184 research outputs found
Conditionals and modularity in general logics
In this work in progress, we discuss independence and interpolation and
related topics for classical, modal, and non-monotonic logics
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Meaning of temperature in different thermostatistical ensembles
Depending on the exact experimental conditions, the thermodynamic properties
of physical systems can be related to one or more thermostatistical ensembles.
Here, we survey the notion of thermodynamic temperature in different
statistical ensembles, focusing in particular on subtleties that arise when
ensembles become non-equivalent. The 'mother' of all ensembles, the
microcanonical ensemble, uses entropy and internal energy (the most
fundamental, dynamically conserved quantity) to derive temperature as a
secondary thermodynamic variable. Over the past century, some confusion has
been caused by the fact that several competing microcanonical entropy
definitions are used in the literature, most commonly the volume and surface
entropies introduced by Gibbs. It can be proved, however, that only the volume
entropy satisfies exactly the traditional form of the laws of thermodynamics
for a broad class of physical systems, including all standard classical
Hamiltonian systems, regardless of their size. This mathematically rigorous
fact implies that negative 'absolute' temperatures and Carnot efficiencies
are not achievable within a standard thermodynamical framework. As an important
offspring of microcanonical thermostatistics, we shall briefly consider the
canonical ensemble and comment on the validity of the Boltzmann weight factor.
We conclude by addressing open mathematical problems that arise for systems
with discrete energy spectrum.Comment: 11 pages, 1 figur
Fidel Semantics for Propositional and First-Order Version of the Logic of CGâ3
Paraconsistent extensions of 3-valued Gödel logic are studied as tools for knowledge representation and nonmonotonic reasoning. Particularly, Osorio and his collaborators showed that some of these logics can be used to express interesting nonmonotonic semantics. CGâ3 is one of these 3-valued logics. In this paper, we introduce Fidel semantics for a certain calculus of CGâ3 by means of Fidel structures, named CGâ3-structures. These structures are constructed from enriched Boolean algebras with a special family of sets. Moreover, we also show that the most basic CGâ3-structures coincide with da CostaâAlvesâ bi-valuation semantics; this connection is displayed through a Representation Theorem for CGâ3-structures. By contrast, we show that for other paraconsistent logics that allow us to present semantics through Fidel structures, this connection is not held. Finally, Fidel semantics for the first-order version of the logic of CGâ3 are presented by means of adapting algebraic tools
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
Reasoning about Action: An Argumentation - Theoretic Approach
We present a uniform non-monotonic solution to the problems of reasoning
about action on the basis of an argumentation-theoretic approach. Our theory is
provably correct relative to a sensible minimisation policy introduced on top
of a temporal propositional logic. Sophisticated problem domains can be
formalised in our framework. As much attention of researchers in the field has
been paid to the traditional and basic problems in reasoning about actions such
as the frame, the qualification and the ramification problems, approaches to
these problems within our formalisation lie at heart of the expositions
presented in this paper
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