toad.social

CubeRootOfTrue<a href="https://sfba.social/@muiren" class="u-url mention" rel="nofollow noopener" target="_blank">@muiren</a> 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?)<a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> <a href="https://mathstodon.xyz/tags/SQL" class="mention hashtag" rel="nofollow noopener" target="_blank">#SQL</a> <a href="https://mathstodon.xyz/tags/K" class="mention hashtag" rel="nofollow noopener" target="_blank">#K</a> <a href="https://mathstodon.xyz/tags/paraconsistent" class="mention hashtag" rel="nofollow noopener" target="_blank">#paraconsistent</a>

CubeRootOfTrue<a href="https://sfba.social/@muiren" class="u-url mention" rel="nofollow noopener" target="_blank">@muiren</a> The Axiom of Weakening is invalid<a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> <a href="https://mathstodon.xyz/tags/RelevanceLogic" class="mention hashtag" rel="nofollow noopener" target="_blank">#RelevanceLogic</a>

CubeRootOfTrue$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.<a href="https://mathstodon.xyz/tags/JustSayNo" class="mention hashtag" rel="nofollow noopener" target="_blank">#JustSayNo</a> <a href="https://mathstodon.xyz/tags/ModalLogic" class="mention hashtag" rel="nofollow noopener" target="_blank">#ModalLogic</a> <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>

CubeRootOfTrue<a href="https://journa.host/@samlitzinger" class="u-url mention" rel="nofollow noopener" target="_blank">@samlitzinger</a> 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<a href="https://mathstodon.xyz/tags/rm3" class="mention hashtag" rel="nofollow noopener" target="_blank">#rm3</a> <a href="https://mathstodon.xyz/tags/RelevanceFallacy" class="mention hashtag" rel="nofollow noopener" target="_blank">#RelevanceFallacy</a>

CubeRootOfTrue<a href="https://beige.party/@RickiTarr" class="u-url mention" rel="nofollow noopener" target="_blank">@RickiTarr</a> 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<a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> <a href="https://mathstodon.xyz/tags/NonBinaryLogic" class="mention hashtag" rel="nofollow noopener" target="_blank">#NonBinaryLogic</a> <a href="https://mathstodon.xyz/tags/hyperrings" class="mention hashtag" rel="nofollow noopener" target="_blank">#hyperrings</a> <a href="https://mathstodon.xyz/tags/chaoticEvil" class="mention hashtag" rel="nofollow noopener" target="_blank">#chaoticEvil</a>

CubeRootOfTrue<a href="https://mstdn.social/@MaryAustinBooks" class="u-url mention" rel="nofollow noopener" target="_blank">@MaryAustinBooks</a> lol 🤣 Did you notice how Frodo is constantly getting screwed over by Merry, Sam, and Pippin? Like when they light the fire to cook and it draws Black Riders from everywhere?Yeeaahhh, they are all "good guys" ... very good NYTNot to mention that wasn't in the books, only the movie. But what is truth, anyway?<a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>

CubeRootOfTrueRobert Rosen talked a lot about the distinction between simple and complex systems. Essentially, simple systems can be built from parts, like a clock. Complex objects, on the other hand, are things with many models. Like time, or music, or a plant. No single description is enough.He went further, saying that when studying a complex system, one can "throw away the matter and study the organization" to learn those things that are essential.He was talking about Category Theory and its use in studying the organization of organisms.A complex system, you might say, is not decomposable into parts. In this manner the complex :: simple distinction is akin to the difference in monoidal :: Cartesian tensor products in a monoidal closed category. Cartesian products have projections, you can take them apart. A monoidal product does not. You can only make inferences about their structure (make models)<a href="https://mathstodon.xyz/tags/robertrosen" class="mention hashtag" rel="nofollow noopener" target="_blank">#robertrosen</a> <a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> <a href="https://mathstodon.xyz/tags/complexsystems" class="mention hashtag" rel="nofollow noopener" target="_blank">#complexsystems</a>

CubeRootOfTrue<a href="https://mastodon.social/@cmconseils" class="u-url mention" rel="nofollow noopener" target="_blank">@cmconseils</a> I prefer 1/3 / 1/3 / 1/3 <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> I hope it happens before 33/33/33 but you know the Earth's rotation *is* slowing down

CubeRootOfTrue<a href="https://universeodon.com/@georgetakei" class="u-url mention" rel="nofollow noopener" target="_blank">@georgetakei</a> <a href="https://www.politico.com/magazine/story/2016/05/donald-trump-2016-contradictions-213869/" rel="nofollow noopener" translate="no" target="_blank">https://www.politico.com/magazine/story/2016/05/donald-trump-2016-contradictions-213869/</a> If it were not for Trump I never would have gotten interested in <a href="https://mathstodon.xyz/tags/relevance" class="mention hashtag" rel="nofollow noopener" target="_blank">#relevance</a> <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#logic</a> <a href="https://mathstodon.xyz/tags/rm3" class="mention hashtag" rel="nofollow noopener" target="_blank">#rm3</a> because I had to ask myself "How can people be so stupid?" and I noticed that binary logic is full of paradoxes, it is easy to lie using binary logic, and a lot of politicians say things in a way that makes use of fallacious binary reasoning.Binary logic is computers yeah, so that's popular and it does certain things really well. But humans don't use binary logic.RM3 actually stands in relation to binary logic the way that the Complex numbers are related to the Reals. In that case, you want to find the square root of negative 1, and $ \mathbb{C} $ arises as a field extensions of $ \mathbb{R} $. RM3 is derived from a field extension of $ \mathbb{Z}_2 $ modulo a boolean expression representing the Liar Paradox $ x^2 + x + 1 $, resulting in $ \mathbb{F}_4 $. The two new truth values are "Both true and false" and "Neither true nor false", and the concept of Truth is replaced with that of Validity.And yeah, after years of studying logic, and discovering that you can solve most relevance fallacies using a non-binary logic, I was like, "but wait he's just lying."

Archivist LizToday has been a big <a href="https://digipres.club/tags/RecordsManagement" class="mention hashtag" rel="nofollow noopener" target="_blank">#RecordsManagement</a> day, so I've got an <a href="https://digipres.club/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>+: 1) Practiced entering retention labels in our QA environment and checked that our guides are up-to-date 2) Discussed with colleagues in another IO about how different RM solutions work 3) Continue refining requirements for a digital preservation system 4) Meet on a possible database archiving use case 5) Attend a CAB meeting on RM features in <a href="https://digipres.club/tags/M365" class="mention hashtag" rel="nofollow noopener" target="_blank">#M365</a>

CubeRootOfTrue<a href="https://mathstodon.xyz/@logicbot" class="u-url mention" rel="nofollow noopener" target="_blank">@logicbot</a> Maybe this is what they mean when they say <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> is "pseudo" relevant. Because strictly syntactically, $ \lnot a \rightarrow (b \vee \lnot b) $ has $ b $ as a consequence but it doesn't appear on the left hand side. Yet the statement is valid in RM3 because $ b \vee \lnot b $ is always valid. Even though it is "irrelevant".

CubeRootOfTrue<a href="https://mathstodon.xyz/@logicbot" class="u-url mention" rel="nofollow noopener" target="_blank">@logicbot</a> This is not even paraconsistently valid. An implication with an impredicative premise. Also not a tautology in relevant logic <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> of course. If b is false.

CubeRootOfTrue<a href="https://mathstodon.xyz/@TaliaRinger" class="u-url mention" rel="nofollow noopener" target="_blank">@TaliaRinger</a> Check out what happens when you form $ \mathbb Z_2 /(x^2+x+1) $. The result is $ \mathbb F_4 $. The two new truth values $ \phi $ and $ \phi + 1 $ behave similarly (there is an isomorphism between them) and if you identify them you get <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>, a sort of complex logic with imaginary truth value (but don't call them that, the complex numbers are perfectly "real" not "imaginary", and so too the truth value "Both").

CubeRootOfTrue$ \huge{\text{The Bellman's Rule}} $"Just the place for a Snark!" the Bellman cried, As he landed his crew with care; Supporting each man on the top of the tide By a finger entwined in his hair. "Just the place for a Snark!" I have said it twice: That alone should encourage the crew. "Just the place for a Snark!" I have said it thrice: What I tell you three times is true.From "The Hunting of the Snark", by Lewis CarrollSo in addition to the other nice properties we've seen, $ \mathbb F_4 $ has another trick up its sleeve: the non-zero elements of the multiplication table form the smallest cyclic group on 3 elements. What this means is that $ a^3 = T $ if $ a \ne F $: \[ \begin{array}{c|c} a & P(a) \\ \hline F & F \\ B & T \\ T & T \\ \end{array}\]which captures the notion of validity in <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> perfectly. There's a lot more to say about this, like, it naturally falls out of the definitions (it's just cubing), it's non-linear, and it's a closure operator. It's called $ P $ as in possible, and also known as the modal operator $ \Diamond $. And it's a monad. Necessity is the comonad $ N(a) =_{def} \lnot P(\lnot a) $, by duality.Now let's define $ a \rightarrow b $ as $ \lnot P(a) \vee b $. Not possible a, OR b. \[ \begin{array}{c|ccc} \rightarrow & F & B & T \\ \hline F & T & T & T \\ B & F & B & T \\ T & F & B & T \\ \end{array}\]This conditional is not explosive. $(a \wedge \lnot a) \rightarrow b$ is:\[\begin{array}{c|ccc} & F & B & T \\ \hline F & T & T & T \\ B & F & B & T \\ T & T & T & T \\ \end{array}\]That is, it is *invalid*. Unlike in the binary case, an inconsistent truth value does NOT imply everything, or reduce the logic to triviality. It's paraconsistent.

CubeRootOfTrueOK, so for <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>, we're going to do a couple things. First, we'll stick with T, B, and F as truth values. Then in the addition and multiplication tables, we ignore the N rows and columns, and change all the remaining Ns into Bs. Multiplication is AND. Addition is XOR. And NOT has the nice property that $ \lnot \lnot a = a $ (it's idempotent).NOT$(a)$ is $ a + T $. \[ \begin{array}{c|c} a & \lnot a \\ \hline F & T \\ B & B \\ T & F \\ \end{array}\]AND is $ \times $:\[ \begin{array}{c|ccc} \wedge & F & B & T \\ \hline F & F & F & F \\ B & F & B & B \\ T & F & B & T \\ \end{array}\]Inclusive OR is defined by DeMorgan duality:\[ \begin{array}{c|ccc} \vee & F & B & T \\ \hline F & F & B & T \\ B & B & B & T \\ T & T & T & T \\ \end{array}\]and the regular conditional $\lnot a \vee b $:\[ \begin{array}{c|ccc} \supset & F & B & T \\ \hline F & T & T & T \\ B & B & B & T \\ T & F & B & T \\ \end{array}\] At this point, we have a basic 3-valued logic. We aren't where we want to be yet, though. This logic is still explosive. The conditional is too weak. This is because we have expanded our notion of Truth. Things are no longer black or white.The conditional is defined as "NOT a OR b". But is it appropriate to use the ordinary negation? $ \lnot B = B $, so are we negating anything?Instead of Truth, we need to think in terms of Validity. In <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>, both T and B are "designated" as valid. So our conditional should be \[ a \rightarrow b =_{def} \lnot \text{valid}(a) \vee b. \]And it turns out that there is a natural way to implement valid$(a)$.

CubeRootOfTrueActually the *first* cut is to notice that if we restrict attention to just 0 and 1, arithmetic in $ \mathbb F_4 $ is exactly the same as in $ \mathbb Z_2 $. In fact, we can recover binary <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#logic</a> in about 12 different ways, depending on which elements we pick for $ \top $ and $ \bot $. But we'll skip that. The next cut is Dunn's 4-valued logic <a href="https://mathstodon.xyz/tags/FOUR" class="mention hashtag" rel="nofollow noopener" target="_blank">#FOUR</a>. We'll get back to <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>I first came across this construction in (French):P. V. GROSJEAN The logic on the splitting field of paradoxes, Seminar of Philosophy and Mathematics, 1988, Issue 7

CubeRootOfTrueBy analogy (lol it's like saying "by <a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a>")I'll start again. By analogy with the way the complex numbers are derived from the real numbers, we can derive a multi-valued logic from binary logic.In the real number case, we quotient $ \mathbb{R} $ by the equivalence $ x^2 + 1 = 0 $. The roots of that equation are $ i $ and $ -i $, and the quotient $ \mathbb{C} $ is a type of field extension called a splitting field, which "splits" the equation into linear factors.In the binary logic case, we start with the Boolean algebra of $ \mathbb{Z}_2 $ and look at the equivalence $ (A \wedge \lnot A) = \top $. In Boole's notation, this is \[ \begin{array}{cl} & x (1 - x) \\ =& x (x + 1) \hspace{.5cm}\text{in } \mathbb Z_2 \\ =& x^2 + x \\ =& 1 \end{array} \]It turns out that the quotient $ \mathbb Z_2 /(x^2+x-1) $ is $ \mathbb F_4 $, the field with 4 elements. The two new elements are $ \phi $ and $ \phi + 1 $, and \[ \phi (\phi+1)=1. \]Note that $ x^2 + x - 1 = x^2 + x + 1 $ in $ \mathbb Z_2 $, but I am perfectly happy to use a minus sign there, since a solution to $ x^2 + x - 1 = 0 $ in the reals is $ (1 + \sqrt{5})/2 = \phi $, which also looks like a 0 and a 1 superimposed on one another, which is proper for Both true and false. 😉 As in the complex number case, there are two roots. They are also inverses of each other, like $ i $ and $ -i $, $ -\phi = \phi + 1 $. Now, a 4-valued logic is all well and good, and there is an example we will look at, but for now it will be simpler to exploit this symmetry between $ \phi $ and $ \phi + 1 $ by identifying them. Thus we make the first cut through our logical crystal, to obtain a 3-valued logic, which is <a href="https://mathstodon.xyz/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>

Bruce Halperin :vbike:I missed this huge news from yesterday: the <a href="https://sfba.social/tags/CaliforniaSupremeCourt" class="mention hashtag" rel="nofollow noopener" target="_blank">#CaliforniaSupremeCourt</a> finally rejected the frivolous lawsuit over <a href="https://sfba.social/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a>! That means the funds (from toll increases on Bay Area bridges) can finally be used on transit projects. <a href="https://www.sfchronicle.com/bayarea/article/california-supreme-court-rejects-lawsuit-against-17742422.php?utm_source=marketing&utm_medium=copy-url-link&utm_campaign=article-share&hash=aHR0cHM6Ly93d3cuc2ZjaHJvbmljbGUuY29tL2JheWFyZWEvYXJ0aWNsZS9jYWxpZm9ybmlhLXN1cHJlbWUtY291cnQtcmVqZWN0cy1sYXdzdWl0LWFnYWluc3QtMTc3NDI0MjIucGhw&time=MTY3NDc5ODc3MTAxNw==&rid=N2Y0NGNhYjMtNzI0OS00OGFmLTkxZTQtMDVhMTI0YjA3NzRk&sharecount=Mw==" rel="nofollow noopener" target="_blank">https://www.sfchronicle.com/bayarea/article/california-supreme-court-rejects-lawsuit-against-17742422.php?utm_source=marketing&utm_medium=copy-url-link&utm_campaign=article-share&hash=aHR0cHM6Ly93d3cuc2ZjaHJvbmljbGUuY29tL2JheWFyZWEvYXJ0aWNsZS9jYWxpZm9ybmlhLXN1cHJlbWUtY291cnQtcmVqZWN0cy1sYXdzdWl0LWFnYWluc3QtMTc3NDI0MjIucGhw&time=MTY3NDc5ODc3MTAxNw==&rid=N2Y0NGNhYjMtNzI0OS00OGFmLTkxZTQtMDVhMTI0YjA3NzRk&sharecount=Mw==</a>

Joy Kreg 🎮&🎲2°) Mes captures de jeux, en images comme en vidéos : <a href="https://ludosphere.fr/tags/MonsterHunter" class="mention hashtag" rel="nofollow noopener" target="_blank">#MonsterHunter</a> <a href="https://ludosphere.fr/tags/MonsterHunterWorld" class="mention hashtag" rel="nofollow noopener" target="_blank">#MonsterHunterWorld</a> <a href="https://ludosphere.fr/tags/MonsterHunterWorldIceborne" class="mention hashtag" rel="nofollow noopener" target="_blank">#MonsterHunterWorldIceborne</a> <a href="https://ludosphere.fr/tags/MHW" class="mention hashtag" rel="nofollow noopener" target="_blank">#MHW</a> <a href="https://ludosphere.fr/tags/MHWI" class="mention hashtag" rel="nofollow noopener" target="_blank">#MHWI</a> <a href="https://ludosphere.fr/tags/Capcom" class="mention hashtag" rel="nofollow noopener" target="_blank">#Capcom</a> <a href="https://ludosphere.fr/tags/Microsoft" class="mention hashtag" rel="nofollow noopener" target="_blank">#Microsoft</a> <a href="https://ludosphere.fr/tags/XboxOne" class="mention hashtag" rel="nofollow noopener" target="_blank">#XboxOne</a> <a href="https://ludosphere.fr/tags/XboxOneX" class="mention hashtag" rel="nofollow noopener" target="_blank">#XboxOneX</a> <a href="https://ludosphere.fr/tags/D%C3%A9sertDesTermites" class="mention hashtag" rel="nofollow noopener" target="_blank">#DésertDesTermites</a> <a href="https://ludosphere.fr/tags/Glavenus" class="mention hashtag" rel="nofollow noopener" target="_blank">#Glavenus</a> <a href="https://ludosphere.fr/tags/%C3%89p%C3%A9eLongue" class="mention hashtag" rel="nofollow noopener" target="_blank">#ÉpéeLongue</a> <a href="https://ludosphere.fr/tags/RangMa%C3%AEtre" class="mention hashtag" rel="nofollow noopener" target="_blank">#RangMaître</a> <a href="https://ludosphere.fr/tags/RM3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RM3</a> <a href="https://ludosphere.fr/tags/RangMa%C3%AEtre3" class="mention hashtag" rel="nofollow noopener" target="_blank">#RangMaître3</a>

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