#rm3 — Public Fediverse posts

People have an implicit bias towards binary logic. Computers are great and all, but have you noticed they break and fail all the time? Binary logic is not the best system. "You are either with us or against us" is a fallacy, they are trying to trick you. 🚥🚦🚥🚦🚥🚦🚥🚦🚥🚦🚥🚦🚥🚦You use "real world logic" every day, and it's 3-valued. They tried binary traffic signals in the beginning but switched very quickly to 3

Also the Liar Paradox is solvable using a field extension which creates (like i^2 = -1) the complex truth value "I don't know"

which you should know

#nonbinary #RM3 #implicitbias #TheLiar

@MartinEscardo This is true. I'm suggesting that this is not just a computer engineering thing. I'm saying that at base, nature itself is non-binary, and we are non-binary, and we abstract things we don't understand into black and white distinctions to make them easier to understand. It's fast and efficient to ignore certain problems. There are still problems with binary logic.

Did you know that SQL is one of the few computer languages to use 3-valued logic? Databases are highly exposed to the problems of inconsistent or missing information. But we can use binary logic to simulate 3-valued logic (MySQL uses the paraconsistent logic LP by Graham Priest, but you can use that to create a fully relevant implication as in RM3, it's just a longer expression)

#rm3 #mysql #relevance #paraconsistent

@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

@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

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

"Shoot down the drones" is a logical fallacy, like "dark matter". Calling it matter doesn't make it matter. Using the word "drone" doesn't mean there are drones. "Shoot down the drones that are looking for the radiation from nuclear weapons" makes it more obvious.

In both #paraconsistent and relevant 3-valued logic (#rm3), inferring from an unknown is invalid. Many logical fallacies are of this type. In relevance logic, inferring towards an unknown is *also* invalid, although it is valid in paraconsistent and binary logic. Those are the so-called "informal" fallacies, aka relevance fallacies, which are in fact formal in multi-valued logics, where you can prove they are invalid. But you need more than binary truth

I am always careful, when discussing the organization of organisms, because saying things like "the whole is not the sum of its parts" makes people think I'm talking about vitalism or religion or something, but I'm talking about non-cartesian products which lack projections #RM3 #SMCC #AppliedCategoryTheory

I am always careful, when discussing the organization of organisms, because saying things like "the whole is not the sum of its parts" makes people think I'm talking about vitalism or religion or something, but I'm talking about non-cartesian products which lack projections #RM3 #SMCC #AppliedCategoryTheory

@maonu It is remarkable. In school we are taught that (sufficiently powerful) mathematical systems are either inconsistent or incomplete. Then everyone runs away screaming from inconsistency, and nobody ever seems to say, hey, wouldn't completeness be nice? If we have a choice ... completeness (and I might add constructive) seems to be a good tradeoff vs something we already do all the time. 3-valued logic is so ubiquitous you don't even notice it

I've always said the 5th fundamental force of the universe is the one that makes you accelerate when the light turns yellow

#rm3 #smcc

@maonu It is remarkable. In school we are taught that (sufficiently powerful) mathematical systems are either inconsistent or incomplete. Then everyone runs away screaming from inconsistency, and nobody ever seems to say, hey, wouldn't completeness be nice? If we have a choice ... completeness (and I might add constructive) seems to be a good tradeoff vs something we already do all the time. 3-valued logic is so ubiquitous you don't even notice it

I've always said the 5th fundamental force of the universe is the one that makes you accelerate when the light turns yellow

#rm3 #smcc

@jcreed @andrejbauer @maxsnew @boarders In #RM3 or other #paraconsistent logics, True, and also Both true and false, are valid. False is not valid.

@dougmerritt It's not so much hand-waving because there is a formal mathematical proof. That's the part a lot of people don't get. This has all been done already, long ago. It's not a question, this is the answer. #RM3 #SMCC Also look up Linear Logic, which has RM3 as a subset. RM3 can be obtained from \( \mathbb Z_2 \) exactly the same way the Complex numbers are obtained from the Reals. The truth value "Both" is exactly like the square root of negative one. It solves the Liar Paradox. Search my posts. The Bellman's rule is right there. But Lovecraft would enjoy it, certainly. Being able to solve relevance fallacies does mean you can write some great fiction --- the best lies are true in binary logic.

@dougmerritt It's not so much hand-waving because there is a formal mathematical proof. That's the part a lot of people don't get. This has all been done already, long ago. It's not a question, this is the answer. #RM3 #SMCC Also look up Linear Logic, which has RM3 as a subset. RM3 can be obtained from \( \mathbb Z_2 \) exactly the same way the Complex numbers are obtained from the Reals. The truth value "Both" is exactly like the square root of negative one. It solves the Liar Paradox. Search my posts. The Bellman's rule is right there. But Lovecraft would enjoy it, certainly. Being able to solve relevance fallacies does mean you can write some great fiction --- the best lies are true in binary logic.

Gödel himself created a three valued logic because he PROVED that the 2-valued case doesn't WORK!

#RM3 #SMCC #categorytheory

Gödel himself created a three valued logic because he PROVED that the 2-valued case doesn't WORK!

#RM3 #SMCC #categorytheory

Rosen stated: "I argue that the only resolution to such problems [of the subject-object boundary and what constitutes objectivity] is in the recognition that closed loops of causation are 'objective'; i.e. legitimate objects of scientific scrutiny. These are explicitly forbidden in any machine or mechanism."

Saying that closed causal loops are objective leads directly to the need for non-binary logic. Binary logic cannot deal with causal loops, which are impredicative, like the set that contains itself. Recent developments in modern category theory make this all clear. We can handle this now with monoidal closed categories, a generalization of the old cartesian categories used in binary logic. #RM3 #LinearLogic #paraconsistent #paradox

Working in the context of Myhill-Aczel constructive set theories ...
These theories are constructive subtheories of classical ZF set theory ... compatible with the classical tradition in the sense that all of their theorems are classically true.
In fact, Constructive Zermelo-Fraenkel (CZF) and Intuitionistic Zermelo-Fraenkel (IZF) give rise to full classical ZF by the simple addition of the principle of the excluded middle. (SEP)

The same sort of thing is true for #RM3. All its theorems are classically true; but classical (binary) logic proves too much! There are many pardoxes and fallacies in binary logic. #Paraconsistent relevance logics solve these problems by eliminating a few key theorems through the use of one or more new truth values (RM3 is a 3-valued logic, T, B, F), but collapse to classical logic if one excludes these middle values.

“And that might mean that you have to deal with people that you disagree with on some things, or many things, or even most things, but you find enough common cause that you can work with them on something.” -- Steve Inskeep

In a world of binary logic, a paraconsistent logic is the bridge you need to communicate
#RM3 #paraconsistent #relevance #logic

@jcastroarnaud In #RM3, Both is considered valid. Validity itself is represented as a #monad, the logical modal operator Possible (\(\Diamond\). In this case \(\Diamond B\) is True.

Classical logic is often represented as the natural logic of open sets. So that the boundary of any set is infinitesimal. 3-valued logic, #paraconsistent logic, #relevant logic, use CLOSED sets. The boundary has a thickness. You don't know if you are in set A, when you are on the boundary. It's A and not A

Think of it as the Halting Problem. It's really the Liar Paradox. If the program halts, then it doesn't, and if it doesn't halt then it halts. It's Both! It halts and it doesn't. This is easy to understand in #RM3, where "Both" is a valid logical value different from either true or false.

Binary logic is bad. But don't worry! It turns out that you don't need to go beyond 3. A 3-valued logic, True, False, and Both, is complete. What about Gödel? Yeah, he showed that binary logic sucks, and then came up with a 3-valued logic that solves the problem mentioned in the Incompleteness theorem. He said, it's either inconsistent or incomplete. Turns out, completeness is a far better property than consistency. Because the world (and the reals) are inconsistent. There are things that are both true and false #dialethism #paraconsistent #logic #turing

Venn diagrams you learned in school probably didn't include everything. But there ARE such things as CLOSED SETS. Things with boundaries, potentially thick ones, representing uncertainty or vagueness in set membership. And you can build a logic from closed sets.

Notice too, that \( (A \wedge\lnot A) \) is different from \( (B \wedge\lnot B) \). In binary logic, all inconsistent sets are indistinguishable. But here, in the picture for \( A \wedge B \), instead of just the usual 4, there are an additional 5 regions (the two on the sides are the same set, but disconnected).

#RM3 #paraconsistent #logic

@jcreed Well, in #RM3, which is a symmetric monoidal closed category,\[ ((A \Rightarrow B) \otimes (B \Rightarrow A)) \Rightarrow ((A \Rightarrow A) \otimes (B \Rightarrow B)) \]is valid. The reverse implication is *not*. This also works for the Cartesian conjunction \( \wedge \) and the #paraconsistent implication \( \rightarrow \).

@ryk047 @andrejbauer

I've written some here (search for #RM3) https://mathstodon.xyz/@CubeRootOfTrue/110775995709463374 about the naturalness of RM3, it is essentially the "complex logic" you get by solving the equation \( A \wedge \lnot A = \top \), akin to \( x^2 + 1 = 0 \) in the reals. In fact it's \( x^2 + x + 1 = 0.\)

Gödel said that blah blah either inconsistent or incomplete. 20th century mathematicians were so horrified at the thought of inconsistency it's been effectively banished (the "law" of excluded middle). We're happy, apparently, with incompleteness. But what about the opposite case!? Gödel himself developed a 3-valued logic, because he obviously understood that if you allow inconsistency, you can have completeness.

Normally inconsistency can't be tolerated because \( (A \wedge \lnot A) \supset B \), you can prove anything from an inconsistency, aka the principle of explosion. Hence the horror.

In a 3-valued logic, there are statements that are inconsistent, but the logic doesn't allow explosion, so everything's under control.

So yes, #paraconsistent and #relevant #logic have very much to do with foundations.

And yes, it's possibly the simplest example of a symmetric closed monoidal category (symmetry is optional), and maybe a useful teaching tool, not to mention that it's a superior logic than 2-valued logic, as it can handle vagueness.

I honestly wonder why the relevance #logic #RM3 is not taught in high school. Not only is it an obvious extension, and still relatively simple, but it works BETTER than regular binary logic!

It is a myth, by the way, that modern computers use binary logic. They don't. Deep within their electrical circuits are logical states like X for Don't Care and N for No Connection. Some have many more than 2 states.

You don't need non-binary logic everywhere! It's pretty useless in an adder, for example. Binary logic is good at stuff like that. As long as your inputs are consistent.

Where non-binary logic really comes in handy is things like the law, or politics, where disentangling the validity of assertions can be hard, because natural language is full of ambiguities and vagueness. You need a #paraconsistent logic.

#SQL is one of the only popular programming languages on the planet to implement #paraconsistent logic. It's quite natural, of course, for a database system that accepts inputs from the real world to have a mechanism for dealing with inconsistent or missing data. That mechanism is the paraconsistent logic LP.

It is thus possible to build a fully relevant #RM3 implication in an SQL Select statement. The three values of the logic are 0, 1, and None. The table t, here, is just a list of all pairs (a, b).

select a, b, (not a) or b, a is false or b,
(a is false or b) and ((not b) is false or not a) from t

a b ~a|b a->b a=>b
----------------------------------------
0 0 1 1 1
0 None 1 1 1
0 1 1 1 1
None 0 None 0 0
None None None None None
None 1 1 1 1
1 0 0 0 0
1 None None None 0
1 1 1 1 1

The last three columns show the ordinary (non-paraconsistent) conditional, the paraconsistent conditional, and the relevant conditional, resp., the latter being implemented by applying the contrapositive to the paraconsistent conditional.

Validity is expressed by "not false". This is an important concept. For the binary conditional, validity is ALSO "not false", but the lack of any other choice makes this concept invisible. Graham Priest has written extensively about this topic, and the semantics of LP.

In particular, \[ (a \wedge \lnot a) \rightarrow b \]is not valid, because None -> 0 is false, and your database won't explode when it hits a None.

Note to the geeks out there who want to add some robustness to their databases: you can usually create a function in SQL like Imp(a, b) or something, so you don't have to constantly write the whole expression out every time.