66 research outputs found
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
Completeness for Flat Modal Fixpoint Logics
This paper exhibits a general and uniform method to prove completeness for
certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form
\gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the
language L\sharp (\Gamma) is obtained by adding to the language of polymodal
logic a connective \sharp\_\gamma for each \gamma \epsilon. The term
\sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the
least fixed point of the functional interpretation of the term \gamma(x,
\varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma,
construct an axiom system which is sound and complete with respect to the
concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We
prove two results that solve this problem. First, let K\sharp (\Gamma) be the
logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint
axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma.
Provided that each indexing formula \gamma satisfies the syntactic criterion of
being untied in x, we prove this axiom system to be complete. Second,
addressing the general case, we prove the soundness and completeness of an
extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an
effective procedure that, given an indexing formula \gamma as input, returns a
finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded
by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever
\Gamma is finite
MacNeille completion and profinite completion can coincide on finitely generated modal algebras
Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by
piecing together results from various authors. Specifically, we show that if
is a residually finite, finitely generated modal algebra such that
has equationally definable principal
congruences, then the profinite completion of is isomorphic to its
MacNeille completion, and is smooth. Specific examples of such modal
algebras are the free -algebra and the free
-algebra.Comment: 5 page
Modal definability in enriched languages
Abstract The paper deals with polymodal languages combined with stan-dard semantics defined by means of some conditions on the frames. So a notion of "polymodal base " arises which provides various enrichments of the classical modal language. One of these enrichments, viz. the base £(R,-R), with modalities over a relation and over its complement, is the paper's main paradigm. The modal definability (in the spirit of van Benthem's correspon-dence theory) of arbitrary and ~-elementary classes of frames in this base and in some of its extensions, e.g., £(R,-R,R-1,_R-1), £(R,-R,=I=) etc., is described, and numerous examples of conditions definable there, as well as undefinable ones, are adduced. 81 Introduction Undoubtedly, first-order languages are reliable and universal tools for formalization. However, in some cases the cost of this universality is not fully acceptable: on the one hand we have the undecidability results, and on the other the fact that the expressive power of first-order languages does no
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