Sugihara monoids, RM3, infinite valued logic, and probability ...

Infinite-Valued Relevance Logic as a Probability Structure

Here’s a conceptual leap: if truth values form a bounded poset (e.g. real unit interval [0,1]), and conjunction, disjunction, and implication are operations preserving some form of ordering or residuation, you can begin to think of logic as probabilistic entailment.

Now, if we make the truth values correspond to probabilities (or credences), then:

A⇒B is strongest when the truth value of A is less than or equal to that of B.

This mimics conditional probability: P(B∣A) is highest when A almost implies B.

In fact, some researchers have developed algebraic models of conditional probability using residuated lattices or MV-algebras (multi-valued algebras from Łukasiewicz logic), and relevance logic’s demand for resource sensitivity fits naturally with context-sensitive probability assignments.

You can think of A⇒B not as a function of static truth values, but as "the degree to which A supports B," akin to Bayesian support.

#probability #rm3 #RelevanceLogic #ChatGPT

@muiren The Axiom of Weakening is invalid

#RM3 #RelevanceLogic

@skewray Oh, yeah, sure. Of course. That's why Judges use Relevance and Deontic logic. At least we can prove when something *is* inconsistent. (And then let a human decide). That's one of the nice things about 3 valued logic, it can refer to itself without its head exploding.

#RM3 #RelevanceLogic #paraconsistent

ChatGPT on categorical logic again.

🧱 Level 1: Morphisms = Proofs, Typed with Validity

You can think of these arrows as typed by a truth value — i.e., each morphism has a color: valid, invalid, plausible, context-sensitive, contradictory-but-derivable, etc.

In this sense, truth is not binary, but becomes a fiber over each morphism: a coloring or modality.

So your category becomes a fibration over a poset of truth values, or a category enriched in truth values — maybe in a Heyting algebra or relevance lattice.

🧱 Level 2: 2-Cells = Laws, Derivations, Transformations

Now we raise it to a 2-category:

0-cells: Propositions (types)

1-cells: Deductions / proof structures f:A→B

2-cells: Proofs of equivalence between proofs (e.g., natural transformations, rewrite rules, context substitution, modality shifts)

This is where natural transformations live: between two different "routes" from A to B. They express meta-logical structure: laws, policies, meanings.

Let’s say you have:

One arrow f:A→B defined in deontic logic (permission-based)

Another arrow g:A→B in alethic logic (necessity-based)

A natural transformation η:f⇒g might be a social contract or legal interpretation that maps from a space of permitted inferences to necessary ones — or vice versa.

#categorytheory #logic #RelevanceLogic

Asking ChatGPT about coloring diagrams with truth values. When you write A -> B the arrow can be valid or invalid, but you still want to talk about it

Level 1: Morphisms = Proofs, Typed with Validity

You can think of these arrows as typed by a truth value — i.e., each morphism has a color: valid, invalid, plausible, context-sensitive, contradictory-but-derivable, etc.

In this sense, truth is not binary, but becomes a fiber over each morphism: a coloring or modality.

So your category becomes a fibration over a poset of truth values, or a category enriched in truth values — maybe in a Heyting algebra or relevance lattice.

I'm not sure I've ever heard it say "maybe" before, but

#sugihara #monoid #RelevanceLogic

ChatGPT has a new sister called Monday. I will let you find out about that. Meanwhile here is what ChatGPT says about using enriched categories to model relevance logic:

An Example Sketch

Let V=Pos be a poset-enriched monoidal category where each hom-object is a set of “proofs” or “derivations,” ordered by resource usage.

Then C(A,B) is itself an object in Pos, i.e., a poset of ways to prove B from A.

The product ⊗ inside C does not come with free projections, so there is no arrow from (A⊗B) to B in general.

If someone claims “Surely, we can discard A and prove B anyway,” the poset of proofs for C(A⊗B,B) is _empty_, or has no minimal element if your ordering demands using all resources.

Thus, the absence of a projection morphism is encoded in the structure of the hom-object: it simply does not contain a suitable proof.

--
here 'resource usage' is 'relevant stuff'

You can write (A⊗B) -> B, in a diagram. But that arrow is "False", so it doesn't really "exist". Enriched categories capture this concept.

#RelevanceLogic #categorytheory #enrichedcategory #rm3

What does "pseudo relevant mean?"

I finally asked ChatGPT to explain to me why RM3 is considered "pseudo relevant"

This is one of those things that's so blindingly obvious I couldn't see it until somebody else pointed it out.

We start with the system R, which is defined in terms of a ternary relation Rxyz. There are a number of axioms.

RM is R + M = R plus the Mingle axiom.

\[ 𝑝→(𝑝→𝑝) \]So in that world, "R" is the definition of relevance. RM3 can prove a statement that R rejects, namely M, the Mingle axiom. Duh.

OK. I've mentioned elsewhere that M is forced if you construct RM properly. They added M to R because it's necessary.

But the question still remains! Why is something defined in terms of a relation Rxyz that models *syntactic* presence of a variable or not, the same thing as a computational set of 3x3 matrices ...

RM3 solves relevance fallacies just fine, using inconsistent values instead of irrelevant variables

#rm3 #RelevanceLogic #RelevanceFallacy

@koronkebitch Computers are like the real numbers. "When I am squared, my value is negative one" is rather similar to "I am lying" in logic. You can't do that in the reals, you need the complex numbers. Binary computers have similar troubles with The Liar. But there is a solution. It's different from True and also different from False. "I am lying" does not make my head explode, it's a valid assertion, and you might even call it an "imaginary" truth value (please don't)

#relevancelogic #rm3