263 research outputs found
Relations between Propositional Normal Modal Logics: an Overview
The modal logic literature is notorious for multiple axiomatizations of the same logic and for conflicting overloading of axiom names. Many of the interesting interderivability results are still scattered over the often hard to obtain classics. We catalogue the most interesting axioms, their numerous variants, and explore the relationships between them in terms of interderivability as both axiom (schema) and as simple formulae. In doing so we introduce the Logics Workbench (LWB, see http://lwbwww.uniba.ch:8080/LWBinfo.html), a versatile tool for proving theorems in numerous propositional (nonclassical) logics. As a side-effect we fulfill a call from the modal theorem proving community for a database of known theorem
Towards the Integration of an Intuitionistic First-Order Prover into Coq
An efficient intuitionistic first-order prover integrated into Coq is useful
to replay proofs found by external automated theorem provers. We propose a
two-phase approach: An intuitionistic prover generates a certificate based on
the matrix characterization of intuitionistic first-order logic; the
certificate is then translated into a sequent-style proof.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
Generic Modal Cut Elimination Applied to Conditional Logics
We develop a general criterion for cut elimination in sequent calculi for
propositional modal logics, which rests on absorption of cut, contraction,
weakening and inversion by the purely modal part of the rule system. Our
criterion applies also to a wide variety of logics outside the realm of normal
modal logic. We give extensive example instantiations of our framework to
various conditional logics. For these, we obtain fully internalised calculi
which are substantially simpler than those known in the literature, along with
leaner proofs of cut elimination and complexity. In one case, conditional logic
with modus ponens and conditional excluded middle, cut elimination and
complexity were explicitly stated as open in the literature
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page
Modal Hybrid Logic
This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory
On Nested Sequents for Constructive Modal Logics
We present deductive systems for various modal logics that can be obtained
from the constructive variant of the normal modal logic CK by adding
combinations of the axioms d, t, b, 4, and 5. This includes the constructive
variants of the standard modal logics K4, S4, and S5. We use for our
presentation the formalism of nested sequents and give a syntactic proof of cut
elimination.Comment: 33 page
Labelled Modal Sequents
In this paper we present a new labelled sequent calculus for modal logic. The proof method works with a more ``liberal'' modal language which allows inferential steps where different formulas refer to different labels without moving to a particular world and there computing if the consequence holds. World-paths can be composed, decomposed and manipulated through unification algorithms and formulas in different worlds can be compared even if they are sub-formulas which do not depend directly on the main connective. Accordingly, such a sequent system can provide a general definition of modal consequence relation. Finally, we briefly sketch a proof of the soundness and completeness results
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