Three values: True, Anti-true, and False. Both True and Anti-true are true in the literal sense i.e. the light is on. Addition:T+T=TA+A=AT+A=FT+F=TA+F=AF+F=FNegation:-T=A-A=T-F=F
>>17022951>Both True and Anti-true are true in the literal senseSo what distinguishes them?>T+A=FSo two statements which "are true in the literal sense" become a false statement when combined?>T+F=TA true statement plus a false statement becomes a true statement?>-T=AThe negation of a true statement is still "true in the literal sense?">-F=FThe negation of a false statement is still false?You might wanna workshop this a touch.
Just go learn what fuzzy logic is already so you can deal with statements that are partially true
>>17022973>So what distinguishes them?Nothing, they are the same, but they cancel each other out. >A true statement plus a false statement becomes a true statement?That part is the same as in binary logic. Why would you pick on that?>The negation of a true statement is still "true in the literal sense?"Yes, you need to cancel it out with Anti-true. >The negation of a false statement is still false?Yes. The negation of nothing is still nothing.
>>17022973Yeah nigger why are you bringing up binary logic into this shit? Guy's introducing axioms, you ain't shit to question the definitions.What you should question is the utility. What problem does this solve, OP?
>>17023006I hope it could allow to make quantum mechanics logical.
>>17023003>>A true statement plus a false statement becomes a true statement?>That part is the same as in binary logic. Why would you pick on that?ooo boy
>>17022951What does this provide you that binary logic doesn't?
>>17022951+ is not a logical operator, RETARD
>>17022951>nu-/sci/ brainlets can't figure out OP's '+' is just an extension of the boolean 'or'>T+T=T>T+F=T>F+F=FThat's the same as a boolean 'or'.>A+A=A>A+F=AThat's analogous to a boolean 'or' but with "anti-truth".>T+A=FThat's weird, though. What does it correspond to in propositional terms? If both elements of a disjunctive clause are true but contradict each other, the clause is false?
>>17023045+ is the OR operator
>>17023054min(max(a + b, -1), 1) =/= a + b, gay OP.
>>17022951I propose a subsequent ternary system:OP, Faggot, and ThreadOP+Faggot = ThreadOP+OP = FaggotOP+Thread = FaggotFaggot+Thread = OPThread+Thread = ThreadFaggot+Faggot = OP
>>17023053some niggas are more comfortable with the wedges for shit like and and or
>>17022951Your operator is idempotent, which is nice, but it is not associative (assuming it is commutative). Consider:> T+(T+A) = T+F = T> (T+T)+A = T+A = FOther counterexamples include (T+A+A), (A+T+T) and (A+A+T). Associativity is often desirable for an "OR"-like operation. You might want to rename it or use different table?
>>17024506Furthermore, this appears to be the only non-associative option of the three possible magmas that have identity, is order 3, idempotent and commutative. The Cayley table for such an operator can be written as> 0 1 2> 1 1 x> 2 x 2You set x=0, and let 0F, 1A, 2T. Alternatively, if you had set x=2, you would've gotten max (or the semilattice chain). This would give you the "OR" in most three-valued logics such as Kleene logic and Łukasiewicz logic. Using x=1 would've given you "min" which also would've also been associative. There may be logics which use non-associative "OR" (maybe something substructural), but I am not familiar with them. If your interested in three-valued logics, you should check out the Wikipedia page for it, you'll see quite a lot of interesting ways people achieve these kinds of structures.
>>17024526Also you stated you were interested in applying this to Quantum physics >>17023008There's a lot of interesting research in Quantum logic to this day, both on the philosophical side and also the physics side. Perhaps you already know, but when we think of classical propositional logic, we tend to think of it as defined with T/F or 0/1, however, this kind of semantical perspective didn't appear until Boole (1847) and further refined with truth tables from Peirce (1883). These days, lattice and order theory has been found to be another particularly useful way of viewing logics (like truth tables).And so in a sense, logics are more generally defined in terms of axioms and identities. These are often formalised in proof systems such as Natural Deduction or Sequent Calculi. Girard's Linear logic is an exotic example which has connections to Quantum physics, Programming Languages and Game Theory. So if you understand Linear logic, you'll be able to play around with Quantum logic pretty easily.
>>17022951If i were a polfag I'd tell you you were not bullied enough as a kid.
>>17024551He won't read all thatHow do you think he ended up here, making such a thread in the first place?
Einstein himself would shed a tear for you OP, and would wish to be even half of the hecking genius you are.
>>17022951it should be true, none, false
>>17022951>T+T=T>-T=A=> T+T-T = T+A=> T = T + AA = 0?>T+A=FF = T?It looks like every element in this group is 0.>>17022975>Just go learn what fuzzy logic is already so you can deal with statements that are partially trueEducate 4chan
>>17024672You assumed cancellation and associativity holds, but that isn't the case. OP's operator has an identity (F) but isn't associative (see >>17024506). The Cayley table of OP's operator is:[eqn]\begin{array}{|c|c c c|} \hline & F & A & T \\ \hline F & F & A & T \\ A & A & A & F \\ T & T & F & T \\ \hline\end{array}[/eqn]The properties are mentioned in >>17024526, and so the calculation you did has two solutions:> [math]T + (T + (-T)) = T + F = T[/math]> [math](T + T) + (-T)) = T + A = F[/math]For cancellation to hold, we need [math]X+Y=X+Z \implies Y=Z[/math], however:> [math]F+A = A+A[/math], but [math]F \neq A[/math]> [math]F+T = T+T[/math], but [math]F \neq T[/math]This means no left/right inverse (i.e. not a quasigroup).
>>17022951(T + A) + A = F + A = AT + (A + A) = T + A = FA = Fthough you could deduce that from -T = A.
>>17024799Thanks. I believe we need associativity for logic system to be applicable to real world, OP should elaborate more.
>>17022951Show us an example of a mathematical equation which is anti-true.
>>17024840>I believe we need associativity for logic system to be applicable to real world>I believeWhy do you believe this?
>>17024973While I'm personally not interested in "real world" applications, philosophically and linguistically, we would expect "or" to be associative. I imagine it could cause some issues if you want to build material implication on top of it. OP's system's material implication is already quite unintuitive:[eqn] \begin{array}{|c|c c c|} \hline \to & F & A & T \\ \hline F & F & A & T \\ A & T & F & T \\ T & A & A & F \\ \hline \end{array} [/eqn]In particular, we have [math]T \to T = F[/math], which isn't entirely "bad" per se, but it is unusual. An interesting example of a 'logic' where [math]A \to A[/math] fails can be found in "Cause as an Implication" (1998) from Sylvan & Da Costa, where [math]\to[/math] is interpreted as "causes". Therefore "A causes A" doesn't always hold.
>>17022951Your logic, to the extant you described it, is also known as the sign function.
>>17025691*e
>>17025635>I'm personally not interested in "real world" applicationsignored. fuck off.
>>17025635Can you switch A and F so it looks more symmetric?
>>17024973Inner gut
>>17025768wdym "switch"? implies isn't commutative?
>>17025635FATAT
>>17022951i've messed around with ternary and n-ary linear logics a bit. it appears to be common to generalize or/and to min/max, but less common to generalize negation as rotation (addition modulo n), however it's possible to recover negation from both min(a,b) + rot(a) and max(a,b) + rot(a). it's a lot of fun to look for rewrite rules in the algebras, lots of stuff that you simply wouldn't come across in boolean because the pattern becomes trivial or outright disappears at n=2.this is something more exotic than a straight extension of boolean. best of luck
>>17027619circuit diagram i drew of a senary negation using min(a,b) to join and addition modulo 7 to cycle values
>>17025635kek...fat
>>17024799>You assumed cancellation and associativity holdsso will you stop using '+' and '-' as logical operators, then?
>>17022975>Just go learn what fuzzy logic is already so you can deal with statements that are partially trueSince author of this post didn't educate 4chan, so I will.Fuzzy logic is a bullshit set of rules that has nothing "logical" in it and is reduced to boolean logic when you try to implement it.
>>17027619Emil Post defined many-valued logics with cyclic negations. So his logic would be defined on [math]\{ 0, 1/2, 1 \}[/math] with negation [math]\lnot_P 0 = 1, \lnot_P 1/2 = 0, \lnot_P 1 = 1/2[/math] There is also Bochvar logics, which don't use min/max.
>>17029388there are lots of ways to bootstrap many-valued logics. the main advantage of min/max + rot is it's universal
>>17022951Oof. It's not a good sign if even the yes-machine isn't on your side.
>>17029783It's true that min/max is probably one of the nicest, in the sense that it has a lot of nice properties for many-valued logics (not sure what you mean by universal). However, logics motivated from philosophy often start off by defining implication. As such, starting with min/max can be undesirable since it often means you have to give up on strong material implication or a definition of not from implication:[eqn]\begin{align*}\forall P, Q\;(\lnot P \lor Q &= P \to Q) \quad \text{(Strong material implication)}\\ \forall P\;(P \to \bot &= \lnot P)\quad \text{(Definition of $\;\lnot\;$ from $\;\to$)}\end{align*}[/eqn]This means we end up with a weaker version of material implication. As an example in Łukasiewicz three-valued logic, he uses min/max but defines [math]p \to q = \min\{1, 1 - p + q\}[/math]. As a result, his implication fails strong material implication:[eqn]0.5 \to 0.5 = \min\{1, 1 - 0.5 + 0.5\} = 1[/eqn]Whereas, in Kleene, we have:[eqn]0.5 \to 0.5 = \min\{1-0.5, 0.5\} = 0.5[/eqn]Both use min/max, but Łukasiewicz loses strong material implication. In a sense, if you use min/max in three-valued logics, you end up with the most "reasonable" logic (assuming strong material implication), which is a Kleene algebra.
>>17029936>not sure what you mean by universali mean if n is the number of values in your many-valued logic, you can use addition modulo n (rotation) and either min or max to construct all possible operations. >>17027620is an example of a negation/flip circuit for n = 7. the vertical bars represent rotation, joins are max (typo in the original post).and as i said, there are lots of ways to do this sort of thing, but for example how would you extend the bochvar ^/v operations to n = 4? it's not clear to me from looking at the truth table that it's the type of operation that naturally grows with additional rows and columns. similarly traditional negation is tricky because it has a fixed point for odd n so it's behavior fundamentally oscillates. cyclic negation is universal because addition/subtraction mod n is the same no matter what n is.not an academic, just a curious ex software dev, so apologies if my terminology is nonstandard.
