3,119 research outputs found
On Modal Logics of Partial Recursive Functions
The classical propositional logic is known to be sound and complete with
respect to the set semantics that interprets connectives as set operations. The
paper extends propositional language by a new binary modality that corresponds
to partial recursive function type constructor under the above interpretation.
The cases of deterministic and non-deterministic functions are considered and
for both of them semantically complete modal logics are described and
decidability of these logics is established
On the Satisfiability of Quasi-Classical Description Logics
Though quasi-classical description logic (QCDL) can tolerate the inconsistency of description logic in reasoning, a knowledge base in QCDL possibly has no model. In this paper, we investigate the satisfiability of QCDL, namely, QC-coherency and QC-consistency and develop a tableau calculus, as a formal proof, to determine whether a knowledge base in QCDL is QC-consistent. To do so, we repair the standard tableau for DL by introducing several new expansion rules and defining a new closeness condition. Finally, we prove that this calculus is sound and complete. Based on this calculus, we implement an OWL paraconsistent reasoner called QC-OWL. Preliminary experiments show that QC-OWL is highly efficient in checking QC-consistency
Many-valued coalgebraic logic over semi-primal varieties
We study many-valued coalgebraic logics with semi-primal algebras of
truth-degrees. We provide a systematic way to lift endofunctors defined on the
variety of Boolean algebras to endofunctors on the variety generated by a
semi-primal algebra. We show that this can be extended to a technique to lift
classical coalgebraic logics to many-valued ones, and that (one-step)
completeness and expressivity are preserved under this lifting. For specific
classes of endofunctors, we also describe how to obtain an axiomatization of
the lifted many-valued logic directly from an axiomatization of the original
classical one. In particular, we apply all of these techniques to classical
modal logic
Semantics and Ontology:\ud On the Modal Structure of an Epistemic Theory of Meaning
In this paper I shall confront three basic questions.\ud
First, the relevance of epistemic structures, as formalized\ud
and dealt with by current epistemic logics, for a\ud
general Theory of meaning. Here I acknowledge M. Dummett"s\ud
idea that a systematic account of what is meaning of\ud
an arbitrary language subsystem must especially take into\ud
account the inferential components of meaning itself. That\ud
is, an analysis of meaning comprehension processes,\ud
given in terms of epistemic logics and semantics for epistemic\ud
notions.\ud
The second and third questions relate to the ontological\ud
and epistemological framework for this approach.\ud
Concerning the epistemological aspects of an epistemic\ud
theory of meaning, the question is: how epistemic logics\ud
can eventually account for the informative character of\ud
meaning comprehension processes. "Information� seems\ud
to be built in the very formal structure of epistemic processes,\ud
and should be exhibited in modal and possibleworld\ud
semantics for propositional knowledge and belief.\ud
However, it is not yet clear what is e.g. a possible world.\ud
That is: how it can be defined semantically, other than by\ud
accessibility rules which merely define it by considering its\ud
set-theoretic relations with other sets-possible worlds.\ud
Therefore, it is not clear which is the epistemological status\ud
of propositional information contained in the structural\ud
aspects of possible world semantics. The problem here\ud
seems to be what kind of meaning one attributes to the\ud
modal notion of possibility, thus allowing semantical and\ud
synctactical selectors for possibilities. This is a typically\ud
Dummett-style problem.\ud
The third question is linked with this epistemological\ud
problem, since it is its ontological counterpart. It concerns\ud
the limits of the logical space and of logical semantics for a\ud
of meaning. That is, it is concerned with the kind of\ud
structure described by inferential processes, thought, in a\ud
fregean perspective, as pre-conditions of estentional\ud
treatment of meaning itself. The second and third questions\ud
relate to some observations in Wittgenstein"s Tractatus.\ud
I shall also try to show how their behaviour limits the\ud
explicative power of some semantics for epistemic logics\ud
(Konolige"s and Levesque"s for knowledge and belief)
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 of Flat Coalgebraic Fixpoint Logics
Modal fixpoint logics traditionally play a central role in computer science,
in particular in artificial intelligence and concurrency. The mu-calculus and
its relatives are among the most expressive logics of this type. However,
popular fixpoint logics tend to trade expressivity for simplicity and
readability, and in fact often live within the single variable fragment of the
mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL,
and the logic of common knowledge. Extending this notion to the generic
semantic framework of coalgebraic logic enables covering a wide range of logics
beyond the standard mu-calculus including, e.g., flat fragments of the graded
mu-calculus and the alternating-time mu-calculus (such as alternating-time
temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We
give a generic proof of completeness of the Kozen-Park axiomatization for such
flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on
Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer
Science, Springer, 2010, pp. 524-53
- …