8,863 research outputs found
Categories for Dynamic Epistemic Logic
The primary goal of this paper is to recast the semantics of modal logic, and
dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We
first review the category of relations and categories of Kripke frames, with
particular emphasis on the duality between relations and adjoint homomorphisms.
Using these categories, we then reformulate the semantics of DEL in a more
categorical and algebraic form. Several virtues of the new formulation will be
demonstrated: The DEL idea of updating a model into another is captured
naturally by the categorical perspective -- which emphasizes a family of
objects and structural relationships among them, as opposed to a single object
and structure on it. Also, the categorical semantics of DEL can be merged
straightforwardly with a standard categorical semantics for first-order logic,
providing a semantics for first-order DEL.Comment: In Proceedings TARK 2017, arXiv:1707.0825
Taking Heisenberg's Potentia Seriously
It is argued that quantum theory is best understood as requiring an
ontological duality of res extensa and res potentia, where the latter is
understood per Heisenberg's original proposal, and the former is roughly
equivalent to Descartes' 'extended substance.' However, this is not a dualism
of mutually exclusive substances in the classical Cartesian sense, and
therefore does not inherit the infamous 'mind-body' problem. Rather, res
potentia and res extensa are proposed as mutually implicative ontological
extants that serve to explain the key conceptual challenges of quantum theory;
in particular, nonlocality, entanglement, null measurements, and wave function
collapse. It is shown that a natural account of these quantum perplexities
emerges, along with a need to reassess our usual ontological commitments
involving the nature of space and time.Comment: Final version, to appear in International Journal of Quantum
Foundation
Categorical Semantics for Functional Reactive Programming with Temporal Recursion and Corecursion
Functional reactive programming (FRP) makes it possible to express temporal
aspects of computations in a declarative way. Recently we developed two kinds
of categorical models of FRP: abstract process categories (APCs) and concrete
process categories (CPCs). Furthermore we showed that APCs generalize CPCs. In
this paper, we extend APCs with additional structure. This structure models
recursion and corecursion operators that are related to time. We show that the
resulting categorical models generalize those CPCs that impose an additional
constraint on time scales. This constraint boils down to ruling out
-supertasks, which are closely related to Zeno's paradox of Achilles
and the tortoise.Comment: In Proceedings MSFP 2014, arXiv:1406.153
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
Previous work on applications of Abstract Differential Geometry (ADG) to
discrete Lorentzian quantum gravity is brought to its categorical climax by
organizing the curved finitary spacetime sheaves of quantum causal sets
involved therein, on which a finitary (:locally finite), singularity-free,
background manifold independent and geometrically prequantized version of the
gravitational vacuum Einstein field equations were seen to hold, into a topos
structure. This topos is seen to be a finitary instance of both an elementary
and a Grothendieck topos, generalizing in a differential geometric setting, as
befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies.
The paper closes with a thorough discussion of four future routes we could take
in order to further develop our topos-theoretic perspective on ADG-gravity
along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references
polished) Submitted to the International Journal of Theoretical Physic
Does God Have a Body? Rāmānuja’s Challenge to the Christian Tradition
The Christian tradition’s core theological assertion is the embodiment of God in the person of Jesus Christ. Yet, even while asserting God’s incarnation in space and time, the tradition has usually denied embodiment unto the Godhead itself. Theologians have based this denial on Jewish iconoclasm, Greek idealism, and inferences from God’s omnipresence, transcendence, and infinity. This speculative essay will argue that Hindu Śrīvaiṣṇava theologian Rāmānuja successfully addresses these concerns. He argues for the embodiment of an omnipresent, transcendent, and infinite personal God. Rāmānuja largely derives his arguments from the Hindu scriptures. Nevertheless, their rational explication and internal coherence render divine embodiment a legitimate theological option for the Christian tradition, whose scriptures present both anthropomorphic and iconoclastic concepts of God. Since Godhead embodiment is ontologically coherent and rationally defensible, Christians must accept or reject it based on axiological grounds, by evaluating the felt consequences of the doctrine in Christian life. For embodied beings, any pastoral theology should commend embodiment within the Godhead
From Simple to Complex and Ultra-complex Systems:\ud A Paradigm Shift Towards Non-Abelian Systems Dynamics
Atoms, molecules, organisms distinguish layers of reality because of the causal links that govern their behavior, both horizontally (atom-atom, molecule-molecule, organism-organism) and vertically (atom-molecule-organism). This is the first intuition of the theory of levels. Even if the further development of the theory will require imposing a number of qualifications to this initial intuition, the idea of a series of entities organized on different levels of complexity will prove correct. Living systems as well as social systems and the human mind present features remarkably different from those characterizing non-living, simple physical and chemical systems. We propose that super-complexity requires at least four different categorical frameworks, provided by the theories of levels of reality, chronotopoids, (generalized) interactions, and anticipation
From Simple to Complex and Ultra-complex Systems:\ud A Paradigm Shift Towards Non-Abelian Systems Dynamics
Atoms, molecules, organisms distinguish layers of reality because of the causal links that govern their behavior, both horizontally (atom-atom, molecule-molecule, organism-organism) and vertically (atom-molecule-organism). This is the first intuition of the theory of levels. Even if the further development of the theory will require imposing a number of qualifications to this initial intuition, the idea of a series of entities organized on different levels of complexity will prove correct. Living systems as well as social systems and the human mind present features remarkably different from those characterizing non-living, simple physical and chemical systems. We propose that super-complexity requires at least four different categorical frameworks, provided by the theories of levels of reality, chronotopoids, (generalized) interactions, and anticipation
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