Decreasing sentences in Simple Type Theory

Publisher: John Wiley & Sons Inc

E-ISSN: 1521-3870|63|5|342-363

ISSN: 0942-5616

Source: MLQ- MATHEMATICAL LOGIC QUARTERLY, Vol.63, Iss.5, 2017-12, pp. : 342-363

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Abstract

AbstractWe present various results regarding the decidability of certain sets of sentences by Simple Type Theory (TST). First, we introduce the notion of decreasing sentence, and prove that the set of decreasing sentences is undecidable by Simple Type Theory with infinitely many zero‐type elements (TST); a result that follows directly from the fact that every sentence is equivalent to a decreasing sentence. We then establish two different positive decidability results for a weak subtheory of TST. Namely, the decidability of ¯((b¯)b¯b), (a subset of Σ1) and ¯STDEC (the set of all sentences x¯φ(x¯), where φ is strictly decreasing). Finally, we present some consequences for the set of existential‐universal sentences. All the above results have direct implications for Quine's theory of “New Foundations” (NF) and its weak subtheory NFO.