How Much Expressive Power Is Needed for Natural Language Temporal Indexicality?

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Abstract

The paper studies how much expressive power beyond the capabilities of the simple Priorean temporal language K t is needed to give proper translation to natural language examples by Kamp and Vlach which are extensively used in the linguistic and philosophical literature as forcing the use of quite expressive languages, all the way up to full two-sorted FOL. It turns out that when examined carefully, the examples in question only require a quite mild Kamp- and Cresswell-style system with now and then operators, or, equivalently, hybrid K t  + ↓ + @. The paper generalizes the earlier results showing that in the propositional case, now and then do not increase the power of K t . For the first-order case, a notion of FOL path bisimulation for first-order \(\mathbf{K}_t^{FO}\) with untensed quantification and equality is defined, and it is demonstrated how to prove that a particular NL sentence cannot be expressed in \(\mathbf{K}_t^{FO}\) through non-preservation under FOL path bisimulation. It is also shown that \(\mathbf{K}_t^{FO}\) plus now and then is still strictly less expressive than HL(↓, @), which is itself much less expressive than the strong systems that were claimed to be needed for NL translation. Thus the paper provides strict lower and upper bounds on the expressivity of the translation language forced by Kamp-Vlach NL sentences, and the upper bound turns out to be much lower than was widely believed in the linguistics community.