Tuesday, 27 November 2018

First-Order Logic with Adverbs

New draft available here. Here is the abstract:
In this paper I introduce a logical treatment of adverbs based on first-order logic with identity (FOL). The guiding idea is that an adverb can come between an individual constant and a predicate, such that its addition to a true statement can result in a false statement - or in another true statement, but a different one. In contrast to prominent logical treatments of adverbs in the literature, the aim is not to model details of natural language, but to add to FOL a flexible, simple device inspired by adverbs, while retaining much of the elegance and the desirable metalogical properties of FOL. In Section 1 I outline the general aim, the notation, and the desired inferential behaviour of adverb-containing propositions. In Sections 2, 3 and 4 I define the extended language, make the necessary addition to its semantics, and update the definition of truth on a model. In Section 5 I exhibit how the consequence relation induced by the semantics gives us the desired inferential behaviour. In Section 6 I explain how a tree system which is sound and complete for FOL can be extended to FOL with adverbs. In Section 7 I give tree proofs of the consequences exhibited in Section 5. In Section 8 I show that the augmented tree system is sound and complete for FOL with adverbs. In a brief Appendix I float the idea of postulates or logical rules for adverbs of interest.

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