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A semantics for nabla
Published online by Cambridge University Press: 17 April 2018
Abstract
We give a semantics for a classical variant of Dale Miller and Alwen Tiu’s logic FOλ∇. Our semantics validates the rule that nabla x implies exists x, but is otherwise faithful to the authors’ original intentions. The semantics is based on a category of so-called nabla sets, which are simply strictly increasing sequences of non-empty sets. We show that the logic is sound for that semantics. Assuming there is a unique base type ι, we show that it is complete for Henkin structures, incomplete for standard structures in general, but complete for standard structures in the case of Π1 formulae, and that includes all first-order formulae.
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Footnotes
†
A variant of this paper was presented at Dale Miller’s 60th birthday. The current presentation is simpler, although essentially equivalent. We have also taken the opportunity to correct a mistake in the proof of Proposition 8.4.
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