Using ordered pairs to transcend the lilitations of knowledge representation with application to quantum logic.
Phil Catabrese
The algebraic construct of 'forming ordered pairs of the members of a set' has been used and reused with amazing success. Among these developments are Integer Fractions to better measure distances with pairs of whole numbers; Dedekind Cuts - an early way to define real numbers with ordered pairs of subsets of fractions; Vectors; Quatemians; and others like complex numbers - ordered pairs of real numbers. In the realm of Boolean Logic, ordered pairs of Boolean propositions (alternately ordered pairs of subsets of some universe set) transcend the limitations of pure Boolean Iogic to represent partially true conditional statements such as those used by conditiona1 probability. Pure Boolean logic is unable to adequately represent uncertain conditional logic, but ordered pairs of BooIean propositions can and do represent this knowledge in the logic of Boolean Fractions, also called Conditional Events. A given B can not be represented by A or not B unless P(AJB) and P(B) are both 1. Otherwise, P(A1B) < P(A or not B). But A given B can be represented by an ordered pair of propositions (AJB). These Boolean Fractions apply to the natural language and logic of A given B, they apply to classical conditional probability calculations, to expert system if-then rules, and now to the logic of quantum measurements, a logic that Boolean algebra could not represent, but which Boolean Fractions can represent without Hilbert space.
IEEE International Conference on Information Reuse and Integration (IRI)•Jan 1, 2004•0 citations