Introduction

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 $p\wedge q'$ is true.

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 non-classical 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 non-classical 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). But soon, it was realized (by Piron) that there exists a sublattice that not modular, rendering the whole lattice non-modular.

Definition

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 $a\le b^{\perp}$ , 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.

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.

Bibliography

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. 823-843, (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).