39,170 research outputs found
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page
Topos Quantum Logic and Mixed States
The topos approach to the formulation of physical theories includes a new
form of quantum logic. We present this topos quantum logic, including some new
results, and compare it to standard quantum logic, all with an eye to
conceptual issues. In particular, we show that topos quantum logic is
distributive, multi-valued, contextual and intuitionistic. It incorporates
superposition without being based on linear structures, has a built-in form of
coarse-graining which automatically avoids interpretational problems usually
associated with the conjunction of propositions about incompatible physical
quantities, and provides a material implication that is lacking from standard
quantum logic. Importantly, topos quantum logic comes with a clear geometrical
underpinning. The representation of pure states and truth-value assignments are
discussed. It is briefly shown how mixed states fit into this approach.Comment: 25 pages; to appear in Electronic Notes in Theoretical Computer
Science (6th Workshop on Quantum Physics and Logic, QPL VI, Oxford, 8.--9.
April 2009), eds. B. Coecke, P. Panangaden, P. Selinger (2010
Coherent frequentism
By representing the range of fair betting odds according to a pair of
confidence set estimators, dual probability measures on parameter space called
frequentist posteriors secure the coherence of subjective inference without any
prior distribution. The closure of the set of expected losses corresponding to
the dual frequentist posteriors constrains decisions without arbitrarily
forcing optimization under all circumstances. This decision theory reduces to
those that maximize expected utility when the pair of frequentist posteriors is
induced by an exact or approximate confidence set estimator or when an
automatic reduction rule is applied to the pair. In such cases, the resulting
frequentist posterior is coherent in the sense that, as a probability
distribution of the parameter of interest, it satisfies the axioms of the
decision-theoretic and logic-theoretic systems typically cited in support of
the Bayesian posterior. Unlike the p-value, the confidence level of an interval
hypothesis derived from such a measure is suitable as an estimator of the
indicator of hypothesis truth since it converges in sample-space probability to
1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly
extended to vector parameters of interest. The derivation of upper and lower
confidence levels from valid and nonconservative set estimators is formalize
On Expressing and Monitoring Oscillatory Dynamics
To express temporal properties of dense-time real-valued signals, the Signal
Temporal Logic (STL) has been defined by Maler et al. The work presented a
monitoring algorithm deciding the satisfiability of STL formulae on finite
discrete samples of continuous signals. The logic has been used to express and
analyse biological systems, but it is not expressive enough to sufficiently
distinguish oscillatory properties important in biology. In this paper we
define the extended logic STL* in which STL is augmented with a signal-value
freezing operator allowing us to express (and distinguish) detailed properties
of biological oscillations. The logic is supported by a monitoring algorithm
prototyped in Matlab. The monitoring procedure of STL* is evaluated on a
biologically-relevant case study.Comment: In Proceedings HSB 2012, arXiv:1208.315
Betting on Quantum Objects
Dutch book arguments have been applied to beliefs about the outcomes of
measurements of quantum systems, but not to beliefs about quantum objects prior
to measurement. In this paper, we prove a quantum version of the probabilists'
Dutch book theorem that applies to both sorts of beliefs: roughly, if ideal
beliefs are given by vector states, all and only Born-rule probabilities avoid
Dutch books. This theorem and associated results have implications for
operational and realist interpretations of the logic of a Hilbert lattice. In
the latter case, we show that the defenders of the eigenstate-value orthodoxy
face a trilemma. Those who favor vague properties avoid the trilemma, admitting
all and only those beliefs about quantum objects that avoid Dutch books.Comment: 26 pages, 3 figures, 1 table; improved operational semantics, results
unchange
Variable types for meaning assembly: a logical syntax for generic noun phrases introduced by most
This paper proposes a way to compute the meanings associated with sentences
with generic noun phrases corresponding to the generalized quantifier most. We
call these generics specimens and they resemble stereotypes or prototypes in
lexical semantics. The meanings are viewed as logical formulae that can
thereafter be interpreted in your favourite models. To do so, we depart
significantly from the dominant Fregean view with a single untyped universe.
Indeed, our proposal adopts type theory with some hints from Hilbert
\epsilon-calculus (Hilbert, 1922; Avigad and Zach, 2008) and from medieval
philosophy, see e.g. de Libera (1993, 1996). Our type theoretic analysis bears
some resemblance with ongoing work in lexical semantics (Asher 2011; Bassac et
al. 2010; Moot, Pr\'evot and Retor\'e 2011). Our model also applies to
classical examples involving a class, or a generic element of this class, which
is not uttered but provided by the context. An outcome of this study is that,
in the minimalism-contextualism debate, see Conrad (2011), if one adopts a type
theoretical view, terms encode the purely semantic meaning component while
their typing is pragmatically determined
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