Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTest sets for tautologies in modular quantum logic
116
0
0.0
(
0
)
Authors
Christian Herrmann
Ask ChatGPT about the research
No Arabic abstract
As defined by Dunn, Moss, and Wang, an universal test set in an ortholattice $L$ is a subset $T$ such that each term takes value $1$, only, if it does so under all substitutions from $T$. Generalizing their result for ortholattices of subspaces of finite dimensional Hilbert spaces, we show that no infinite modular ortholattice of finite dimension admits a finite universal test set. On the other hand, answering a question of the same authors, we provide a countable universal test set for the ortholattice of projections of any type II$_1$ von Neumann algebra factor as well as for von Neumanns algebraic construction of a continuous geometry. These universal test sets consist of elements having rational normalized dimension with denominator a power of $2$.
rate research
Read More
We show that numerous distinctive concepts of constructive mathematics arise automatically from an antithesis translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically constructivize classical definitions, handling the resulting bookkeeping automatically.
The purpose of this note is to discuss some of the questions raised by Dunn, J. Michael; Moss, Lawrence S.; Wang, Zhenghan in Editors introduction: the third life of quantum logic: quantum logic inspired by quantum computing.
Probability logic has contributed to significant developments in belief types for game-theoretical economics. We present a new probability logic for Harsanyi Type spaces, show its completeness, and prove both a de-nesting property and a unique extension theorem. We then prove that multi-agent interactive epistemology has greater complexity than its single-agent counterpart by showing that if the probability indices of the belief language are restricted to a finite set of rationals and there are finitely many propositional letters, then the canonical space for probabilistic beliefs with one agent is finite while the canonical one with at least two agents has the cardinality of the continuum. Finally, we generalize the three notions of definability in multimodal logics to logics of probabilistic belief and knowledge, namely implicit definability, reducibility, and explicit definability. We find that S5-knowledge can be implicitly defined by probabilistic belief but not reduced to it and hence is not explicitly definable by probabilistic belief.
We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation.
A logic satisfies the interpolation property provided that whenever a formula {Delta} is a consequence of another formula {Gamma}, then this is witnessed by a formula {Theta} which only refers to the language common to {Gamma} and {Delta}. That is, the relational (and functional) symbols occurring in {Theta} occur in both {Gamma} and {Delta}, {Gamma} has {Theta} as a consequence, and {Theta} has {Delta} as a consequence. Both classical and intuitionistic predicate logic have the interpolation property, but it is a long open problem which intermediate predicate logics enjoy it. In 2013 Mints, Olkhovikov, and Urquhart showed that constant domain intuitionistic logic does not have the interpolation property, while leaving open whether predicate Godel logic does. In this short note, we show that their counterexample for constant domain intuitionistic logic does admit an interpolant in predicate Godel logic. While this has no impact on settling the question for predicate Godel logic, it lends some credence to a common belief that it does satisfy interpolation. Also, our method is based on an analysis of the semantic tools of Olkhovikov and it is our hope that this might eventually be useful in settling this question.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
