94 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
Logics for modelling collective attitudes
We introduce a number of logics to reason about collective propositional
attitudes that are defined by means of the majority rule. It is well known that majoritarian
aggregation is subject to irrationality, as the results in social choice theory and judgment
aggregation show. The proposed logics for modelling collective attitudes are based on
a substructural propositional logic that allows for circumventing inconsistent outcomes.
Individual and collective propositional attitudes, such as beliefs, desires, obligations, are
then modelled by means of minimal modalities to ensure a number of basic principles. In
this way, a viable consistent modelling of collective attitudes is obtained
Sequent calculi and decidability for intuitionistic hybrid logic
AbstractIn this paper we study the proof theory of the first constructive version of hybrid logic called Intuitionistic Hybrid Logic (IHL) in order to prove its decidability. In this perspective we propose a sequent-style natural deduction system and then the first sequent calculus for this logic. We prove its main properties like soundness, completeness and also the cut-elimination property. Finally we provide, from our calculus, the first decision procedure for IHL and then prove its decidability
Multiple Conclusion Rules in Logics with the Disjunction Property
We prove that for the intermediate logics with the disjunction property any
basis of admissible rules can be reduced to a basis of admissible m-rules
(multiple-conclusion rules), and every basis of admissible m-rules can be
reduced to a basis of admissible rules. These results can be generalized to a
broad class of logics including positive logic and its extensions, Johansson
logic, normal extensions of S4, n-transitive logics and intuitionistic modal
logics
Fixed-point elimination in the intuitionistic propositional calculus
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic
models of the Intuitionistic Propositional Calculus-always exist, even when
these algebras are not complete as lattices. The reason is that these extremal
fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equivalent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed-point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal
Itauto: An Extensible Intuitionistic SAT Solver
We present the design and implementation of itauto, a Coq reflexive tactic for intuitionistic propositional logic. The tactic inherits features found in modern SAT solvers: definitional conjunctive normal form; lazy unit propagation and conflict driven backjumping. Formulae are hash-consed using native integers thus enabling a fast equality test and a pervasive use of Patricia Trees. We also propose a hybrid proof by reflection scheme whereby the extracted solver calls user-defined tactics on the leaves of the propositional proof search thus enabling theory reasoning and the generation of conflict clauses. The solver has decent efficiency and is more scalable than existing tactics on synthetic benchmarks and preliminary experiments are encouraging for existing developments
On formal aspects of the epistemic approach to paraconsistency
This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for both BLE and LET J . The meanings of the connectives of BLE and LET J , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LET J is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed
Refutation Systems : An Overview and Some Applications to Philosophical Logics
Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics
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