A Refined Geometry of Logic

David Miller
2009 Principia: An International Journal of Epistemology  
In order to measure the degree of dissimilarity between elements of a Boolean algebra, the author's (1984) proposed to use pseudometrics satisfying generalizations of the usual axioms for identity. The proposal is extended, as far as is feasible, from Boolean algebras (algebras of propositions) to Brouwerian algebras (algebras of deductive theories). The relation between Boolean and Brouwerian geometries of logic turns out to resemble in a curious way the relation between Euclidean and
more » ... dean geometries of physical space. The paper ends with a brief consideration of the problem of the metrization of the algebra of theories. In (1984) I proposed an axiom system for a function d on a poset L that in a rather straightforward manner generalizes and strengthens the usual elementary axioms for identity. The value of d(a, c) can be understood as one way, by no means the only one, of grading the degree of dissimilarity or diversity of the elements a and c of L. The function d is not required to be real-valued, but that is its most natural interpretation, in which case, as is easily demonstrated, it satisfies the usual requirements of a pseudometric operation. The system was applied to an arbitrary Boolean algebra L, for which it was proved to be equivalent, given some appropriate definitions, to two other axiomatic systems: , a different and more familiar set of axioms for a pseudometric operation d; and , the standard system of axioms for an unnormalized measure or positive valuation function µ. As was remarked, the close relation between the systems and in Boolean algebras (in which every strictly positive unnormalized measure is also a positive isotone valuation) is well known. The paper also drew attention to a number of variations and generalizations, some of them first mentioned in Miller (1979). Several of the derivations, for example, but not all of them, can be carried out when L is assumed not to be a Boolean algebra, but only a lattice. It was noted that if L is a lattice, the system , together with the assumption that the pseudometric d is a metric (that is, d(a, c) = 0 only if a = c), compels L to be modular (Birkhoff 1967, Theorem X.2). It was observed also that if one of the axioms in is strengthened from an inequality to an identity, then L Principia 13(3): 339-56 (2009).
doi:10.5007/1808-1711.2009v13n3p339 fatcat:m6wxxijvbfb75hk43bfn745xz4