A phase semantics for polarized linear logic and second order conservativity

Masahiro Hamano, Ryo Takemura

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

This paper presents a polarized phase semantics, with respect to which the linear fragment of second order polarized linear logic of Laurent [15] is complete. This is done by adding a topological structure to Girard's phase semantics [9]. The topological structure results naturally from the categorical construction developed by Hamano-Scott [12]. The polarity shifting operator ↓ (resp. ↑) is interpreted as an interior (resp. closure) operator in such a manner that positive (resp. negative) formulas correspond to open (resp. closed) facts. By accommodating the exponentials of linear logic, our model is extended to the polarized fragment of the second order linear logic. Strong forms of completeness theorems are given to yield cut-eliminations for the both second order systems. As an application of our semantics, the first order conservativity of linear logic is studied over its polarized fragment of Laurent [16]. Using a counter model construction, the extension of this conservativity is shown to fail into the second order, whose solution is posed as an open problem in [16]. After this negative result, a second order conservativity theorem is proved for an eta expanded fragment of the second order linear logic, which fragment retains a focalized sequent property of [3].

Original languageEnglish
Pages (from-to)77-102
Number of pages26
JournalJournal of Symbolic Logic
Volume75
Issue number1
DOIs
Publication statusPublished - 2010 Mar

All Science Journal Classification (ASJC) codes

  • Philosophy
  • Logic

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