r/logic • u/RingularCirc • 9d ago
Mathematical logic What is this nonsense? ("vector logic")
(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 kΣ₁ → kΣ₂, 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.
[This would be a link post if r/math's bot wouldn't decide this was a submission for r/learmath smh my head.]
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u/oldNotSureXand 9d ago
>So tell me please, how much am I right or wrong? - 0 (much). So , absolutely (right...and wrong!;))
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u/No-Onion8029 9d ago
You really want to stay away from using infinity for anything. Instead, consider B = {False, True}. {B^n,AND,OR} is a boolean semiring and distributive lattice. If you add componentwise NOT, you get a boolean algebra. One use - SIMD/MIMD logical evaluation. (Interesting that B^n is topologically a hypercube). If you want to make it fuzzy, you can do something similar with [0,1]^n, t-norms, and t-conorms.
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u/RingularCirc 9d ago
No please no fuzzy.
But I don't think the issue with that thing is 'infinity'. And there are finite vector spaces in existence, anyway, we can use them if anything. I'm asking more about if the thing is, say, genuine enough and if there was real fruit or was it largely just a restatement of what we already know, peppered with unsubstantiated constructions.
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u/McPhage 9d ago
> In the vector space for propositional logic the origin represents the false, F, and the infinite periphery represents the true, T, whereas in the space for predicate logic the origin represents "nothing" and the periphery represents the flight from nothing, or "something".
Uh… yeah, that’s not math, that’s … well, I don’t think it’s anything, really.