Multivalued logic to transform potential into actual objects (Q2454637)

This inspiring article deals with an interpretational problem of first-order multi-valued logics. Namely, in some fuzzy model, a formula \(\exists x \alpha(x)\) may be fully true (i.e.~hold to the degree 1) although there is no \(d\) such that \(\alpha(d)\) is fully true. The reason is simply that the existential quantifier is interpreted by the supremum operator. The author refers to this situation as ``potential existence'', as opposed to ``actual existence'', which means that a ``witness'' of the truth of \(\exists x \alpha(x)\) does exist. Two illustrative examples are elaborated. The first one is the so-called wide set theory, which axiomatizes the graded property INF of being infinite; it has a model consisting of finite sets, and the formula \(\exists x\text{INF}(x)\) then provides the example. As the main result, it is shown that any model for a first-order multi-valued logic possesses an elementary extension in which potential and actual existence are equivalent, provided that the logical connectives are continuous. To this end, the ultrapower construction is appropriately adapted. The fact that using ultrafilters easily leads away from intuition is admitted. However, an approach is proposed how to overcome the problem, the two examples given as the background. Namely, a quotiont of a substructure of the ultrapower is considered whose elements can, in view of the examples, well be identified with processes of abstraction.