Proof-Theoretic Semantics for a Natural Language Fragment

We propose a Proof-Theoretic Semantics (PTS) for a fragment E 0 (delineated below, and extended in the sequel) of Natural Language (NL). This semantics is intended to be incorporated into type-logical grammars(TLG) [4], constituting an alternative to the traditional model-theoretic semantics (MTS), originating in Montague’s seminal work [3], used in TLG. The essence of our proposal is: 1. For sentences, replace truth conditions (in arbitrary models) by canonical derivability conditions (from suitable assumptions). In particular, this involves a “dedicated” proof-system (in natural deduction form), based on which the derivability conditions obtain. The system should be harmonious, in that its rules satisfy certain balance between introduction and elimination, in order to qualify as meaning conferring. Two notions of harmony are shown to be satisfied by the proposed rules. The approach put forward here is different from a related one by Ranta (e.g., [9]), who relates NL constructs to constructive type-theory in Martin-Löf’s theory. 2. For sub-sentential phrases, down to lexical units (words), replace their denotations (in arbitrary models) as conferring meaning, by their contributions to the meanings (i.e. derivability conditions) of sentences in which they occur. This adheres to Frege’s context principle, the latter made more specific by the incorporation into a TLG. For lack of space, extraction of sub-sentential phrase meanings is not shown here. To the best of our knowledge, there has been no attempt to develop PTS as part of a grammar for NL. The following quotation from [13] (p. 2) emphasizes this lack of applicability to NL, the original reason for considering PTS to start with: