A Lexical Extension of Montague Semantics

Montague\u27s linguistic theory provides a completely formalized account of language in general and natural language in particular. It would appear to be especially applicable to the problem of natural language understanding by computer systems. However the theory does not deal with meaning at the lexical level. As a result, deduction in a system based on Montague semantics is severely restricted. This paper considers lexical extension of Montague semantics as a way to remove this restriction. Representation of lexical semantics by a logic program or semantic net is complex. An alternative representation, called a semantic space, is described. This alternative lacks the expressiveness of a logic program but it offers conceptual simplicity and intrinsically parallel structure

A Theory of Lexical Semantics

The linguistic theory of Richard Montague (variously referred to as Montague Grammar or Montague Semantics) provides a comprehensive formalized account of natural language semantics. It appears to be particularly applicable to the problem of natural language understanding by computer systems. However the theory does not deal with meaning at the lexical level. With few exceptions, lexical items are treated simply as unanalyzed basic expressions. As a result, comparison of distinct lexical meanings or of semantic expressions containing these lexical meanings falls outside the theory. In this paper, I attempt to provide a compatible compatible theory of lexical semantics which may serve as an extension of Montague Semantics

A Variable-Free Logic for Mass Terms

This paper presents a logic appropriate for mass terms, that is, a logic that does not presuppose interpretation in discrete models. Models may range from atomistic to atomless. This logic is a generalization of the author\u27s work on natural language reasoning. The following claims are made for this logic. First, absence of variables makes it simpler than more conventional formalizations based on predicate logic. Second, capability to deal effectively with discrete terms, and in particular with singular terms, can be added to the logic, making it possible to reason about discrete entities and mass entities in a uniform manner. Third, this logic is similar to surface English, in that the formal language and English are well-translatable, making it particularly suitable for natural language applications. Fourth, deduction performed in this logic is similar to syllogistic, and therefore captures an essential characteristic of human reasoning

Inexpressiveness of First-Order Fragments

It is well-known that first-order logic is semi-decidable. Therefore, first-order logic is less than ideal for computational purposes (computer science, knowledge engineering). Certain fragments of first-order logic are of interest because they are decidable. But decidability is gained at the cost of expressiveness. The objective of this paper is to investigate inexpressiveness of fragments that have received much attention

Resolution without Unification

Resolution as an inference procedure forms the basis of most automated theorem-proving and reasoning systems. The most costly constituent of the resolution procedure in its conventional form is unification. This paper describes PCS, a first-order language in which resolution-based inference can be conducted without unification. PCS resembles the language of elementary logic with the difference that singular predicates supplant individual constants and functions. The result is a uniformity in the treatment of individual constants, functions and predicates. An especially costly part of unification is the occur check. Since unification is unnecessary for resolution in PCS, the occur check is completely circumvented. The conditions that would invoke an occur check are properly represented however. In this sense, resolution in PCS can be viewed as a refinement of conventional resolution. PCS does not have an identity relation. Nonetheless, identity can be expressed in PCS and deduction with identical can be performed

Surface Reasoning

Surface reasoning is defined to be deduction conducted in the surface language in terms of certain primitive logical relations. The surface language is a spoken or written natural language (in this paper, English), in contrast to a base language or “deep structure sometimes hypothesized to explain natural language phenomena. The primitive logical relations are inclusion, exclusion and overlap between classes of entities. A calculus for surface reasoning is presented. Then a model for reasoning in this calculus is developed. The model is similar to but more general than syllogistic. In this model, reasoning is represented as construction of fragments (subposets) of lattices. Elements of the lattices are expressions denoting classes of individuals. Strategies to streamline the reasoning process are described. Criteria for strategy selection are proposed

Binary Resolution in Surface Reasoning

Intuition suggests the hypothesis that everyday human reasoning is conducted in the written or spoken natural language, rather than in some disparate representation into which the surface language is translated. An examination of human reasoning reveals patterns of inference that parallel binary resolution. But any standard implementation of resolution requires Skolemization. Skolemization would seem an unlikely component of human reasoning. This appears to contradict the hypothesis that human reasoning takes place at the surface. To reconcile these observations, this paper develops a new rule of inference, which operates on surface expressions directly. This rule is shown to produce results which exactly parallel those produced by Skolemization and resolution. It extends the notion of \u27surface reasoning\u27 that was defined in previous papers. Several examples are given to illustrate its use in surface reasoning

A Lexical Extension of Montague Semantics

This paper presents a model theory of lexical semantics that is compatible with theories in the Montagovian tradition. Lexical expressions are modeled as subsets or “subspaces” in a “semantic spaces”. A unique representation is defined for subspaces of the semantic space. This unique representation is called the normal form of the lexical denotation. A Boolean algebra of normal forms is developed, in which lexical entailment is Boolean inclusion. The presentation in the body of the paper is informal, making use of examples to illustrate the theory and to indicate the range of applicability. Formal definitions and proofs in support of the presentation are given in the Appendix

On the Question ‘Do We Need Identity?’

Sommers posed the question \u27Do We Need Identity?\u27 and answered in the negative. According to Sommers, the need for a special identity relation resulted from an arbitrary distinction between concept and object introduced by Frege and retained in modern predicate logic (MPL). This is reflected in the syntactic distinction between predicate and individual constant. Traditional formal logic (TFL) does not respect this distinction and, as a consequence, has no need for a special identity relation. But Sommers\u27 position has not gained general acceptance. On the contrary, it has received considerable criticism. While it is conceded that TFL can express the identity of individual constants, it is quickly pointed out that this falls far short of providing the expressiveness of the logical identity relation. But the precise extent of the deficit in expressiveness, if indeed there is any deficit, has not been determined. It appears that Sommers\u27 position on identity has not been adequately formalized to permit such a determination. This paper formalizes and extends Sommers\u27 position on identity. This formalization is compared with MPL to define precisely the difference in expressive power. The conclusion is that it has less expressive power than MPL, but nonetheless does provide essentially all the expressiveness of the logical identity relation. The formal language defined for this investigation is similar to the language of MPL. The similarity will not only facilitate comparison, but perhaps will also make this formal language more palatable to readers whose experience and/or predisposition favors MPL

A Logic for Natural Language

This paper describes a language called £N whose structure mirrors that of natural language. £N is characterized by absence of variables and individual constants. Singular predicates assume the role of both individual constants and free variables. The role of bound variables is played by predicate functors called selection operators. Like natural languages, £N is implicitly many-sorted. £N does not have an identity relation. Its expressive power lies between the predicate calculus without identity and the predicate calculus with identity. The loss in expressiveness relative to the predicate calculus with identity however is not significant. Deduction in £N is intended to parallel reasoning in natural language, and therefore is termed surface reasoning. In contrast to deduction in a disparate underlying logic such as clausal form, each step of a proof in £N has a direct counterpart in the surface language. A sound and complete axiomatization is given. Derived rules, corresponding to monotonicity and conservativity of quantifiers and to unification and resolution in conventional logic, are presented. Several problems are worked to illustrate reasoning in £N