74 research outputs found
Canonical formulas for k-potent commutative, integral, residuated lattices
Canonical formulas are a powerful tool for studying intuitionistic and modal
logics. Actually, they provide a uniform and semantic way to axiomatise all
extensions of intuitionistic logic and all modal logics above K4. Although the
method originally hinged on the relational semantics of those logics, recently
it has been completely recast in algebraic terms. In this new perspective
canonical formulas are built from a finite subdirectly irreducible algebra by
describing completely the behaviour of some operations and only partially the
behaviour of some others. In this paper we export the machinery of canonical
formulas to substructural logics by introducing canonical formulas for
-potent, commutative, integral, residuated lattices (-).
We show that any subvariety of - is axiomatised by canonical
formulas. The paper ends with some applications and examples.Comment: Some typo corrected and additional comments adde
Degrees of the finite model property: the antidichotomy theorem
A classic result in modal logic, known as the Blok Dichotomy Theorem, states
that the degree of incompleteness of a normal extension of the basic modal
logic is or . It is a long-standing open problem
whether Blok Dichotomy holds for normal extensions of other prominent modal
logics (such as or ) or for extensions of the intuitionistic
propositional calculus . In this paper, we introduce the notion
of the degree of finite model property (fmp), which is a natural variation of
the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem
that the degree of fmp of a normal extension of remains or
. In contrast, our main result establishes the following
Antidichotomy Theorem for the degree of fmp for extensions of :
each nonzero cardinal such that or is realized as the degree of fmp of some extension of
. We then use the Blok-Esakia theorem to establish the same
Antidichotomy Theorem for normal extensions of and
The Kuznetsov-GerÄŤiu and Rieger-Nishimura logics
We give a systematic method of constructing extensions of the Kuznetsov-Gerčiu logic KG without the finite model property (fmp for short), and show that there are continuum many such. We also introduce a new technique of gluing of cyclic intuitionistic descriptive frames and give a new simple proof of Gerčiu’s result [9, 8] that all extensions of the Rieger-Nishimura logic RN have the fmp. Moreover, we show that each extension of RN has the poly-size model property, thus improving on [9]. Furthermore, for each function f: \omega -> \omega, we construct an extension Lf of KG such that Lf has the fmp, but does not have the f-size model property. We also give a new simple proof of another result of Gerčiu [9] characterizing the only extension of KG that bounds the fmp for extensions of KG. We conclude the paper by proving that RN.KC = RN + (¬p \vee ¬¬p) is the only pre-locally tabular extension of KG, introduce the internal depth of an extension L of RN, and show that L is locally tabular if and only if the internal depth of L is finite
Coalgebraic Geometric Logic: Basic Theory
Using the theory of coalgebra, we introduce a uniform framework for adding
modalities to the language of propositional geometric logic. Models for this
logic are based on coalgebras for an endofunctor on some full subcategory of
the category of topological spaces and continuous functions. We investigate
derivation systems, soundness and completeness for such geometric modal logics,
and we we specify a method of lifting an endofunctor on Set, accompanied by a
collection of predicate liftings, to an endofunctor on the category of
topological spaces, again accompanied by a collection of (open) predicate
liftings. Furthermore, we compare the notions of modal equivalence, behavioural
equivalence and bisimulation on the resulting class of models, and we provide a
final object for the corresponding category
A New Game Equivalence and its Modal Logic
We revisit the crucial issue of natural game equivalences, and semantics of
game logics based on these. We present reasons for investigating finer concepts
of game equivalence than equality of standard powers, though staying short of
modal bisimulation. Concretely, we propose a more finegrained notion of
equality of "basic powers" which record what players can force plus what they
leave to others to do, a crucial feature of interaction. This notion is closer
to game-theoretic strategic form, as we explain in detail, while remaining
amenable to logical analysis. We determine the properties of basic powers via a
new representation theorem, find a matching "instantial neighborhood game
logic", and show how our analysis can be extended to a new game algebra and
dynamic game logic.Comment: In Proceedings TARK 2017, arXiv:1707.0825
Bitopological Duality for Distributive Lattices and Heyting Algebras
We introduce pairwise Stone spaces as a natural bitopological generalization of Stone spaces—the duals of Boolean algebras—and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important for the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, co-Heyting algebras, and bi-Heyting algebras, thus providing two new alternatives of Esakia’s duality
Games for topological fixpoint logic
Topological fixpoint logics are a family of logics that admits topological models and where the fixpoint operators are defined with respect to the topological interpretations. Here we consider a topological fixpoint logic for relational structures based on Stone spaces, where the fixpoint operators are interpreted via clopen sets. We develop a game-theoretic semantics for this logic. First we introduce games characterising clopen fixpoints of monotone operators on Stone spaces. These fixpoint games allow us to characterise the semantics for our topological fixpoint logic using a two-player graph game. Adequacy of this game is the main result of our paper. Finally, we define bisimulations for the topological structures under consideration and use our game semantics to prove that the truth of a formula of our topological fixpoint logic is bisimulation-invariant
An Algebraic Approach to Inquisitive and DNA-Logics
This article provides an algebraic study of the propositional system InqB of inquisitive logic. We also investigate the wider class of DNA-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, DNA -varieties. We prove that the lattice of DNA-logics is dually isomorphic to the lattice of DNA -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff's classic variety theorems. We also introduce locally finite DNA -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of InqB is dually isomorphic to the ordinal omega + 1 and give an axiomatisation of these logics via Jankov DNA -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].(1)Peer reviewe
Positive Modal Logic Beyond Distributivity
We develop a duality for (modal) lattices that need not be distributive, and
use it to study positive (modal) logic beyond distributivity, which we call
weak positive (modal) logic. This duality builds on the Hofmann, Mislove and
Stralka duality for meet-semilattices. We introduce the notion of
-persistence and show that every weak positive modal logic is
-persistent. This approach leads to a new relational semantics for weak
positive modal logic, for which we prove an analogue of Sahlqvist
correspondence result
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