quantum logic
Introduction
In classical physics, results regarding experiments performed on physical propositions^{} obey classical propositional logic. When a ball is tossed, one can, at any given time $t$, measure its position $x(t)$ and velocity $v(t)$ with great precision almost “simultaneously”. If we know where the ball is, then we know, at that location $x(t)$, whether it is moving at $v(t)$ or not.
Let’s translate^{} the above sentence^{} into a logical statement. Suppose $p$ is the statement that “the ball is at $x(t)$”, and $q$ is the statement that “the ball is traveling at velocity $v(t)$”. Then last sentence in the previous paragraph becomes
if $p$ is true, then either $p\wedge q$ is true or $p\wedge {q}^{\prime}$ is true.
Put it more succinctly, we have the tautology^{}
$$p\to (p\wedge q)\vee (p\wedge {q}^{\prime}).$$ 
Converted to the language^{} of Boolean algebra^{} (via Lindenbaum’s construction (http://planetmath.org/LindenbaumAlgebra)), it says
$$p\le (p\wedge q)\vee (p\wedge {q}^{\prime})$$ 
which is clearly true, since the right hand side is equal to $p$ (the distributive property of a Boolean algebra).
However, when we downsize the scale of the system (the ball) to something at the subatomic level, say, an electron, we enter the realm of nonclassical physics (quantum mechanics), the story is much different. Because of the uncertainty principle, the nature of being able to measure everything simultaneously is gone here. Even if we know the location $x(t)$, we will not be able to know its velocity $v(t)$. In other words, the above inequality no longer holds. A new logic is needed for reasoning in nonclassical physics, in particular, quantum theory^{}.
In the 1930’s, Birkhoff and von Neumann introduced a new kind of logic in dealing with quantum mechanics in their paper ”The logic of quantum mechanics”. In that paper, physical constructs were converted into mathematical ones. For example, a physical observable is nothing more than a Hermitian operator acting on an infinite dimensional Hilbert space^{} over the complex numbers. And it turns out that many of the concepts quantum mechanics can be explained in terms of doing math on operators in a Hilbert space. Furthermore, the logic of quantum mechanics corresponds to that of the set of closed subspaces of an infinite dimensional Hilbert space. This set is a lattice^{} and it is not a Boolean algebra, because its lack of distributivity, and therefore does not correspond to classical propositional logic. They tried to axiomatize this lattice as a complete^{} complemented modular lattice^{} (with atoms). It was soon realized (by Piron) that this lattice is not even modular.
Definition
Today, there are several inequivalent definitions of quantum logic^{}, but all of which are in one form or another a generalization^{} of a Boolean algebra (the algebraic form of classical logic). Initially, a quantum logic is none other than the lattice of projection operators over an infinitedimensional separable Hilbert space. Since its introduction, the definition has been generalized, stripping away unnecessary details (atomicity for example) while keeping the essentials.
One of the more general definitions says that a quantum logic is an ordered pair $(L,F)$, where $L$ is an orthomodular poset $L$, and $F$ is a set of functions on $L$, called states, taking values in the unit inverval $[0,1]$, satisfying the following conditions:

1.
$f(0)=0$ for each $f\in F$,

2.
$f(1)=1$ for each $f\in F$,

3.
if $a\le {b}^{\u27c2}$, then $f(a\vee b)=f(a)+f(b)$ for each $f\in F$

4.
if $f(a)\le f(b)$ for each $f\in F$, then $a\le b$. We say that $F$ is full.
One can think of $L$ as a physical system (an electron, for example), and elements of $L$ are propositions concerning the system. $F$ can be regarded as a set of probability measures on the propositions. Conditions 1 and 2 say that $1$ and $0$ are propositions of $L$ that are always true or always false, regardless of its states. Condition 3 is equivalent^{} to the wellknown law of finite additivity of a probability measure.
Remark. One can define the notion of compatibility of a pair of elements (propositions) in a quantum logic: $a,b\in L$ are compatible if they orthogonally commute, that is, there exist pairwise orthogonal^{} elements $c,d,e\in L$ such that $a=c\vee e$ and $b=d\vee e$. It turns out that compatible propositions exactly parallel those physical propositions which can be experimentally tested “simultaneously”. A quantum logic $L$ in which every two propositions are compatible is said to be a classical logic. It is not hard to show that a classical logic is a Boolean algebra.
References
 1 L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
 2 G. Birkhoff, Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
 3 G. Birkhoff, J. von Neumann, The logic of quantum mechanics, Ann. of Math. 37, pp. 823843, (1936).
 4 D. W. Cohen, An Introduction to Hilbert Space and Quantum Logic, Springer, (1989).
 5 P. Mittelstaedt, Quantum Logic, D. Reidel Publishing Company, (1978).
Title  quantum logic 
Canonical name  QuantumLogic 
Date of creation  20130322 16:49:09 
Last modified on  20130322 16:49:09 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03G12 
Synonym  compatible propositions 
Related topic  LatticeOfProjections 
Related topic  QuantumLogicsTopoi 
Related topic  TopicEntryOnFoundationsOfMathematics 
Related topic  ETAS 
Related topic  QuantumTopos 
Related topic  QuantumGroupsAndVonNeumannAlgebras 
Defines  state 
Defines  full set of states 
Defines  compatible 
Defines  classical logic 