Modal Aggregation and the Theory of Paraconsistent Filters
Peter Apostoli · Mathematical logic quarterly · 1996
Abstract This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k‐filter of propositions. A k‐filter on a non‐empty set I is a collection of subsets of I which (i) contains I, (ii) is closed under supersets on I, and (iii) contains ∪{Xi ≤ Xj : 0 ≤ i < j ≤ k} whenever it contains the subsets X0,…, Xk. The mathematical content of the proof that weakly aggregative modal logic is complete relative to k‐ary frame theory, the standard semantic idiom for weakly aggregative modal logic (see [1]) is presented in language‐independent terms as a representation theorem for k‐filters: every non‐trivial k‐filter is included in the union of ≤ k non‐trivial filters. The elementary theory of k‐filters is developed and then applied in the form of an ultrafilter extension result for k‐ary frame theory. Mathematics Subject Classification: 03B45.