@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?)
@muiren The Axiom of Weakening is invalid
$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.
@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
@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
@MaryAustinBooks 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 NYT
Not to mention that wasn't in the books, only the movie. But what is truth, anyway?
@cmconseils I prefer 1/3 / 1/3 / 1/3 #RM3 I hope it happens before 33/33/33 but you know the Earth's rotation *is* slowing down
A monad on a poset is a closure operator https://hansriess.com/monads-on-a-poset/
This means that in a non-binary logic, a closure operator will reduce any truth value to either True or False. This leads directly to modal operators such as Possible(x) and Necessary(x), which differ only in their handling of intermediate truth values between 0 and 1, like .618.
Deontic 3-valued logic, likewise, admits Ought(x) and Permissible(x), which mean that an agent either _must_ do x (Ought) or is _allowed_ to do x (Permissible). The truth values are exactly the same as for Necessary\(\sim\)Possible, but with slightly different semantics. #RM3 captures all of this.
@georgetakei https://www.politico.com/magazine/story/2016/05/donald-trump-2016-contradictions-213869/
If it were not for Trump I never would have gotten interested in #relevance #logic #rm3 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."
Almost end of the day, but three things today:
1) edited an internal standard on retention schedules, 2) worked on requirements for a #digitalPreservation system, 3) presented to managers on roles needed for #recordsManagement functions in #M365. #DP3 / #RM3
@logicbot Maybe this is what they mean when they say #RM3 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".
@MireilleNappert @archivist_Liz Hmmm. Maybe you could do an #RM3?
@TaliaRinger 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 #RM3, 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").
@standefer I suppose, when one is dealing with notions of vagueness in #logic, that it should be no surprise that vagueness starts creeping in everywhere. For example, instead of saying that a particular logical implication is non-monotonic, we can use the logic to *define* the ordering on the underlying set of truth values (the lattice). The figure shows the "orderings" induced by the #RM3 implication (top) and the Sugihara implication (bottom) for \( n = 5 \). Here, the "arrow" on the lattice is not \( \le \). In general as \( n \) increases, only the innermost (resp. outermost) pairs of arrows remain "increasing only", for the two implications. Odd \( n \) only. Other implications are apparently "reversible", although there is the matter of what lattice values are to be considered "valid".
We've got a 2-valued logic, a very nice 3-valued logic called #RM3, and "A useful four-valued logic", N. D. Belnap, 1977. The obvious question is, what about \(n\)-valued logics, for \( n \ge 5 \)? Sadly, the very nice structures we've seen don't generalize. This is often the case, as when we go from the reals to the complex numbers we lose ordering, and when we go to the quaternions we lose commutativity. So it is with this family of logics. Going from 3 to 4 values breaks ordering, but enough structure remains to have something workable (and non-monotonic logics are definitely a thing). Beyond 4 though, we lose more important structure. In Kalman, J. A. (1958) "Lattices with involution," Kalman (no, not Kálmán), showed that these three, plus the trivial lattice \( D_1 \), are the only distributive lattices with involution, the involution being the negation \( \lnot \lnot A = A \). Any larger \( D_n \) decomposes into the smaller ones. Any element of a lattice that satisfies \( \lnot x=x \) is called a 'zero', and Kalman showed that for this setup, there can be only 0, 1, or 2 zeros, corresponding to the 3 lattices shown in the figure.
Furthermore there is a simple proof in Busaniche and Cignoli (2011), "Remarks on an algebraic semantics for paraconsistent Nelson’s logic," that #RM3, the \( D_3 \) lattice with involution, is *unique*.
It is still possible to construct logics with more than 4 (even infinite) truth values, but you lose monotonicity, or other nice properties. #RM3 obeys the law of small numbers; its unique structure follows from it being the *smallest*. The monoid multiplication in \( \mathbb F_4 \) contains the smallest symmetric group \( S_3 \) on 3 elements, aka the Bellman's Rule
\( \huge{\text{The Bellman's Rule}} \)
"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 Carroll
So 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 #RM3 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.
OK, so for #RM3, 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 #RM3, 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\((a)\).
Graham Priest has written extensively on Paraconsistent Logic. This is logic that is not explosive, i.e. \[ (A \wedge \lnot A) \rightarrow B \]is invalid. This is a tautology in binary logic, meaning from an inconsistent premise you can prove anything (that's bad). It's invalid in #RM3, RM3 is not explosive and thus paraconsistent. But we need to be more explicit because RM3 has more than one conditional.
The usual \( \lnot A \vee B \) is actually explosive, it's the same as the binary \( \supset \) conditional.
The first non-explosive conditional in RM3 is the same as the conditional that is often associated with Priest's LP, the Logic of Paradox. It uses validity instead of truth in the conditional: \[ A \rightarrow B := \lnot P(A) \vee B \]where \( P \) is the modal "possible" operator, which also indicates validity in RM3. However, it doesn't satisfy the contrapositive, and it does satisfy the projections from the product(s), in other words Weakening.
The full Relevant Conditional \[ A \Rightarrow B :=(A \rightarrow B) \wedge (\lnot B \rightarrow \lnot A) \] is both paraconsistent and relevant, it is neither explosive nor does it satisfy Weakening (\( A \otimes B \Rightarrow A \) is invalid).
Thus, explosiveness is related to relevance via the contrapositive. Inconsistency needs to be shut down in both cases.
