A New Arithmetically Incomplete First-Order Extension of Gl All Theorems of Which Have Cut Free Proofs - Tourlakis, George (Bulletin of the Section of Logic, 2016)
Reference [12] introduced a novel formula to formula translation tool (“formula-tors”) that enables syntactic metatheoretical investigations of first-order modallogics, bypassing a need to convert them first into Gentzen style logics in order torely on cut elimination and the subformula property. In fact, the formulator tool,as was already demonstrated in loc. cit., is applicable even to the metatheoreticalstudy 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 → ☐∀ A of the logics M3and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this ax<span style="font-size: 9.0pt; mso-bidi-font-size: 11.0pt; line-height: 115%; font-family: 'Times New Roman',serif; mso-hansi-font-family: Georgia; mso-bidi-font-family: Georgia; color: #231f20; mso-font-width: 115%;" lang="EN-US">iom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to <span style="font-size: 9.0pt; mso-bidi-font-size: 11.0pt; line-height: 115%; font-family: 'Times New Roman',serif; mso-hansi-font-family: Georgia; mso-bidi-font-family: Georgia; color: #231f20; mso-font-width: 110%;" lang="EN-US">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 <span style="font-size: 9.0pt; mso-bidi-font-size: 11.0pt; line-height: 115%; font-family: 'Times New Roman',serif; mso-hansi-font-family: Georgia; mso-bidi-font-family: Georgia; color: #231f20; mso-font-width: 110%;" lang="EN-US">for GL and QGL without going through Gentzen versions of these logics (compare with the more complexproofs in [2,8]).