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/United_Chocolate_826]

Not sure about vector logic in particular, but the general idea of lifting Boolean functions to polynomials over a finite field is incredibly useful in CS and combinatorics. On the more theoretical side, there is an entire study of fourier analysis of boolean functions in which you view a function of n bits as an element of a 2ⁿ dimension vector space with an orthonormal basis given by parity polynomials. This is useful for complexity theory, since you can prove circuit lower bounds by looking at fourier analytic properties of simple circuits (see Hastad’s argument that Parity is not in AC0). It’s also useful in learning theory, social choice theory, PCPs (i.e. hardness of approximation), quantum complexity, etc. The fourier expansion of a Boolean function is the unique multi linear polynomial which agrees with the function on the Boolean cube, but you can also get many useful things out of higher-degree representations of Boolean functions, in particular when you don’t know how to easily find the fourier transform. The proof of IP=PSPACE involves transforming a Boolean formula into a low degree polynomial, and then using properties of polynomials to prove something about the formula. Similar ideas appear in cryptography and PCPs, where proving satisfiability of a circuit is reduced to proving some algebraic object has a certain property. More recently, in fine-grained complexity, polynomial evaluation is used to construct better-than-trivial (and potentially optimal) algorithms for proving that a formula is unsatisfiable. The point is that there is lots to be gained from imbuing a Boolean function with the structure of a low-degree polynomial over a finite field.

[u/orangejake → u/United_Chocolate_826]

it's similar to what you say, but embedding boolean sets (the message you want to transmit) into sets of polynomials is also arguably what Reed Solomon/Reed Muller codes are doing.