6 research outputs found
A History of Until
Until is a notoriously difficult temporal operator as it is both existential
and universal at the same time: A until B holds at the current time instant w
iff either B holds at w or there exists a time instant w' in the future at
which B holds and such that A holds in all the time instants between the
current one and w'. This "ambivalent" nature poses a significant challenge when
attempting to give deduction rules for until. In this paper, in contrast, we
make explicit this duality of until to provide well-behaved natural deduction
rules for linear-time logics by introducing a new temporal operator that allows
us to formalize the "history" of until, i.e., the "internal" universal
quantification over the time instants between the current one and w'. This
approach provides the basis for formalizing deduction systems for temporal
logics endowed with the until operator. For concreteness, we give here a
labeled natural deduction system for a linear-time logic endowed with the new
operator and show that, via a proper translation, such a system is also sound
and complete with respect to the linear temporal logic LTL with until.Comment: 24 pages, full version of paper at Methods for Modalities 2009
(M4M-6
Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus
Development of a contraction-free BI sequent calculus, be the
contraction-freeness implicit or explicit, has not been successful in
the literature. We address this problem by presenting such a sequent
system. Our calculus involves no structural rules. It should be an
insight into non-formula contraction absorption in other non-classical
logics. Contraction absorption in sequent calculus is associated to
simpler cut elimination and to efficient proof searches
Semantic Incompleteness of del Cerro and Herzig's Hilbert System for a Combination of Classical and Intuitionistic Propositional Logic
This paper shows Hilbert system (C+J)-, given by del Cerro and Herzig (1996) is semantically incomplete. This system is proposed as a proof theory for Kripke semantics for a combination of intuitionistic and classical propositional logic, which is obtained by adding the natural semantic clause of classical implication into intuitionistic Kripke semantics. Although Hilbert system (C+J)- contains intuitionistic modus ponens as a rule, it does not contain classical modus ponens. This paper gives an argument ensuring that the system (C+J)- is semantically incomplete because of the absence of classical modus ponens. Our method is based on the logic of paradox, which is a paraconsistent logic proposed by Priest (1979)