980s–2015) on the logical expressive power of natural language. It extends the tools and concepts of model theory as used in (higher order) predicate logic to the study of natural language semantics. It focuses on boolean structure, generalized quantification (separated from variable binding), covering some cases of anaphora. Different categories — predicates, adjective, quantifiers — are modeled by non-isomorphic boolean lattices.

Of empirical linguistic interest is the expressibility of many natural classes of quantifiers defined in terms of their logical (automorphism invariant) properties. Some of these correlate with classes used syntactically in generative grammar. In other cases we find general (possibly universal) constraints on possible quantifier denotations in natural language.

Also of novel logical interest are entailment paradigms that depend on relations between pairs or triples of generalized quantifier denoting expressions, ones that are in some cases inherently vague. In addition we note novel binary quantifiers that lie beyond the "Frege boundary" in that they are provably not identical to any iterated application of unary quantifiers.

Of philosophical interest is the existence of models which make the same sentences true as standard models but which lack a universe and hence, seemingly, a notion of "reference". Moreover, these models generalize to ones in which we can represent (some) intensional expressions without the use of novel ontological objects, such as "possible worlds" or "propositions".

  • The Linguistic Prescience of George Boole
  • Logic as Universal Grammar
  • Classifying Nominal Quantifiers in English
  • Generalized Quantifiers and Logical Expressive Power
  • The Midpoint Theorems
  • Three Linguistic Generalizations
  • Mathematical Analysis of English Quantifiers
  • Characterizing Syntactic Categories Semantically?
  • English is Inherently Sortal
  • Logical Operators and Structure Maps
  • Beyond the Frege Boundary
  • Eliminating the Universe

Readership: Researchers in mathematical logic and artificial intelligence.
Natural Logic;Logical Expressive Power (of Natural Language);Boolean Structure;Generalized Quantifiers and Anaphora;Formal Semantics;Entailment Paradigms in Natural Language0Key Features:
  • The work applies well understood boolean notions to an extended study of the meaning system of natural language — of interest then to computational linguistics and philosophical and mathematical studies of linguistically motivated logical systems
  • Its treatment of non-extensional evaluative adjectives is novel and ontologically more parsimonious than standard intensional systems
  • And the novel entailment paradigms involving multiple arguments invites further investigation into the study of inference in natural language