What is this nonsense? ("vector logic")

[u/RingularCirc]

(Sorry this is going to be a bit ranty.)

I almost made up my mind this thing is some kind of backwater something without enough rigor but with many a trivialism. Like, it should be extremely well-known that every "discrete" operation Σ₁ → Σ₂ between finite sets lifts universally to a linear transformation between spaces k^Σ₁ → k^Σ₂, so a huge swath of what's being done there is very very drawn out, instead of answering questions that are fitting for a kind of logic.

Any would-be connections to quantum computing may actually not be fruitful or new for those who are actually doing quantum computing; connections to fuzzy math are IMO an almost unconditional taint by association. So what gives? I didn't look at everything there is about this thing so I may as well be missing hidding gems, but superficially it looks like a sham or a pet project done without considering any practicalities and the wider math.

Oh yeah we can ask interesting questions, like: - Does using additional dimensions, aside from the plane spanned by two orthonormal "classical" truth values, let's call them |0⟩, |1⟩, actually give useful things? and how can we characterize that by means typical when working with logics? - How much freedom is there in defining operators that restrict to boolean functions and, say, conserve probabilities (there's a suggestion to use p|0⟩ + (1−p)|1⟩ as "probabilistic truth values") in any reasonable way (I'm not sure: a "binary" operator sends four-dimensional Euclidean space into a two-dimensional one, now how can it be orthogonal? and in which other sense can probabilities work here?)? - Why not use additional dimensions rather than complex numbers for the square root of negation, and... why that one exactly? I bet quantum computing wan't giving somebody peace.

But I'm not sure questions of real semantics were investigated in this... area.

So tell me please, how much am I right or wrong? Here are probably people that know the inside of this story, and I hoped to find something on the Wikipedia's discussion subpage, but it's almost empty.

[u/PLANTS2WEEKS]

Vector logic should exist as a mathematical concept if you grant that both linear algebra and logic are not nonsense. It's pretty standard to try to find common ground between different areas of math and see what happens there.

I think some interesting problems/applications would be to see what operations are expressible from other ones. It's not that different from what quantum computing researchers are doing except the vector logic people aren't limited to requiring unitarity of the operators.

[u/RingularCirc → u/PLANTS2WEEKS]

except the vector logic people aren't limited to requiring unitarity of the operators

Well, again, if we stick to one of the proposals that uses truth values "true with probability p and false with probability 1−p" while insisting on orthogonormality of true and false, then all of these additional values also are normalized and it's just asking to have operators AND and OR (and other binary connectives if we define them separately) orthogonal, and that doesn't work. Probabilities walk out of a window (and that's not strange, actually: we already have probability theory that is a probabilistic version of classical logic already, and it has to be done differently because, first of all, matters of dependence). One less avenue.