283 research outputs found
Relating Sequent Calculi for Bi-intuitionistic Propositional Logic
Bi-intuitionistic logic is the conservative extension of intuitionistic logic
with a connective dual to implication. It is sometimes presented as a symmetric
constructive subsystem of classical logic.
In this paper, we compare three sequent calculi for bi-intuitionistic
propositional logic: (1) a basic standard-style sequent calculus that restricts
the premises of implication-right and exclusion-left inferences to be
single-conclusion resp. single-assumption and is incomplete without the cut
rule, (2) the calculus with nested sequents by Gore et al., where a complete
class of cuts is encapsulated into special "unnest" rules and (3) a cut-free
labelled sequent calculus derived from the Kripke semantics of the logic. We
show that these calculi can be translated into each other and discuss the
ineliminable cuts of the standard-style sequent calculus.Comment: In Proceedings CL&C 2010, arXiv:1101.520
Towards a Proof Theory of G\"odel Modal Logics
Analytic proof calculi are introduced for box and diamond fragments of basic
modal fuzzy logics that combine the Kripke semantics of modal logic K with the
many-valued semantics of G\"odel logic. The calculi are used to establish
completeness and complexity results for these fragments
An Intuitionistic Formula Hierarchy Based on High-School Identities
We revisit the notion of intuitionistic equivalence and formal proof
representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic
(minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms
corresponding to the high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible proof rules
only, and where the invertible proof rules have been replaced by a formula
normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of
Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the
non-invertible proof rules as strict inequalities between exponential
polynomials; a careful combinatorial treatment is given in order to establish
this fact.
Finally, we extend the exponential polynomial analogy to the first-order
quantifiers, showing that it gives rise to an intuitionistic hierarchy of
formulas, resembling the classical arithmetical hierarchy, and the first one
that classifies formulas while preserving isomorphism
A proof-theoretic study of bi-intuitionistic propositional sequent calculus
Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication usually called âexclusionâ. A standard-style sequent calculus for this logic is easily obtained by extending multiple-conclusion sequent calculus for intuitionistic logic with exclusion rules dual to the implication rules (in particular, the exclusion-left rule restricts the premise to be single-assumption). However, similarly to standard-style sequent calculi for non-classical logics like S5, this calculus is incomplete without the cut rule. Motivated by the problem of proof search for propositional bi-intuitionistic logic (BiInt), various cut-free calculi with extended sequents have been proposed, including (i) a calculus of nested sequents by GorĂ© et al., which includes rules for creation and removal of nests (called ânest rulesâ, resp. âunnest rulesâ) and (ii) a calculus of labelled sequents by the authors, derived from the Kripke semantics of BiInt, which includes âmonotonicity rulesâ to propagate truth/falsehood between accessible worlds.
In this paper, we develop a proof-theoretic study of these three sequent calculi for BiInt grounded on translations between them. We start by establishing the basic meta-theory of the labelled calculus (including cut-admissibility), and use then the translations to obtain results for the other two calculi. The translation of the nested calculus into the standard-style calculus explains how the unnest rules encapsulate cuts. The translations between the labelled and the nested calculi reveal the two formats to be very close, despite the former incorporating semantic elements, and the latter being syntax-driven. Indeed, we single out (i) a labelled calculus whose sequents have a âlabel in focusâ and which includes ârefocusing rulesâ and (ii) a nested calculus with monotonicity and refocusing rules, and prove these two calculi to be isomorphic (in a bijection both at the level of sequents and at the level of derivations).ERDF through the Estonian Centre of Excellence in Computer Science (EXCS), by the Estonian Science Foundation under grant no. 6940; COST action CA15123 EUTYPES.info:eu-repo/semantics/publishedVersio
Proof Theory of Finite-valued Logics
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics
Dialectica Categories for the Lambek Calculus
We revisit the old work of de Paiva on the models of the Lambek Calculus in
dialectica models making sure that the syntactic details that were sketchy on
the first version got completed and verified. We extend the Lambek Calculus
with a \kappa modality, inspired by Yetter's work, which makes the calculus
commutative. Then we add the of-course modality !, as Girard did, to
re-introduce weakening and contraction for all formulas and get back the full
power of intuitionistic and classical logic. We also present the categorical
semantics, proved sound and complete. Finally we show the traditional
properties of type systems, like subject reduction, the Church-Rosser theorem
and normalization for the calculi of extended modalities, which we did not have
before
Bi-Classical Connexive Logic and its Modal Extension: Cut-elimination, completeness and duality
In this study, a new paraconsistent four-valued logic called bi-classical connexive logic (BCC) is introduced as a Gentzen-type sequent calculus. Cut-elimination and completeness theorems for BCC are proved, and it is shown to be decidable. Duality property for BCC is demonstrated as its characteristic property. This property does not hold for typical paraconsistent logics with an implication connective. The same results as those for BCC are also obtained for MBCC, a modal extension of BCC
Focused Proof-search in the Logic of Bunched Implications
The logic of Bunched Implications (BI) freely combines additive and
multiplicative connectives, including implications; however, despite its
well-studied proof theory, proof-search in BI has always been a difficult
problem. The focusing principle is a restriction of the proof-search space that
can capture various goal-directed proof-search procedures. In this paper, we
show that focused proof-search is complete for BI by first reformulating the
traditional bunched sequent calculus using the simpler data-structure of nested
sequents, following with a polarised and focused variant that we show is sound
and complete via a cut-elimination argument. This establishes an operational
semantics for focused proof-search in the logic of Bunched Implications.Comment: 18 pages conten
Intuitionistic G\"odel-L\"ob logic, \`a la Simpson: labelled systems and birelational semantics
We derive an intuitionistic version of G\"odel-L\"ob modal logic ()
in the style of Simpson, via proof theoretic techniques. We recover a labelled
system, , by restricting a non-wellfounded labelled system for
to have only one formula on the right. The latter is obtained using
techniques from cyclic proof theory, sidestepping the barrier that 's
usual frame condition (converse well-foundedness) is not first-order definable.
While existing intuitionistic versions of are typically defined over
only the box (and not the diamond), our presentation includes both modalities.
Our main result is that coincides with a corresponding
semantic condition in birelational semantics: the composition of the modal
relation and the intuitionistic relation is conversely well-founded. We call
the resulting logic . While the soundness direction is proved using
standard ideas, the completeness direction is more complex and necessitates a
detour through several intermediate characterisations of .Comment: 25 pages including 8 pages appendix, 4 figure
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