# 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. Jul 15 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.) Jul 15 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. Jul 15 at 13:51
• I mean an example of a "twofold fragmenting of a related set of propositions". Jul 15 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. Jul 15 at 15:00

## 1 Answer

In logic we have some propositions, and we are interested in some special functions from the set of propositions to the set of truth values {T, F}. For example we want the functions to commute with the AND and OR operations. You are proposing replacing the truth values with a multiset, that contains two copies of T and considering functions that can map propositions to one of the two copies. That's not a multiset. That's just a set with a bad choice of notation.

A multiset over the universe Ω for example is defined as a function Ω → ℕ. In your case the universe contains the symbol T and maybe some other stuff, and your truth multiset is the function F with F(T) = 2 and F(x) = 0 for xT. Looking at it like this, it is not at all clear what a map from a set of propositions into this multiset is supposed to mean.

• The function on [T, T] is not a multiset "just like that" but only [T, T] itself is (let T = 1 and go to [1, 1], for example). And on one level, I'm wondering if we could replace the usual notion of falsity as a qualitatively distinct element of {T, F}. On a metalinguistic level, we'd like to show how the mismatch parameter for mappings to [T1, T2] lines up with our intuitive notion of falsity, although I don't know if that's a stable proposal (hence my question); or, perhaps worse, there'd be a vicious circle where the mismatch-talk depends on a prior sense of falsity-talk. Jul 15 at 17:41
• As far as AND and OR go (or IF, for that matter), I've no clue yet how we'd work them out in this case and for now I don't want to presuppose that I'm trying to maintain the relevant commutativity (my most recent analysis of many-valued logic suggested over 7 truth values, if not arbitrarily many all-things-considered, and I got a decent truth-table system for AND/OR/IF out of that, but not something I've "proven" entirely useful/adequate). Jul 15 at 17:46
• It sounds like you don't understand the definition of a multiset. Jul 15 at 18:18
• Project Euclid has "Multiset Theory" which says: "A multiset is a collection of elements in which elements are allowed to repeat." I don't see that there's much, if any, of an absolute definition beyond that. Now, I upvoted your answer since I've seen that you've made contributions to the MathSE/OF and so I assume you are familiar with these topics, but you still need to cite your sources here or else I can't continue with that assumption. Jul 15 at 18:40
• I'm still going to need a source accompanied by a justification for preferring the given definition over the generic definition from the PE article. Jul 15 at 18:44