Description

• Title:
  A New Arithmetically Incomplete First- Order Extension of Gl All Theorems of Which Have Cut Free Proofs
• Creator:
  Tourlakis, George
• Contributor:
  janciu@uni.lodz.pl
• Description:
  Reference [12] introduced a novel formula to formula translation tool (“formulators”) that enables syntactic metatheoretical investigations of first-order modal logics, bypassing a need to convert them first into Gentzen style logics in order to rely on cut elimination and the subformula property. In fact, the formulator tool, as was already demonstrated in loc. cit., is applicable even to the metatheoretical study of logics such as QGL, where cut elimination is (provably, [2]) unavailable. This paper applies the formulator approach to show the independence of the axiom schema _A ! _8xA of the logics M3 and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics (compare with the more complex proofs in [2, 8]).;
  This research was partially supported by NSERC grant No. 8250.
• Object availability:
  free access
• Rights:
  © Copyright by Authors, Łódź 2016; © Copyright for this edition by Uniwersytet Łódzki, Łódź 2016
• Date:
  2017.05.16;
  2016
• Type:
  Article
• Language:
  English
• Format:
  application/pdf
• Publisher:
  Wydawnictwo Uniwersytetu Łódzkiego
• Relation:
  Bulletin of the Section of Logic;1;
  45
• Subject:
  Modal logic;
  GL;
  first-order logic;
  proof theory;
  cut elimination;
  reflection property;
  disjunction property;
  quantified modal logic;
  QGL;
  arithmetical completeness
• Identifier:
  0138-0680;
  http://hdl.handle.net/11089/21621 ;
  2449-836X;
  10.18778/0138-0680.45.1.02
• Data provider:
  Repozytorium Uniwersytetu Łódzkiego
• Can I use it?:
  seek permission
• Type:
  image