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Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics

100%

Suzuki N.-Y.

Bulletin of the Section of Logic

2017

tom 46

nr 1/2

We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author’s previous paper [16]. In the present paper, the Kripke sheaf semantics is used.

A New Arithmetically Incomplete First-Order Extension of Gl All Theorems of Which Have Cut Free Proofs

63%

Tourlakis G.

Bulletin of the Section of Logic

2016

tom 45

nr 1

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 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 complexproofs in [2,8]).

Ograniczanie wyników

2 Bulletin of the Section of Logic

1 Suzuki N.-Y.

1 Tourlakis G.

1 2017

1 2016

Wyniki wyszukiwania

Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics

A New Arithmetically Incomplete First-Order Extension of Gl All Theorems of Which Have Cut Free Proofs