Abstract

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions'', ``substances'' or ``substantial forms''. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances'' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae νNϕ of formulae ν of a normal modal theory Tm based on T, such that the counterpart `ξi' of a the modal variable `xi' of L(Tm) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T).The article's basic completeness-result (2.13) then establishes that ϕvarphi; is a theorem of Tm iff [[νN(ϕ) is aconsequenceof νN(Tm) for each extension N of T which is a subtheory of the canonical generic theory (ultrafilter) u]] = 1 – or equivalently, that Tm is consistent iff[[there is anextension N of T such that N is a subtheory of the canonical generic theory u, and νN(ϕ) for all ϕ in Tm]] > 0.The proof of thiscompleteness-result also shows that an N which provides a countermodel for a modally unprovable ϕ – or equivalently, a closed set in the Stone space St(T) in the sense of V(C) – is intertranslatable with an `accessibility'-relation of a closely related Kripke-semantics whose `worlds' are generic extensions of an initial universe V via C.This interrelation providesa fairly precise rationale for the semantics' recourse to C-valued structures, and exhibits a sense in which the boolean-valued context is sharp.