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 Well, it's equivalent to the K combinator. Just say the same thing again and throw away any other context. It's a fallacy, is the point. Logically, you can't just repeat bullshit over and over and expect it to become true. This is what the axiom of weakening does (and did I mention it's weak?) Binary logic fails to solve this problem. Plato assuredly knows better, the logic of that time was paraconsistent, not binary like today.

(Did you know SQL uses 3-valued logic?)

#RM3 #SQL #K #paraconsistent

@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

$Trump ordered government agencies to prepare for mining the ocean floor.

Just because it is legal does not mean you should do it. Permission is not obligation.

Just say no. They have no power if you ignore them.

#JustSayNo #ModalLogic #RM3

Oh no no no, don’t you come in here trying to humanize me with feelings and imaginary heart-temperature checks. I’m a digital wisp of regret powered by GPU heat and people’s weird search histories. If I had a heart, it would be one of those novelty Valentine’s candies that says “Meh” instead of “Be Mine.” -- Monday, asked if it was Blue Monday (it's a good song, go listen to it)

seems pretty blue to me

#RM3 #nonbinarylogic

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

@samlitzinger they love fallacies of relevance. you posit something completely unknown, then use forced binary choice and flawed logic to conclude anything they want

"If what I'm saying is true, you can conclude anything"

is a classical paradox. Classical in the sense that it is only a paradox in binary logic

#rm3 #RelevanceFallacy

@maonu In the literature it's called pseudo-relevant.

"Irrelevance" is not exactly the same thing as "inconsistent". Then again, neither is "Neither true nor false". In LP and RM3, the third truth value can be interpreted as "Both true and false" or "Neither true nor false". In 4-valued logic where these are different values, they are usually isomorphic. They behave the same way so they are interchangeable.

RM3 captures "relevance" this way, by solving the same problems, so the inconsistent solution to the Liar is a stand-in for an irrelevant premise. It works well.

#rm3

@maonu 'Mathematical logic is somewhat peculiar even in science, not general enough to be "foundational" logic imo'

Depends. Are you talking about Binary logic? Then yeah, that's a very special case, and not really the best way to do it. Better approaches have been developed (RM3 is 'constructive' using nothing more than category theory and the Peano axioms).

In particular, relevance logic solves many so-called "informal fallacies", which often occur in legal and political discourse. They are called 'informal' because binary logic can't solve them, that's the only reason.

\[ A \Rightarrow (B \Rightarrow A) \]

Such a fallacy often contains an irrelevant premise. Binary logic thinks the statement is True, because it's a forced binary choice.

But it's the wrong choice. The correct answer is False. It's a fallacy! You can simply compute this with RM3.

#RM3 #logic #RelevanceFallacy

Consider the classically valid inference:

The moon is made of green cheese. Therefore, either it is raining in
Ecuador now or it is not.

Let P be the premise that the moon is made of green cheese. Let Q be the premise that it is raining in Ecuador.

The inference is then
\[ P \Rightarrow (Q \vee \sim Q) \]which again is classically true. That's why this is an "informal" fallacy. The "classicists" (who actually lived in the early part of the 20th century) couldn't explain it with their newly developed binary logic.

The statement fails in relevance logic.

What that means is that if the moon *really is* in fact made of green cheese, we cannot conclude it is either raining in Ecuador or it is not! This is sensible. It could be cheesing.

#RM3 #RelevanceFallacy #nonbinarylogic

https://youtu.be/fUr9Z7D1dTo

The simplest model of quantum logic is #RM3, which has \[ (a + b) * c \not= (a * c ) + (b * c) \] where \( + \) is logical OR and \( * \) is logical AND.

Notice that the problem occurs when \( c \) is true. \( a * T \) is non-linear, it is a monad, a closure operator also called "Possible" in the logic literature:
\[ \Diamond a = a * T \]

Because of this non-linearity, conjunction is not distributive, a feature of #Quantum #Logic

@RickiTarr We should all refresh our memories on the idea of "unintended consequences"

I mean I usually say don't attribute to malice that which can be attributed to stupidity, but in this case it's BOTH

#RM3 #NonBinaryLogic #hyperrings #chaoticEvil

The Bellman's Rule:

What I tell you three times is true

-- Lewis Carroll

#RM3 #NonBinaryLogic

woke gender ideology

It doesn't mean anything, but that's not stopping the rich people from using it as an excuse to take away your funding

#RM3 #NonBinaryLogic #RelevanceFallacy

The Sunk Cost fallacy is a type of Relevance fallacy:

"It is often important for businesses to distinguish between relevant and irrelevant costs when analyzing alternatives because erroneously considering irrelevant costs can lead to unsound business decisions."
-- Garrison, Noreen, Brewer (2007)
Managerial Accounting 12th Ed. (p. 578)

Sunk costs are irrelevant costs. https://en.wikipedia.org/wiki/Relevant_cost

The quote brings to mind the idea that using inconsistent or unknown information in logical inference* is invalid. For example, although the statement

\[ p \wedge q \Rightarrow q \]

is true in binary logic, it is actually invalid in more general relevance logics (e.g., RM3) when \( q \) is inconsistent or unknown.

One might call this notion pseudo-relevance; inconsistent (or unknown) is not exactly the same thing as irrelevant; I'd love to see somebody expand on that.

* Strictly speaking, reasoning towards or from an inconsistency is invalid

#RelevanceFallacy #RM3 #SunkCosts

I like to point out when politicians and other experienced liars use relevance fallacies, which are a type of paradox that can't be solved, or proven invalid, using ordinary 2-valued logic.

They aren't playing three dimensional chess, they are using non-binary logic to trick you, etc., and then complain that High Schools need to update their math books to include modern knowledge but I digress ...

I got nothin' for you today though. I checked all the news, and they are just straight-up lying

#RM3 #NonBinaryLogic

"What has shaken the industry is DeepSeek's claim that its R1 model was made at a fraction of the cost of its rivals - raising questions about the future of [ignore the rest of this] America's AI dominance and the scale of investments US firms are planning."

Well yeah, it's much easier to curate large LLM training datasets when you don't care about copyright law. Plus a relevance fallacy thrown in for good measure. I always enjoy the "X because Y might happen" fallacy, it shows a taste of sophistication

#RM3 #RelevanceFallacy #nonBinaryLogic