Geometric viewpoint on the quantization of a fuzzy logic

Within the Hamiltonian framework, the propositions about a classical physical system are described in the Borel σ-algebra of a symplectic manifold (the phase space) where logical connectives are the standard set operations. Considering the geometric formulation of quantum mechanics we give a description of quantum propositions in terms of fuzzy events in a complex projective space equipped with Kähler structure (the quantum phase space) obtaining a quantized version of a fuzzy logic by deformation of the product t-norm

Bell-type inequalities for bivariate maps on orthomodular lattices

Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our invesigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.Comment: 14 pages, no figure

A Context-theoretic Framework for Compositionality in Distributional Semantics

Techniques in which words are represented as vectors have proved useful in many applications in computational linguistics, however there is currently no general semantic formalism for representing meaning in terms of vectors. We present a framework for natural language semantics in which words, phrases and sentences are all represented as vectors, based on a theoretical analysis which assumes that meaning is determined by context. In the theoretical analysis, we define a corpus model as a mathematical abstraction of a text corpus. The meaning of a string of words is assumed to be a vector representing the contexts in which it occurs in the corpus model. Based on this assumption, we can show that the vector representations of words can be considered as elements of an algebra over a field. We note that in applications of vector spaces to representing meanings of words there is an underlying lattice structure; we interpret the partial ordering of the lattice as describing entailment between meanings. We also define the context-theoretic probability of a string, and, based on this and the lattice structure, a degree of entailment between strings. We relate the framework to existing methods of composing vector-based representations of meaning, and show that our approach generalises many of these, including vector addition, component-wise multiplication, and the tensor product.Comment: Submitted to Computational Linguistics on 20th January 2010 for revie

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