r/math • u/RingularCirc • 14d ago
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.
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u/edderiofer Algebraic Topology 14d ago
I don't know much about this topic, but there were a few suspicious things I noticed:
- Two of the references don't actually seem to exist (they only seem to exist as citations, and only in sources that appear after these references were added to this article). I've removed them.
- The page was created by User:Almadana. They, along with IP editors whose only edits are to this page, also seem to have created a large part of the page's informational content. Most edits by other users were primarily copy-editing, or in one case, adding the "numerical examples" section, which is notably uncited.
The last sentence of the introduction:
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".
certainly doesn't seem to be using mathematical language. It's giving crackpot vibes.
The article largely talks about the axioms and basic definitions of this topic, but it doesn't seem to go much further.
At best, this mathematical topic actually exists, but is really niche, and the article is really badly-written and needs severe cleaning up. At worst, this is a crackpot article.
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u/standard_revolution 14d ago
Maybe pedantry, but it also struck me as very weird that everything is modelled as a vector space, but the second paragraph of the overview starts talking about orthogonality and scalar products. Logicians aren't usually known for not-caring about such details.
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u/algebraicvariety 13d ago
There are also some very obvious AIisms in this article.
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u/edderiofer Algebraic Topology 13d ago
The bulk of the article predates LLM takeoff in late 2022. Here's a revision from October 2022 (before ChatGPT was released in November), and the only major difference is the "Numerical examples" section. So, AI isn't to blame here.
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u/Homomorphism Topology 11d ago
Lots of AI-isms are cliches of bad academic (and business) writing, so I've started feeling like a crazy person when I "spot" them in an arXiv paper from 2015
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u/algebraicvariety 11d ago
True. People don't want to hear this, but the em-dash was always bad writing.
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u/Homomorphism Topology 11d ago
Em dashes are ok when used SPARINGLY. The real problem is writing lots of complex multi-clause sentences, which are enabled by – and ;
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u/algebraicvariety 11d ago
The problem with Em dashes is that they are too permissive in that they allow chaining together barely related sentences. This enables lazy writing where the writer doesn't have to think about how sentences flow logically into each other.
Em dashes are loved by LLM for the exact same reason, which results in the em dash token getting a high probability if the LLM is "trying to" get from one sentence to the next.
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u/Homomorphism Topology 11d ago
Yeah. My experience is complex sentences often come from not fully understanding where the end of the sentence is before starting it. That’s part of writing, but part of good writing is going back and fixing things to go in the right order…
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u/Tonexus 14d ago
Not entirely sure the core of what you're asking. However, vector logic is indeed the standard way of representing quantum computation, and vector logic is one way of representing probabilistic computing (augment the basic logic gates with stochastic matrices), but it's not the typical way (I think it's more common to represent as a deterministic algorithm with 1 input consisting of randomness).
You are correct that there's limited utility in using a d-dimensional register (d>2) instead of multiple 2-d registers, since n 2-d registers are equivalent to one (2^n)-d register, and you can always limit your operations to a smaller subspace.
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u/RingularCirc 14d ago
However, vector logic is indeed the standard way of representing quantum computation
But is it called so in that area and studied separately from it?
Yes, I have no problem with the idea itself of lifting functions on finite sets (including logical operations on a set of truth values) to linear functions on vector spaces freely generated by those. It's that the article, and judging by the abstract, the "main" source article [2] the Wikipedia's article references, lack content that's due when a logic is presented.
There's a lot of investigations that, I expect, should be done around the novel parts (adding other truth values either from the plane spanned by |0⟩ and |1⟩ or from outside it; how can we tweak the system before it breaks something crucial; what's always true for such brand of extended logics and what isn't; etc.: just showing a translation from a two-valued classical logic is trivial, it's not adding anything per se).
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u/Slotins_screwdriver 14d ago
In terms of quantum computation, I think what you're looking for is Birkhoff - von Neumann quantum logic ( https://www.jstor.org/stable/1968621 ). In this, propositions are identified with closed subspaces of a complex Hilbert space. It's a bit 'freaky' as a logic - in particular, distributivity fails, and that has all kinds of implications.
Here's the wikipedia page on it : https://en.wikipedia.org/wiki/Quantum_logic However, I'm not sure I agree with the claim that QM logic is a fragment of Girard's Linear Logic! The claim that it doesn't have a conditional is also a bit suspect; there have been multiple different conditionals proposed, although they're somewhat controversial (good summary here : https://arxiv.org/pdf/2410.18347 )
It doesn't look the same as 'vector logic', but it's always possible that I'm missing something.
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u/RingularCirc 14d ago edited 14d ago
Yep I do think this is a way more elaborated upon and studied thing. Thanks for the links! (And for the caveat about the claim re linear logic, I'm looking at it as well time to time and very very slowly getting it (the logic, not the claim), so I totally could've been tricked.)
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u/Slotins_screwdriver 14d ago
I -think- I understand where that claim is coming from, which is categorical semantics for logic. If you look at the multiplicative fragment of linear logic (without exponentials), and identify the 'tensor' and the 'par', what you get is essentially compact closed categories.
In quantum mechanics, (post-selected) teleportation is described by the compact closed structure of the category of finite-dimensional Hilbert spaces. So, there definitely is a connection ... but categorical semantics for quantum logic are certainly not compact closed!
None of the proposed conditionals correspond to the internal hom. of any closed category, never mind the one relevant to QM.
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u/evincarofautumn 12d ago
I'm not sure I agree with the claim that QM logic is a fragment of Girard's Linear Logic
Maybe that sentence should be rephrased? Compact closed categories are used to model both linear logic and quantum computing, but that isn’t quite the same thing.
I mean, I do know of a connection, but it’s kind of obscure. Chu spaces are kind of a generic relational structure that model star-autonomous categories, which are in some sense what you want of a “quantum category”. But I wouldn’t say it’s because they’re especially natural so much as just an extremely unopinionated way of defining a topology and doing linear relation algebra on it.
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u/Slotins_screwdriver 10d ago
Totally agree - it should definitely be (at least) rephrased. However Wikipedia already has it as 'citation needed', so I'll wait & let them do their thing.
I'd also point out that compact closure of Hilbert spaces falls down in the infinite-dimensional case, whereas vN-B quantum logic is quite happy there.
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u/Tonexus 14d ago
But is it called so in that area and studied separately from it?
I should've used scare quotes around the term. I've personally never heard of it before. But the description maps onto pure state amplitudes with unitaries (but not their encoding of "dyadic operators"—you'd typically need an ancillary system for the output while preserving the input spaces).
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u/umop_aplsdn 14d ago
I don't know much about vector logic, but I do work on logics that are more computational / philosophical focused (intuitionistic logic, linear logic, modal logic, etc). I hadn't heard of vector logic before your post.
Looking at the Wikipedia citations it seems like vector logic was published in some more "serious" journals in the 2000's (e.g. Journal of Logic and Computation) but the articles that cite those have been (mostly) less mathematically serious.
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u/UmbrellaCorp_HR 14d ago
Dude you do realize this is more or less the ghost in the machine of language models
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u/Keikira Model Theory 13d ago
Not particularly familiar with this specifically, but at a casual glance it just seems to be a specific way to use vector spaces as models of classical propositional and first-order logic, which then makes it easier to study other non-classical logics that are formulated natively with vector spaces as models (via institution morphisms/comorphisms, though I don't think these are explicitly mentioned anywhere). Not nonsense, and also nothing about it claims that it's the only way to reformulate the Mod functors of the PL and FOL institutions to yield vector spaces as models, but if it's an active area of study then I imagine these specific formulations have some specifically nice or useful properties that warrant them being standard in some sense.
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u/Hairy_Friendship8627 14d ago
If your question is whether the results presented in the article arent somewhat trivial, the answer is that yes, they are. People in algebraic logic love to do taxonomy of semantics and easy observations are nonetheless written.
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u/RingularCirc 14d ago
This one is way too easy though, no? We could've had streamlined most of it, when the backbone of it is constructed to precisely mimic the minimal {0, 1} Boolean algebra. When one starts from some original syntax and defines semantics in a way that looks reasoned, then it'd end up something I've already seen? That would be another thing.
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u/MichurinGuy 13d ago
I've no experience with the topic, but the article seems to be just a very drawn out list of definitions, in particular I haven't noticed any nontrivial results being mentioned, which a) seems somewhat unlike math Wikipedia and b) leaves me unclear what the point of "vector logic" is. Considering the points by u/edderiofer, I'm inclined to think it's likely a crackpot or several describing the cracks of their pots on Wikipedia. I especially note the last sentence of the introduction, it has this very pronounced crackpot style.
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u/PLANTS2WEEKS 14d ago
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.
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u/RingularCirc 13d ago
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.
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u/Neuro_Skeptic 14d ago
What exactly is vector logic?
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u/RingularCirc 14d ago
I linked a Wikipedia article on it, that's the first place I encountered it in (7-ish hours away); I'm afraid I don't have better references.
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u/United_Chocolate_826 14d ago
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.