Quasi-Implicative Lattices and the Logic of Quantum Mechanics

نویسنده

  • Gary M. Hardegree
چکیده

Mittelstaedt has defined the class of quasi-implicative lattices and shown that an ortholattice (orthocomplemented lattice) is quasi-implicative exactly if it is orthomodular (quasi-modular). He has also shown that the quasi-implication operation is uniquely determined by the quasi-implicative conditions. One of Mittelstaedt's conditions, however, seems to lack immediate intuitive motivation. Consequently, this paper seeks to provide a number of reformulations of the quasi-implicative conditions which are more intuitively plausible. Three sets of conditions are examined, and it is shown that each set of conditions is both necessary and sufficient to ensure that an ortholattice is orthomodular, and each set of conditions uniquely specifies the implication operation to be Mittelstaedt's quasi-implication. Various properties of the quasi-implication are then investigated. In particular, it is shown that the quasi-implication fails to satisfy a number of laws associated with the classical material conditional. Various weakenings of these laws, satisfied by the quasi-implication, are also discussed.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Semi-G-filters, Stonean filters, MTL-filters, divisible filters, BL-filters and regular filters in residuated lattices

At present, the filter theory of $BL$textit{-}algebras has been widelystudied, and some important results have been published (see for examplecite{4}, cite{5}, cite{xi}, cite{6}, cite{7}). In other works such ascite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory inthe more general setting of residuated lattices is done, generalizing thatfor $BL$textit{-}algebras. Note that fil...

متن کامل

Semi-linear Varieties of Lattice-Ordered Algebras

We consider varieties of pointed lattice-ordered algebras satisfying a restricted distributivity condition and admitting a very weak implication. Examples of these varieties are ubiquitous in algebraic logic: integral or distributive residuated lattices; their {·}-free subreducts; their expansions (hence, in particular, Boolean algebras with operators and modal algebras); and varieties arising ...

متن کامل

RADICAL OF FILTERS IN RESIDUATED LATTICES

‎In this paper‎, ‎the notion of the radical of a filter in‎ ‎residuated lattices is defined and several characterizations of‎ ‎the radical of a filter are given‎. ‎We show that if F is a‎ ‎positive implicative filter (or obstinate filter)‎, ‎then‎ ‎Rad(F)=F‎. ‎We proved the extension theorem for radical of filters in residuated lattices‎. ‎Also‎, ‎we study the radical‎ ‎of filters in linearly o...

متن کامل

Quantum Logic and Quantum Reconstruction

Quantum logic understood as a reconstruction program had real successes and genuine limitations. This paper offers a synopsis of both and suggests a way of seeing quantum logic in a larger, still thriving context. keywords: quantum logic, quantum mechanics, lattice theory, orthomodular lattices, POVMs

متن کامل

A Rule-Based Logic for Quantum Information

The logic modelled by orthomodular lattices, usually referred to (in a rather controversial way) as quantum logic, constitutes the main formalism for the logical study of quantum mechanics. First introduced in the thirties by von Neumann and Birkho [Birkho and von Neumann, 1936], the logic modelled by orthomodular lattices (usually referred to, in a rather controversial way, as quantum logic) h...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2012