Abstract
The frequency of intensional and non-first-order definable operators in natural languages constitutes a challenge for automated reasoning with the kind of logical translations that are deemed adequate by formal semanticists. Whereas linguists employ expressive higher-order logics in their theories of meaning, the most successful logical reasoning strategies with natural language to date rely on sophisticated first-order theorem provers and model builders. In order to bridge the fundamental mathematical gap between linguistic theory and computational practice, we present a general translation from a higher-order logic frequently employed in the linguistics literature, two-sorted Type Theory, to first-order logic under Henkin semantics. We investigate alternative formulations of the translation, discuss their properties, and evaluate the availability of linguistically relevant inferences with standard theorem provers in a test suite of inference problems stated in English. The results of the experiment indicate that translation from higher-order logic to first-order logic under Henkin semantics is a promising strategy for automated reasoning with natural languages.The paper is accompanied by the source code (cf. SUPP. FILES) of the grammar and reasoning architecture described in the paper.
Highlights
One of the big challenges for applying automated inference to natural language input comes from a stark discrepancy in the preferred logical languages in theoretical semantics on the one hand and in computational semantics on the other
In contrast to these established linguistic theories, the most successful logical reasoning strategies with natural language rely on theorem provers and model builders for first-order logic
To cope with the discrepancy, previous work by Bos and Markert (2006) on applying first-order inference tools to natural language in part approximated intensions and higher-order quantification in firstorder logic, and in part ignored their role in language. This strategy restricts the fragment of natural language that can be treated, and it forces computational semanticists to recast the theories of formal linguists in a different logical language
Summary
The frequency of intensional and non-first-order definable operators in natural languages constitutes a challenge for automated reasoning with the kind of logical translations that are deemed adequate by formal semanticists. Whereas linguists employ expressive higher-order logics in their theories of meaning, the most successful logical reasoning strategies with natural language to date rely on sophisticated first-order theorem provers and model builders. The results of the experiment indicate that translation from higher-order logic to first-order logic under Henkin semantics is a promising strategy for automated reasoning with natural languages. Rather than being tailored to specific linguistic applications, the present proposal provides a general translation of full higher-order logic, and the fine-grained semantic representations of the formal semantics literature are accepted as input without any modification This means that higher-order representations of challenging natural language facts can be developed independently of implementations in formalisms familiar to semanticists without having to worry about a possible manual reduction to first-order logic, and the original representations may serve as input to automated reasoning. Logical constants such as → ̇ and ≡ ̇ are rendered in infix notation
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