Open AccessOmega-inconsistency without cuts and nonstandard models
Author(s) -
Andreas Fjellstad
Publication year - 2016
Publication title -
australasian journal of logic
Language(s) - English
Resource type - Journals
ISSN - 1448-5052
DOI - 10.26686/ajl.v13i5.3900
Subject(s) - deontic logic , transitive relation , sequent calculus , sequent , logical consequence , omega , mathematics , natural deduction , arithmetic , predicate (mathematical logic) , calculus (dental) , discrete mathematics , computer science , artificial intelligence , philosophy , epistemology , mathematical proof , programming language , linguistics , combinatorics , medicine , geometry , dentistry
This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee (1985) for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency and shows thus, pace Cobreros et al.(2013), that the result in McGee (1985) does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic.