Proportoids
Pith reviewed 2026-05-24 11:09 UTC · model grok-4.3
The pith
Proportoids are sets equipped with a 4-ary analogical proportion relation satisfying axioms, with homomorphisms preserving the relation exactly and kernels forming congruences.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Proportoids consist of a set with a 4-ary analogical proportion relation a:b::c:d obeying suitable axioms. Homomorphisms H between proportoids satisfy a:b::c:d if and only if Ha:Hb::Hc:Hd, and the kernel of such an H is a congruence. Analogies are mappings A such that a:b::Aa:Ab for all a and b, with methods to compute partial analogies. Various relations between these functions are introduced and analyzed.
What carries the argument
The proportoid, defined as a set with a 4-ary analogical proportion relation satisfying axioms, together with homomorphisms that preserve the relation iff and analogies that map proportional pairs.
If this is right
- Homomorphisms of proportoids have kernels that are congruences.
- Analogies allow computation of partial mappings that respect proportions.
- Relations between functions like homomorphisms and analogies on proportoids can be classified by their properties.
- This framework supports building a mathematical theory of analogical proportions.
Where Pith is reading between the lines
- Quotient proportoids formed by congruences could simplify complex analogical structures.
- Algorithms for finding analogies might be developed using the partial analogy construction.
- These axioms could be tested in knowledge bases to see if they capture real-world analogies effectively.
- Proportoids might connect to categorical approaches in AI for modeling similarity.
Load-bearing premise
There exists a suitable but unspecified set of axioms for the 4-ary analogical proportion relation that is mathematically coherent and faithful to the intuitive notion of analogical proportions.
What would settle it
Demonstrating that no set of axioms for the 4-ary relation can be both consistent and capture standard examples of analogical proportions such as '2 is to 4 what 3 is to 6' without collapsing into triviality would falsify the existence of proportoids as useful structures.
read the original abstract
Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning, which itself is at the core of artificial intelligence. This paper contributes to the mathematical foundations of analogical proportions in the axiomatic tradition as initiated -- in the tradition of the ancient Greeks -- by Yves Lepage two decades ago. More precisely, we first introduce the name ``proportoid'' for sets endowed with a 4-ary analogical proportion relation satisfying a suitable set of axioms. We then study study different kinds of proportion-preserving mappings and relations and their properties. Formally, we define homomorphisms of proportoids as mappings $\mathsf H$ satisfying $a:b::c:d$ iff $\mathsf Ha: \mathsf Hb:: \mathsf Hc: \mathsf Hd$ for all elements and show that their kernel is a congruence. Moreover, we introduce (proportional) analogies as mappings $\mathsf A$ satisfying $a:b:: \mathsf Aa: \mathsf Ab$ for all elements $a$ and $b$ in the source domain and show how to compute partial analogies. We then introduce a number of useful relations between functions (including homomorphisms and analogies) on proportoids and study their properties. In a broader sense, this paper is a further step towards a mathematical theory of analogical proportions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces 'proportoids' as sets equipped with a 4-ary analogical proportion relation satisfying a suitable set of axioms. It defines homomorphisms of proportoids as bi-directional relation-preserving maps and claims their kernels are congruences. It further defines proportional analogies as maps A such that a:b :: Aa:Ab, discusses computation of partial analogies, and examines various relations between functions including homomorphisms and analogies on proportoids.
Significance. This axiomatic approach to analogical proportions could advance the mathematical foundations of analogical reasoning in AI, building on prior work by Lepage. The study of proportion-preserving mappings and their properties has potential to formalize aspects of analogical inference. However, the significance is contingent on the specific axioms ensuring non-trivial structures, which are not detailed in the provided text.
major comments (2)
- [Abstract and introduction] The axioms for the 4-ary proportion relation are referred to as 'suitable' but never explicitly stated (abstract). This is load-bearing for the central claims, as without them it cannot be determined whether the universal relation satisfies the axioms, which would make every function a homomorphism and an analogy, rendering the kernel congruence the full relation and collapsing the theory to triviality.
- [Definition of homomorphisms] The claim that the kernel of a homomorphism is a congruence (abstract) depends on the unspecified axioms; no derivation details or axiom list are provided to confirm it holds non-vacuously.
minor comments (1)
- [Abstract] Abstract contains repeated word: 'we then study study different kinds'.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the constructive comments on the manuscript. We address the two major comments below.
read point-by-point responses
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Referee: [Abstract and introduction] The axioms for the 4-ary proportion relation are referred to as 'suitable' but never explicitly stated (abstract). This is load-bearing for the central claims, as without them it cannot be determined whether the universal relation satisfies the axioms, which would make every function a homomorphism and an analogy, rendering the kernel congruence the full relation and collapsing the theory to triviality.
Authors: We agree that the abstract's reference to a 'suitable set of axioms' without listing them creates ambiguity and prevents immediate verification of non-triviality. The revised manuscript will explicitly state the axioms (currently introduced in the body) already in the abstract and introduction, together with a short argument that the chosen axioms exclude the universal relation as a model. revision: yes
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Referee: [Definition of homomorphisms] The claim that the kernel of a homomorphism is a congruence (abstract) depends on the unspecified axioms; no derivation details or axiom list are provided to confirm it holds non-vacuously.
Authors: The claim relies on the specific axioms; the current text does not supply either the axiom list or the derivation in the abstract. In the revision we will insert the axiom list and a concise proof that the kernel is a congruence under those axioms, thereby confirming the result is non-vacuous. revision: yes
Circularity Check
No significant circularity; axiomatic framework derives mapping properties independently
full rationale
The paper defines proportoids axiomatically via a 4-ary relation satisfying a suitable (unspecified) set of axioms, then defines homomorphisms (preserving the relation bidirectionally) and analogies (satisfying a:b::Aa:Ab), and derives properties such as kernels being congruences. No self-definitional loops, no fitted parameters renamed as predictions, no load-bearing self-citations, and no imported uniqueness theorems or ansatzes. The derivation chain is self-contained within the axiomatic setup and does not reduce any claimed result to its own inputs by construction.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption A suitable set of axioms for the 4-ary analogical proportion relation
invented entities (3)
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proportoid
no independent evidence
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homomorphism of proportoids
no independent evidence
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proportional analogy
no independent evidence
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AbsoluteFloorClosure.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Definition 2. A proportoid is a pair P = (P, ::) ... satisfying (1) a:b::a:b (reflexivity), (2) symmetry, (3) inner symmetry.
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Definition 9. Homomorphism H satisfies a:b::c:d iff Ha:Hb::Hc:Hd; Theorem 20: ker H is a congruence.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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