@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?)
@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.
When, however, you have an enemy, then do not requite him good for evil:
for that would shame him. Instead, prove that he did some good for you.
-- Also Sprach Zarathustra
"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
seeing these posts on HN about 1/0==1, maybe I should also write a blog post about #paraconsistent #logic and #arithmetic
The answer is obviously 1/0==[+\inf, -\inf] (yeah, that's at least positive infinity, and at most negative infinity) and 1/0<=0 is simulatenously both true and false :blobfoxdealwithit:
(That's modal interval arithmetic and Belnap--Dunn logic to reason about inconsistency without the principle of explosion.)
Matemático e filósofo Newton da Costa, criador da lógica paraconsistente, morre aos 94 anos
Mathematician and philosopher Newton da Costa, creator of paraconsistent logic, dies at the age of 94
https://www.youtube.com/watch?v=8gKKabtLA_U
#brazil #Brasil #philosophy #filosofia #logic #logica #paraconsistencia #paraconsistent #Science #ciencia
@jcreed @andrejbauer @maxsnew @boarders In #RM3 or other #paraconsistent logics, True, and also Both true and false, are valid. False is not valid.
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
@dougmerritt @AmenZwa Legal jurisprudence is famously logically inconsistent. Binary logic is useless. One must understand #paraconsistent and #relevance #logic if you want to be a lawyer.
@leemph Philosophically this is a dialethic logic, see lots more here, https://plato.stanford.edu/entries/dialetheism/ by Graham Priest, who is the canonical reference. Mathematically, well, search for closed set logic #paraconsistent e.g. W. James 96, although actually if anybody knows of a less technical reference that'd be swell.
Really it's just the idea that something can be both true and false, like the Liar Paradox, "This sentence is false." If it's false, it's true, and if it's true, it's false. So it's both. The third logical value is the boundary. Some call it \( \phi \), some call it O (other), or B (both), or I (inconsistent), or even N (neither true nor false; it's symmetric).
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).
@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 \( \rightarrow \).
Aloitin helposta päästä eli mahdottomista maailmoista. Tuo Vacekin IEP-entry oli lähinnä raapaisu, mutta muistin sitä kautta opiskelleeni joskus parakonsistenttia logiikkaa SEP:stä, ja löytyi kevyempi IEP-tekstikin. Kertailen nyt niitä ja annan välillä ajatuksen harhailla. Parasta iltapuhdetta. 🤗
https://plato.stanford.edu/entries/logic-paraconsistent/
Relevanttia aluetta kaikille, joita johdonmukaisuus sitoo, mutta ei voi välttyä sisäisiltä ristiriidoilta.
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.
@andrejbauer (I am assuming classical logic, but nothing really changes if we switch to intuitionistic logic, just replace Boolean algebras with Heyting algebras.)
This is a bit like saying "I am assuming euclidean geometry but nothing really changes if we switch to spherical." Because there are more than 2 kinds of geometry, and intuitionistic logic is not the only other choice here. Is RH "neither true nor false" or perhaps RH is "both true and false"? You need a dialetheic logic, a #paraconsistent logic. And in that case, these are both valid truth values, and RH would be formally undecidable. But more than that, asking for it to have no truth value at all is different from just not being true and also not being false. It might be similar to Graham Priest's "ineffable" logical value. #logic #dialetheism #twothingscanbetrue
