#relevancelogic — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #relevancelogic, aggregated by home.social.
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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.
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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.
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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.
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@muiren The Axiom of Weakening is invalid
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@muiren The Axiom of Weakening is invalid
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@muiren The Axiom of Weakening is invalid
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@muiren The Axiom of Weakening is invalid
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@muiren The Axiom of Weakening is invalid
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@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.
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@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.
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@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.
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@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.
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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.
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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.
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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.
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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.
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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.
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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
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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
-
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
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@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)
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@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)
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@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)
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@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)
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@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)