218 research outputs found
Cut-elimination for the modal Grzegorczyk logic via non-well-founded proofs
We present a sequent calculus for the modal Grzegorczyk logic Grz allowing
non-well-founded proofs and obtain the cut-elimination theorem for it by
constructing a continuous cut-elimination mapping acting on these proofs.Comment: WOLLIC'17, 12 pages, 1 appendi
Failure of interpolation in the intuitionistic logic of constant domains
This paper shows that the interpolation theorem fails in the intuitionistic
logic of constant domains. This result refutes two previously published claims
that the interpolation property holds.Comment: 13 pages, 0 figures. Overlaps with arXiv 1202.1195 removed, the text
thouroughly reworked in terms of notation and style, historical notes as well
as some other minor details adde
Failure of interpolation in the intuitionistic logic of constant domains
This paper shows that the interpolation theorem fails in the intuitionistic
logic of constant domains. This result refutes two previously published claims
that the interpolation property holds.Comment: 13 pages, 0 figures. Overlaps with arXiv 1202.1195 removed, the text
thouroughly reworked in terms of notation and style, historical notes as well
as some other minor details adde
Cyclic proof systems for modal fixpoint logics
This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one
Modal Logics that Bound the Circumference of Transitive Frames
For each natural number we study the modal logic determined by the class
of transitive Kripke frames in which there are no cycles of length greater than
and no strictly ascending chains. The case is the G\"odel-L\"ob
provability logic. Each logic is axiomatised by adding a single axiom to K4,
and is shown to have the finite model property and be decidable.
We then consider a number of extensions of these logics, including
restricting to reflexive frames to obtain a corresponding sequence of
extensions of S4. When , this gives the famous logic of Grzegorczyk, known
as S4Grz, which is the strongest modal companion to intuitionistic
propositional logic. A topological semantic analysis shows that the -th
member of the sequence of extensions of S4 is the logic of hereditarily
-irresolvable spaces when the modality is interpreted as the
topological closure operation. We also study the definability of this class of
spaces under the interpretation of as the derived set (of limit
points) operation.
The variety of modal algebras validating the -th logic is shown to be
generated by the powerset algebras of the finite frames with cycle length
bounded by . Moreover each algebra in the variety is a model of the
universal theory of the finite ones, and so is embeddable into an ultraproduct
of them
Syntax without Abstract Objects
In line with the nominalistic denial of the existence of abstract objects, a basic theory of syntax for formal languages is developed and shown to satisfy certain fundamental requirements
Characteristic formulas over intermediate logics
We expand the notion of characteristic formula to infinite finitely
presentable subdirectly irreducible algebras. We prove that there is a
continuum of varieties of Heyting algebras containing infinite finitely
presentable subdirectly irreducible algebras. Moreover, we prove that there is
a continuum of intermediate logics that can be axiomatized by characteristic
formulas of infinite algebras while they are not axiomatizable by standard
Jankov formulas. We give the examples of intermediate logics that are not
axiomatizable by characteristic formulas of infinite algebras. Also, using the
Goedel-McKinsey-Tarski translation we extend these results to the varieties of
interior algebras and normal extensions of S
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