31 research outputs found
Sequent Calculus in the Topos of Trees
Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b provability logic,
has been applied in type systems and program logics to capture guarded
recursion. Birkedal et al modelled this modality via the internal logic of the
topos of trees. We show that the semantics of the propositional fragment of
this logic can be given by linear converse-well-founded intuitionistic Kripke
frames, so this logic is a marriage of the intuitionistic modal logic KM and
the intermediate logic LC. We therefore call this logic
. We give a sound and cut-free complete sequent
calculus for via a strategy that decomposes
implication into its static and irreflexive components. Our calculus provides
deterministic and terminating backward proof-search, yields decidability of the
logic and the coNP-completeness of its validity problem. Our calculus and
decision procedure can be restricted to drop linearity and hence capture KM.Comment: Extended version, with full proof details, of a paper accepted to
FoSSaCS 2015 (this version edited to fix some minor typos
On subreducts of subresiduated lattices and logic
Subresiduated lattices were introduced during the decade of 1970 by Epstein
and Horn as an algebraic counterpart of some logics with strong implication
previously studied by Lewy and Hacking. These logics are examples of
subuintuitionistic logics, i.e., logics in the language of intuitionistic logic
that are defined semantically by using Kripke models, in the same way as
intuitionistic logic is defined, but without requiring of the models some of
the properties required in the intuitionistic case. Also in relation with the
study of subintuitionistic logics, Celani and Jansana get these algebras as the
elements of a subvariety of that of weak Heyting algebras.
Here, we study both the implicative and the implicative-infimum subreducts of
subresiduated lattices. Besides, we propose a calculus whose algebraic
semantics is given by these classes of algebras. Several expansions of this
calculi are also studied together to some interesting properties of them
Classical model existence and left resolution
By analyzing what are necessary conditions in the proof [4] of the classical model existence theorem CME (every consistent set has a classical model), we present the left resolution Gentzen systems R(¬,-), which proof-theoretically characterize CME
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
A unified relational semantics for intuitionistic logic, basic propositional logic and orthologic with strict implication
In this paper, by slightly generalizing the notion of 'proposition' in
'Propositional Logic and Modal Logic - A Connection via Relational Semantics'
by Shengyang Zhong, we propose a relational semantics of propositional language
with bottom, conjunction and imlication, which unifies the relational semantics
of intuitionistic logic, Visser's basic propositional logic and orthologic with
strict implication. We study the semantic and syntactic consequence relations
and prove the soundness and completeness theorems for eight propositional
logics