Parfaits Miroirs De L'univers'': A `Virtual' Interpretation of Leibnizian Metaphysics

Abstract

In "Virtual Modality" (in press) a long technical article I developed an adequate metalogical semantics for modal extensions of a recursively axiomatisable first-order theory T, and called the motivations for this se mantic framework 'Leibnizian". The purpose of this somewhat less formal essay will be to offer a partial justification my invocation of one of the more venerable names in the history of western philosophy. To this end, let T be such an incomplete axiomatisable theory, and C the completion of T's Lindenbaum algebra the (Dedekind-MacNeill) completion of the the boolean algebra of formulas of T modulo T-provable equivalence. In the present context, C may be concretely realised as the Stone algebra, or boolean algebra of regular open sets in T's Stone space, where the latter is construed as the (metatheoretically defined) space of Henkin interpretations of T. Detailed information about Stone algebras and Stone spaces (so named for their twentieth-century discoverer and investigator, Marshall Stone), as well as the notions of boolean-valued 'randomness' which underlie these results, may be found in (Bell) and (Bell and Machover), (Jech) and other articles, textbooks and monographs cited in (Boos 1998).