511 research outputs found
Belief as Willingness to Bet
We investigate modal logics of high probability having two unary modal
operators: an operator expressing probabilistic certainty and an operator
expressing probability exceeding a fixed rational threshold . Identifying knowledge with the former and belief with the latter, we may
think of as the agent's betting threshold, which leads to the motto "belief
is willingness to bet." The logic for has an
modality along with a sub-normal modality that extends
the minimal modal logic by way of four schemes relating
and , one of which is a complex scheme arising out of a theorem due to
Scott. Lenzen was the first to use Scott's theorem to show that a version of
this logic is sound and complete for the probability interpretation. We
reformulate Lenzen's results and present them here in a modern and accessible
form. In addition, we introduce a new epistemic neighborhood semantics that
will be more familiar to modern modal logicians. Using Scott's theorem, we
provide the Lenzen-derivative properties that must be imposed on finite
epistemic neighborhood models so as to guarantee the existence of a probability
measure respecting the neighborhood function in the appropriate way for
threshold . This yields a link between probabilistic and modal
neighborhood semantics that we hope will be of use in future work on modal
logics of qualitative probability. We leave open the question of which
properties must be imposed on finite epistemic neighborhood models so as to
guarantee existence of an appropriate probability measure for thresholds
.Comment: Removed date from v1 to avoid confusion on citation/reference,
otherwise identical to v
PDL as a Multi-Agent Strategy Logic
Propositional Dynamic Logic or PDL was invented as a logic for reasoning
about regular programming constructs. We propose a new perspective on PDL as a
multi-agent strategic logic (MASL). This logic for strategic reasoning has
group strategies as first class citizens, and brings game logic closer to
standard modal logic. We demonstrate that MASL can express key notions of game
theory, social choice theory and voting theory in a natural way, we give a
sound and complete proof system for MASL, and we show that MASL encodes
coalition logic. Next, we extend the language to epistemic multi-agent
strategic logic (EMASL), we give examples of what it can express, we propose to
use it for posing new questions in epistemic social choice theory, and we give
a calculus for reasoning about a natural class of epistemic game models. We end
by listing avenues for future research and by tracing connections to a number
of other logics for reasoning about strategies.Comment: 10 pages, Poster presentation at TARK 2013 (arXiv:1310.6382)
http://www.tark.or
Yet More Modal Logics of Preference Change and Belief Revision
We contrast Bonanno's `Belief Revision in a Temporal Framework'
\cite{Bonanno07:briatfTV} with preference change and belief revision
from the perspective of dynamic epistemic logic (DEL).
For that, we extend the
logic of communic
Perception and Change in Update Logic
Abstract Three key ways of updating one's knowledge are (i) perception of states
of affairs, e.g., seeing with one's own eyes that something is the case, (ii) recep-
tion of messages, e.g., being told that something is the case, and (iii) drawing new
conclusions from known facts. If one represents knowledge by means of Kripke
models, the implicit assumption is that drawing conclusions is immediate. This as-
sumption of logical omniscience is a useful abstraction. It leaves the distinction
between (i) and (ii) to be accounted for. In current versions of Update Logic (Dy-
namic Epistemic Logic, Logic of Communication and Change) perception and mes-
sage reception are not distinguished. This paper proposes an extension of Update
Logic that makes this distinction explicit. The logic deals with three kinds of up-
dates: announcements, changes of the world, and observations about the world in
the presence of witnesses. The resulting logic is shown to be complete by means of
a reduction to epistemic propositional dynamic logic by a well known method
Varieties of Belief and Probability
For reasoning about uncertain situations, we have probability theory, and we have logics
of knowledge and belief. How does elementary probability theory relate to epistemic
logic and the logic of belief? The paper focuses on the notion of betting belief, and
interprets a language for knowledge and belief in two kinds of models: epistemic neighbourhood
models and epistemic probability models. It is shown that the first class of
models is more general in the sense that every probability model gives rise to a neighbourhood
model, but not vice versa. The basic calculus of knowledge and betting belief
is incomplete for probability models. These formal results were obtained in Van Eijck
and Renne [9]
Dynamic reasoning without variables
A variable free notation for dynamic logic is proposed which takes its cue from De Bruijn's variable free notation for lambda calculus. De Bruijn indexing replaces variables by indices which indicate the distance to their binders. We propose to use reverse De Bruijn indexing, which works almost the same, only now the indices refer to the depth of the binding operator in the formula. The resulting system is analysed at length and applied to a new rational reconstruction of discourse representation theory. It is argued that the present system of dynamic logic without variables provides an explicit account of anaphoric context and yields new insight into the dynamics of anaphoric linking in reasoning. A calculus for dynamic reasoning with anaphora is presented and its soundness and completeness are established
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