# Truth-value multisets with only one type of element

Suppose we had a multiset of truth-values [T, T], and that was it. Letting those be indexed as T1 and T2, suppose a twofold fragmenting of a related set of propositions (maybe not a set of all-propositions-whatsoever) and say that propositions from fragment A are true only when they map to T1 and B's elements go with T2. Then, however, sometimes elements of B map to T1, say: and that is how falsity arises, here (mismatches of T-maps).

Is that enough to have a theory of falsehood in a truth-value multiset, or would attempts to translate/encode [T, T] as {T, F} not work out like that at all?

Precedent:

Čulina[21] includes the following:

I don't know how well the following can be applied, here, or to the request for an example of how to separate propositions into two sets for the purposes of attempting the rest of the idea, but:

This occurs in a discussion of a one-dimensional example of an n-dimensional type of "oppositional geometries." In writings about such graph-like things that I am previously familiar with, the main author (Alessio Moretti) focuses on a need to use bi-simplexes to "geometrically complete" the graphs (I think he says that he doesn't want the geometries to be taken for just graphs with logical flavoring, so maybe we should take them as generalized over a variable graph-theoretic type that ranges over the conceptual differences between graphs, multigraphs, and hypergraphs; but otherwise I'm not sure why the geometries are sufficiently distinct from samples of those structures). I think some of the simplexes that Moretti uses are at a flipped angle compared to the ones they're paired with in whole bi-simplexes, but otherwise, having them side-by-side might be like having [A, A].

• Give an example. Commented Jul 15, 2023 at 13:28
• @Daron offhand (I thought of truth-value multisets just yesterday, so I haven't had time to think through the possible "experiments in logic" very much) let's say ethics propositions really are non-factual/non-realist in some robust manner, except here it turns out that they're factual/realist for a different T-multielement than propositions about non-ethical topics are. (I don't think the bare concept of facts is strong enough to support the "fact/value" or "is/ought" distinction, but stronger abstract definitions of facts might support those.) Commented Jul 15, 2023 at 13:48
• Otherwise, the idea seems like it could be used in a semantics for substructural logics with premise repetition. There's a reliable contributor here who I think is invested in questions of substructural logic, I'm thinking that he'll be familiar with information pertinent to my question. So maybe such an idea has already been used like so, and that's what I'll find out. Commented Jul 15, 2023 at 13:51
• I mean an example of a "twofold fragmenting of a related set of propositions". Commented Jul 15, 2023 at 14:40
• @Daron well "ethical propositions vs. non-ethical propositions" does that trick too, doesn't it? IIRC, Russell thought that the set of propositional functions outweighed the set of objects (or something along those lines) and we might go on to think that there are "too many" propositions, that "all of them" form a proper class rather than a proper set, but then I suppose we could work with a proper class of ordered pairs of propositions nevertheless, maybe. Commented Jul 15, 2023 at 15:00