Logical Embeddings for Argument Analysis

# Logical Embeddings for Argument Analysis
Source: [https://arxiv.org/html/2608.15325](https://arxiv.org/html/2608.15325)
Leander HeldringAffiliation:Kellogg School of ManagementAffiliation:Northwestern UniversityAffiliation:2211 Campus Drive, Evanston, IL 60208Email:[leander\.heldring@kellogg\.northwestern\.edu](mailto:)Santiago TorresAffiliation:Department of EconomicsAffiliation:MITAffiliation:50 Memorial Drive, Cambridge, MA 02142Email:[storresp@mit\.edu](mailto:)

###### Abstract

We propose a new framework for machine\-learning\-oriented argument analysis tasks\. Our proposal involves replacing traditional contextualized word embeddings used in most NLP tasks withlogical embeddings, an alternative encoding that directly exploits argumentation structures\. In essence,logical embeddingsencapsulate the logical semantics of an argument, allowing for a better representation of its meaning\. Supporting these embeddings is a mathematical logic\-based similarity measure that offers a transparent notion of proximity and is guaranteed to satisfy several desirable theoretical properties that current cosine similarity\-based contextualized word embeddings cannot assure\. This similarity measure induces a positive semi\-definite kernel on the set of arguments, enabling us to uniquely define logical embeddings using the theory of Reproducing Kernel Hilbert Spaces \(RKHS\)\. Moreover, we prove that this encoding is optimal, in the sense that no logical information is lost in the process\. As with other RKHS applications,logical embeddingscan be used in numerous supervised and unsupervised tasks\. We provide an implementation of the method and aim to test it against literature benchmarks\. Additionally, we demonstrate thatlogical embeddingsoutperform most standard embedding methods on a classification task\.

## 1Introduction

The advent of contextualized word embeddings has revolutionized the field of Natural Language Processing \(NLP\)\([27](https://arxiv.org/html/2608.15325#bib.bib25);[18](https://arxiv.org/html/2608.15325#bib.bib6);[11](https://arxiv.org/html/2608.15325#bib.bib12)\)\. One area that has significantly benefited from this innovation is argument and debate analysis, which features tasks such as argument detection, classification, similarity, and generation\.

However, the adoption of contextualized word embeddings for argument learning tasks has not been without limitations, especially when assessing argument similarity\. In argument similarity, state of the art methods often confuse similarity in argumentation, with similarity in other linguistic features\. For example, in Figure[1](https://arxiv.org/html/2608.15325#S1.F1), we report three arguments from the IBM\-ArgQ\-6\.3kArgs dataset\([26](https://arxiv.org/html/2608.15325#bib.bib7)\)\. For the argument at the top, we promptgpt\-4o\-minito assign a full\-argument logical\-overlap score between the reference argument and each of the two candidate arguments shown at the bottom\. The exact prompt used for this exercise is reported in Appendix[B\.3\.1](https://arxiv.org/html/2608.15325#A2.SS3.SSS1)\. While the right hand side pair is clearly more similar, the left hand side pair gets a higher similarity\. This is fundamentally because language models attend to more than the logical overlap between arguments\.

Figure 1:Misleading full\-text similarity scores\.![Refer to caption](https://arxiv.org/html/2608.15325v1/Figures/fig_1_comparison_argument.png)Notes: The figure shows how full\-text similarity scores could not correspond to argument equivalence\. The left argument receives a high score because it shares the same abstract concept of necessity despite concerning a different domain\. The right argument instead have a lower score despite topic overlap because it is relative to a more specific context\. This motivates distinguishing between surface level semantic similarity and argument level equivalence\.Notably, although the distinction between the pairs is clear to a human, the contextualized word embeddings represent the second pair as more similar vectors than the first pair as witnessed by a higher similarity score, causing the classifier to make errors\.

This example is an instance of the point made by several authors \(e\.g\.[11](https://arxiv.org/html/2608.15325#bib.bib12)\) that contextualized word embeddings might not be the optimal way to represent arguments\. To humans, “Argumentative reasoning is based on justifying a plausible conclusion with arguments in its favour”\([5](https://arxiv.org/html/2608.15325#bib.bib14), p\. iv\)\. At its core, therefore, argument similarity rests ultimately in assessing the proximity of logical reasoning\. In addition, similarity metrics based on the cosine similarity of embeddings have additional problems due to the lack of a notion of the size of an embedding\([25](https://arxiv.org/html/2608.15325#bib.bib20)\)\.

In sum, while contextualized word embeddings are a great general\-purpose technology for NLP tasks, they may be suboptimal for argument learning tasks\. Moreover, their derived similarity metrics often fail to comply with desirable properties for such tools, such as the basic monotonicity requirement \(i\.e\., two more similar arguments should receive a higher similarity score than two less similar arguments\)\. This, in turn, can lead to errors in downstream tasks\.

In this paper, we propose a new framework to represent and assess similarity between arguments:logical embeddings\. As its name implies,logical embeddingsprovide a way to encode only the logical content of arguments, thus avoiding many shortcomings of contextualized word embeddings\. This approach has three desirable properties\. First,logical embeddingsareoptimal embeddingsin the sense that no logical information is lost in the representation\. Second, the supporting similarity metric is ensured to satisfy theoretically desirable properties of a proximity notion, such as monotonicity and the triangle inequality\. Third, as we will illustrate, logical embeddings provide a powerful tool to analyze complex argumentation schemes\.

Our starting point is to introduce into the argument similarity literature a notion of similarity from mathematical logic\([5](https://arxiv.org/html/2608.15325#bib.bib14);[23](https://arxiv.org/html/2608.15325#bib.bib15)\)\. We outline the full necessary mathematical structure in the next section\. In essence, this metric builds upon the definition of an argument as a logical object, composed of a set of premises and a conclusion, with a logical association that allows the latter to follow from the former \(for a similar treatment from linguistics, see[9](https://arxiv.org/html/2608.15325#bib.bib9)\)\. Argument similarity is then defined as the fraction of overlapping premises, modulo logical equivalence, together with the overlap of the logical consequences derived from its conclusion\. The resulting similarity metric has numerous theoretical properties, among which are monotonicity and triangle inequality, that allow it tometricizelogic\. Following[5](https://arxiv.org/html/2608.15325#bib.bib14)and[10](https://arxiv.org/html/2608.15325#bib.bib19), we call this similarity measure the Syntactic\-Semantic Jaccard measure\.

We prove that the Syntactic\-Semantic Jaccard similarity measure is a positive semi\-definite kernel on the set of arguments\. Following the literature on Reproducing Kernel Hilbert Spaces \(RKHS\)\([3](https://arxiv.org/html/2608.15325#bib.bib4)\), such a property defines unique embeddings of arguments via their premise\-conclusion decomposition\. This gives rise tological embeddings\. We further prove that, because of the desirable properties of the similarity metric that spans them,logical embeddingspreserve all logical information\. This stems from the fact that the Syntactic\-Semantic Jaccard similarity measure is a characteristic kernel\([7](https://arxiv.org/html/2608.15325#bib.bib2);[8](https://arxiv.org/html/2608.15325#bib.bib1)\), which guarantees an injective map from an argument to its embedding\. As with other RKHS applications,logical embeddingscan be used for many supervised and unsupervised tasks \(see[22](https://arxiv.org/html/2608.15325#bib.bib26)and references therein\)\.

We discuss how to implement the proposed framework in practice, and test performance\. In short, we propose a three\-step procedure\. First, frontier Large Language Models \(LLMs\) can productively be used to decompose an argument into its constituent premises and conclusion\. Second, an LLM maps a conclusion into its logical consequences\. Third, we then train an entailment model to measure pairwise entailment among premises and \(sub\)conclusions\. Steps 1 and 2 are essentially retrieval tasks, and standard methods perform adequately for this purpose\. Our next steps involve decomposing conclusions into their logical consequences\. This is important and best illustrated by example\. The conclusions "The earth is round" and "The earth is round and blue" are not logically equivalent but have common implications—both imply that the earth is round—making them partially similar\. LLMs perform very well on this task, and we propose an implementation\. Finally, we train a standard entailment model to compute logical similarity between premises and conclusions in their Conjunctive Normal Form \(CNF\)\. These steps allow for the computation of the Syntactic\-Semantic Jaccard measure, thereby enabling the practical use oflogical embeddings\.

Finally, we test the performance of logical embeddings on a standard classification task, using the IBM\-ArgQ\-6\.3kArgs corpus\([26](https://arxiv.org/html/2608.15325#bib.bib7)\)\. Using this corpus, we construct a Gram matrix of pairwise argument\-similarity scores and use its reduced\-dimensional representation as a logical embedding for each argument\. We compare these embeddings against standard semantic embedding baselines, including averaged GloVe vectors, BERT\-base, RoBERTa\-base, SBERT, and OpenAI text\-embedding\-3\-small embeddings\. Across logistic and linear regressions, as well as a small neural network and a random forest, embeddings outperform the semantic baselines\. We then concatenate logical embeddings with each semantic embedding to combine logical and semantic information\. The best performance is obtained either by logical embeddings alone or by logical embeddings combined with OpenAI text\-embedding\-3\-small embeddings\.

Our paper makes three key contributions\. First, we introduce logical embeddings\. We show that the minimal structure imposed on similarity computation by imposing the premise\-conclusion structure and the Syntactic\-Semantic Jaccard measure provide several desirable properties for resulting similarity score\. Second, we prove that the similarity matrix resulting from the Syntactic\-Semantic Jaccard measure on a set of arguments is an injective positive semi\-definite kernel, and we, therefore, prove the optimality of the ‘logical embeddings’ induced by this map\. Finally, we propose an implementation pipeline and test the performance of logical embeddings against standard benchmarks\.

## 2Logical embeddings

This section develops the framework used in the rest of the paper\. The construction proceeds in three steps\. First, we model an argument as a collection of premises paired with a conclusion\. Second, we define a logic\-aware similarity function on pairs of arguments\. Third, we prove that this similarity function is a positive semidefinite kernel\. Accordingly, this kernel induces a feature map into a reproducing kernel Hilbert space \(RKHS\)\. We call that feature map a logical embedding\.

### 2\.1Arguments as premise–conclusion pairs

We work with a propositional logic\(ℒ,⊢\)\(\\mathcal\{L\},\\vdash\), whereℒ\\mathcal\{L\}is the language of formulas and⊢\\vdashis the logical consequence relation\. Appendix[A\.1](https://arxiv.org/html/2608.15325#A1.SS1)gives the formal construction of these objects\.

The definitions in this section use only the standard syntax of propositional logic\. Atomic propositions are statements that can be assigned a truth value\. Formulas are built from atomic propositions using the Boolean connectives: negation\(¬\)\(\\neg\), conjunction\(∧\)\(\\land\), disjunction\(∨\)\(\\lor\), implication\(→\)\(\\rightarrow\), and biconditionality\(↔\)\(\\leftrightarrow\)\. Thus, ifppandqqare atomic propositions, thenp∧qp\\land q,p→qp\\rightarrow q, and¬p∨q\\neg p\\lor qare formulas\. Finally, a literal is an atomic proposition or its negation\.

The relation⊢\\vdashdenotes logical consequence\. ForΓ⊆ℒ\\Gamma\\subseteq\\mathcal\{L\}andϕ∈ℒ\\phi\\in\\mathcal\{L\}, the expressionΓ⊢ϕ\\Gamma\\vdash\\phimeans that every truth assignment satisfying all formulas inΓ\\Gammaalso satisfiesϕ\\phi\.

For formulasϕ,ψ∈ℒ\\phi,\\psi\\in\\mathcal\{L\}, we say thatϕ\\phiandψ\\psiare logically equivalent, and writeϕ≡ψ\\phi\\equiv\\psi, when they entail each other:ϕ⊢ψ\\phi\\vdash\\psiandψ⊢ϕ\\psi\\vdash\\phi\. Hereϕ⊢ψ\\phi\\vdash\\psiabbreviates\{ϕ\}⊢ψ\\\{\\phi\\\}\\vdash\\psi\. Equivalently,ϕ≡ψ\\phi\\equiv\\psiif, and only if,ϕ\\phiandψ\\psihave the same truth conditions\. We denote the equivalence class ofϕ\\phiunder≡\\equivby\[ϕ\]≡=\{ψ∈ℒ:ψ≡ϕ\}\[\\phi\]\_\{\\equiv\}=\\\{\\psi\\in\\mathcal\{L\}:\\psi\\equiv\\phi\\\}\.

###### Definition 1\(Argument\)\.

An argument is a pair

whereΦ⊆ℒ\\Phi\\subseteq\\mathcal\{L\}is a finite set of premises andϕ∈ℒ\\phi\\in\\mathcal\{L\}is a conclusion such thatΦ⊢ϕ\\Phi\\vdash\\phi\. The premise set is required to be consistent, meaning that premises do not contradict each other, and non\-redundant, meaning that no premise is dispensable for derivingϕ\\phi\. The set of all arguments is denotedArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\.

The point of this representation is to map natural\-language arguments into a common premise–conclusion structure that captures their logical content\. Under this structure, an argument is represented by the premises it uses and the conclusion those premises support\. This makes it possible to compare arguments on the basis of their logical components rather than their wording\. Appendix[A\.1](https://arxiv.org/html/2608.15325#A1.SS1)details the formal consistency and non\-redundancy conditions\. Section[3](https://arxiv.org/html/2608.15325#S3)describes how we approximate this structure from text using NLP models\.

This perspective determines an equivalence relation on arguments\. Two arguments are identical for our purposes when they use the same premises up to logical equivalence and have logically equivalent conclusions\. For a set of formulasΦ\\Phi, define

\[Φ\]≡=\{\[ξ\]≡:ξ∈Φ\},\[\\Phi\]\_\{\\equiv\}=\\\{\[\\xi\]\_\{\\equiv\}:\\xi\\in\\Phi\\\},the set of equivalence classes represented by the formulas inΦ\\Phi\.

###### Definition 2\(Argument equivalence\)\.

For argumentsa=\(Φ,ϕ\)a=\(\\Phi,\\phi\)andb=\(Ψ,ψ\)b=\(\\Psi,\\psi\), we writea≈ba\\approx bif

\[Φ\]≡=\[Ψ\]≡andϕ≡ψ\.\[\\Phi\]\_\{\\equiv\}=\[\\Psi\]\_\{\\equiv\}\\qquad\\text\{and\}\\qquad\\phi\\equiv\\psi\.

Thus, two arguments may be expressed differently and still be equivalent\. For example, the premises “John is Susan’s brother” and “Susan is John’s sister” are worded differently but have the same truth conditions: each is true if and only if the other is true\. The equivalence relation therefore identifies arguments by mutual entailment, not by wording\.

### 2\.2A syntactic–semantic similarity between arguments

We now define the similarity function that measures logical proximity between arguments and induces the logical embedding\. The construction follows[5](https://arxiv.org/html/2608.15325#bib.bib14)and[23](https://arxiv.org/html/2608.15325#bib.bib15)\. It compares arguments along two dimensions: the premises they use and the logical content of their conclusions\.

The main difficulty lies in comparing the conclusions\. We do not compare formulas by the full set of their logical consequences, since that set is typically infinite and contains many equivalent reformulations\. Instead, following[5](https://arxiv.org/html/2608.15325#bib.bib14), we compare conclusions by fixing a finite set of non\-redundant consequences\.

We use conjunctive normal form \(CNF\) to fix these representatives\. A formula is in CNF if it is a conjunction of one or more clauses, each of which is a disjunction of one or more literals\. Informally, a CNF formula is obtained by joining “or” statements with “and” connectives\. For example,p→\(q∧r\)p\\to\(q\\land r\)is logically equivalent to the CNF formula

\(¬p∨q\)∧\(¬p∨r\)\.\(\\neg p\\lor q\)\\land\(\\neg p\\lor r\)\.This restriction loses no expressive power: every propositional formula is logically equivalent to some CNF formula\([19](https://arxiv.org/html/2608.15325#bib.bib17)\)\.

For a formulaϕ\\phi, letCNℱ⁡\(ϕ\)\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)be its set of logical consequences in CNF form\. This is the finite set of CNF formulas that are entailed byϕ\\phi, use only the literals relevant toϕ\\phi, and are listed only once up to logical equivalence\.111Appendix[A\.1](https://arxiv.org/html/2608.15325#A1.SS1)gives the formal construction of this set\.

For example,

CNℱ⁡\(p∧q\)=\{p,q,p∨q,p∧q\}\.\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(p\\land q\)=\\\{p,q,p\\lor q,p\\land q\\\}\.Indeed,p∧qp\\land qentailspp, entailsqq, entailsp∨qp\\lor q, and entails itself\. In this case, the relevant literals areppandqq, and the displayed set lists the distinct CNF consequences over those literals\.

Given two argumentsa=\(Φ,ϕ\)a=\(\\Phi,\\phi\)andb=\(Ψ,ψ\)b=\(\\Psi,\\psi\), define their premise similarity, or syntactic similarity, by the Jaccard overlap of their premise equivalence classes:

ssyn​\(Φ,Ψ\)=\|\[Φ\]≡∩\[Ψ\]≡\|\|\[Φ\]≡∪\[Ψ\]≡\|,s\_\{\\mathrm\{syn\}\}\(\\Phi,\\Psi\)=\\frac\{\|\[\\Phi\]\_\{\\equiv\}\\cap\[\\Psi\]\_\{\\equiv\}\|\}\{\|\[\\Phi\]\_\{\\equiv\}\\cup\[\\Psi\]\_\{\\equiv\}\|\},with the convention that the ratio is11when both sets are empty\.

Define the conclusion similarity, or semantic similarity, by the Jaccard overlap of the sets of logical consequences in CNF form:

ssem​\(ϕ,ψ\)=\|CNℱ⁡\(ϕ\)∩CNℱ⁡\(ψ\)\|\|CNℱ⁡\(ϕ\)∪CNℱ⁡\(ψ\)\|,s\_\{\\mathrm\{sem\}\}\(\\phi,\\psi\)=\\frac\{\|\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\cap\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\psi\)\|\}\{\|\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\cup\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\psi\)\|\},again with the convention that the ratio is11when both sets are empty\.

The syntactic–semantic Jaccard similarity ofaaandbbis the convex combination

simσ⁡\(a,b\)=σ​ssyn​\(Φ,Ψ\)\+\(1−σ\)​ssem​\(ϕ,ψ\),0<σ<1\.\\operatorname\{sim\}^\{\\sigma\}\(a,b\)=\\sigma s\_\{\\mathrm\{syn\}\}\(\\Phi,\\Psi\)\+\(1\-\\sigma\)s\_\{\\mathrm\{sem\}\}\(\\phi,\\psi\),\\qquad 0<\\sigma<1\.Accordingly, the similarity measure has a direct interpretation: two arguments are similar when they rely on logically equivalent premises and when their conclusions have logical consequences in common\. The parameterσ\\sigmacontrols the relative weight placed on shared premises versus shared conclusions\.

[5](https://arxiv.org/html/2608.15325#bib.bib14)shows that the syntactic–semantic Jaccard similarity satisfies two groups of desirable properties for comparing logical arguments\. The first group concerns general consistency: the measure is maximized on identical arguments, symmetric in its arguments, compatible with a triangle\-inequality principle, and invariant under substitution of maximally similar arguments\. The second group concerns sensitivity to argumentative content: the measure is zero when arguments share no relevant content, positive when they share some content, and strictly increases with greater overlap in premises or in the logical consequences of their conclusions\. Appendix[A\.5](https://arxiv.org/html/2608.15325#A1.SS5)states the full list of properties formally\.

The following result is a direct consequence of[1](https://arxiv.org/html/2608.15325#bib.bib13)\.

###### Theorem 1\(Logical faithfulness\)\.

For any0<σ<10<\\sigma<1and any argumentsa,b∈Arg⁡\(ℒ\)a,b\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\),

simσ⁡\(a,b\)=1if and only ifa≈b\.\\operatorname\{sim\}^\{\\sigma\}\(a,b\)=1\\quad\\text\{if and only if\}\\quad a\\approx b\.

Theorem[1](https://arxiv.org/html/2608.15325#Thmtheorem1)shows that the similarity measure is faithful to the underlying logical representation of arguments\. Two arguments have maximal similarity if and only if they are logically equivalent: they have the same premises, up to logical equivalence, and logically equivalent conclusions\. Thus,simσ\\operatorname\{sim\}^\{\\sigma\}treats argument equivalence as the condition for maximal similarity, while still assigning graded values to partial overlap in premises and conclusion consequences\.

This property distinguishes the proposed similarity from generic embedding similarities, such as cosine similarity between text embeddings\. Indeed, similarity is determined by logical equivalence, not by word choice, topic overlap, or stylistic resemblance\. The measure therefore attends to the logical structure of the argument rather than to its wording\.

### 2\.3From similarity to logical embeddings

We now show that the syntactic–semantic Jaccard similarity induces an embedding of arguments into a Hilbert space\. The key step is the following additional property of the similarity measure\.

###### Theorem 2\(Kernel property\)\.

For any0<σ<10<\\sigma<1,simσ\\operatorname\{sim\}^\{\\sigma\}is a positive semidefinite kernel onArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\.

By the Moore–Aronszajn theorem\([3](https://arxiv.org/html/2608.15325#bib.bib4)\), the similarity measuresimσ\\operatorname\{sim\}^\{\\sigma\}determines a reproducing kernel Hilbert spaceℋσ\\mathcal\{H\}\_\{\\sigma\}and a canonical feature map

Θσ:Arg⁡\(ℒ\)→ℋσ\\Theta\_\{\\sigma\}:\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\to\\mathcal\{H\}\_\{\\sigma\}such that, for alla,b∈Arg⁡\(ℒ\)a,b\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\),

⟨Θσ​\(a\),Θσ​\(b\)⟩ℋσ=simσ⁡\(a,b\)\.\\left\\langle\\Theta\_\{\\sigma\}\(a\),\\Theta\_\{\\sigma\}\(b\)\\right\\rangle\_\{\\mathcal\{H\}\_\{\\sigma\}\}=\\operatorname\{sim\}^\{\\sigma\}\(a,b\)\.We callΘσ​\(a\)\\Theta\_\{\\sigma\}\(a\)the logical embedding of the argumentaa\. Appendix[A\.6](https://arxiv.org/html/2608.15325#A1.SS6)gives a brief review of reproducing kernel Hilbert spaces\.

This construction gives the embedding a direct interpretation\. The vectorΘσ​\(a\)\\Theta\_\{\\sigma\}\(a\)encodes the logical content ofaa: two arguments receive the same embedding exactly when they are equivalent under the premise–conclusion representation\. This encoding is relational\. The position ofaain the Hilbert space is determined by howaarelates to every other possible argument through the similarity functionsimσ⁡\(a,⋅\)\\operatorname\{sim\}^\{\\sigma\}\(a,\\cdot\)\. Thus, the geometry of the embedding is governed by the premise\-conclusion structure of arguments\.

Moreover, the RKHS embedding admits a finite\-sample approximation\. Given a corpus of argumentsa1,…,ana\_\{1\},\\ldots,a\_\{n\}, the formal construction defines its Gram matrix, that is, the matrix of pairwise similarities

Ki​jσ=simσ⁡\(ai,aj\)=⟨Θσ​\(ai\),Θσ​\(aj\)⟩ℋσ\.K^\{\\sigma\}\_\{ij\}=\\operatorname\{sim\}^\{\\sigma\}\(a\_\{i\},a\_\{j\}\)=\\left\\langle\\Theta\_\{\\sigma\}\(a\_\{i\}\),\\Theta\_\{\\sigma\}\(a\_\{j\}\)\\right\\rangle\_\{\\mathcal\{H\}\_\{\\sigma\}\}\.In practice, we estimate this matrix from natural\-language text by approximating the premise–conclusion structure of each argument and the corresponding syntactic–semantic similarity\. LetK^σ\\widehat\{K\}^\{\\sigma\}denote the resulting empirical kernel matrix\. We then apply kernel PCA toK^σ\\widehat\{K\}^\{\\sigma\}to obtain finite\-dimensional coordinates for the observed arguments\.222Appendix[A\.7](https://arxiv.org/html/2608.15325#A1.SS7)reviews the theoretical basis of this procedure\.These coordinates provide a low\-dimensional approximation to the ideal RKHS embeddings on the observed corpus\.

Finally, the logical embedding preserves all information relevant to the formal argument representation, up to logical equivalence\. In other words, arguments that are not equivalent under≈\\approxreceive distinct embeddings, whereas equivalent arguments receive the same embedding\. The next theorem formalizes this property\.

###### Theorem 3\(No loss of logical information\)\.

For any0<σ<10<\\sigma<1, letℋσ\\mathcal\{H\}\_\{\\sigma\}be the RKHS induced bysimσ\\operatorname\{sim\}^\{\\sigma\}, and let

Θσ:Arg⁡\(ℒ\)→ℋσ,Θσ​\(a\)=simσ⁡\(⋅,a\)\\Theta\_\{\\sigma\}:\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\to\\mathcal\{H\}\_\{\\sigma\},\\qquad\\Theta\_\{\\sigma\}\(a\)=\\operatorname\{sim\}^\{\\sigma\}\(\\cdot,a\)be the canonical feature map\. Then, for anya,b∈Arg⁡\(ℒ\)a,b\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\),

Θσ​\(a\)=Θσ​\(b\)if and only ifa≈b\.\\Theta\_\{\\sigma\}\(a\)=\\Theta\_\{\\sigma\}\(b\)\\quad\\text\{if and only if\}\\quad a\\approx b\.Consequently,Θσ\\Theta\_\{\\sigma\}induces an injective map

Θ¯σ:Arg\(ℒ\)/≈→ℋσ,Θ¯σ\(\[a\]≈\)=Θσ\(a\)\.\\overline\{\\Theta\}\_\{\\sigma\}:\\operatorname\{Arg\}\(\\mathcal\{L\}\)/\{\\approx\}\\to\\mathcal\{H\}\_\{\\sigma\},\\qquad\\overline\{\\Theta\}\_\{\\sigma\}\(\[a\]\_\{\\approx\}\)=\\Theta\_\{\\sigma\}\(a\)\.

## 3NLP implementation of logical embeddings

While logical embeddings and their supporting similarity measures have desirable theoretical properties, their utility is limited without implementation to real data\. This section discusses how to compute the logic\-based similarity measures from Section[2\.2](https://arxiv.org/html/2608.15325#S2.SS2)using existing NLP and Machine Learning tools\.

In essence, we propose a three\-step procedure to computing Syntactic\-Semantic Jaccard similarity measures:

1. 1\.Training an entailment model\.
2. 2\.Using a large language model \(LLM\) to decompose an argument into its constituent premises and conclusion\.
3. 3\.Using an LLM to map a conclusion into its CNF\.

We further detail each step in detail in the upcoming sections\.

### 3\.1Training an entailment model

The key to establishing logical overlap is determining whether two propositions,ppandqq, are logically equivalent\. This task can be further decomposed into establishing thatppentailsqqand thatqqentailspp\. Thus, a model that can determine whether the first proposition in an ordered pair logically implies the second can also predict logical equivalence\.

Establishing logical equivalence is a well\-defined NLP task known as Recognizing Textual Entailment \(RTE\), first introduced by[12](https://arxiv.org/html/2608.15325#bib.bib21)\. This task has received significant attention and benefits from large training datasets such as the Stanford Natural Language Inference \(SNLI\) dataset, which contains 570,000 human\-written, manually labeled English sentence pairs\. Moreover, existing Machine Learning tools have proven highly effective at this task, achieving accuracies exceeding0\.90\.9in several studies\([28](https://arxiv.org/html/2608.15325#bib.bib22);[17](https://arxiv.org/html/2608.15325#bib.bib23);[29](https://arxiv.org/html/2608.15325#bib.bib24)\)\. Consequently, logical entailment classification can be satisfactorily accomplished via zero\-shot learning or fine\-tuning of these models\.

### 3\.2Argument decomposition

The rest of the paper treats this construction as an ideal target\. In practice, natural\-language arguments do not come with explicit premise sets, formal conclusions, or canonical consequence sets\. Our empirical task is therefore to approximate the ideal logical embedding from text and to evaluate whether the resulting representations are useful for downstream argument\-related tasks\.

Premise and conclusion extraction is a well\-studied task in argumentation NLP settings\([13](https://arxiv.org/html/2608.15325#bib.bib11);[16](https://arxiv.org/html/2608.15325#bib.bib10);[24](https://arxiv.org/html/2608.15325#bib.bib8)\)\. However, existing approaches often rely on sophisticated models that can now be superseded by large language models \(LLMs\)\. For example, we have found good results using a chained prompt as follows:

Phase 1:You are a skilled logician\. Given a text, extract its conclusion\. A conclusion is a logical result of the relationship between the premises\. Conclusions serve as the thesis of the argument\.Inputs:TextPhase 2:You are a skilled logician\. Given a text and its conclusion, extract its premises\.Inputs:Text \+ Conclusion from Phase 1\.

### 3\.3Conclusion CNF formulation

The last step consists of mapping a conclusion into its CNF representation\. This is the hardest subtask, but can also be achieved with high precision by modern LLMs\. In our applications we have used the following prompt with success\.

Convert the following text into its Conjunctive Normal Form \(CNF\)\. Identify the individual propositions and combine them into disjunctions of complete and meaningful propositions\. Ensure each proposition contains at least a noun and a verb and is unique\.Inputs:Conclusion

### 3\.4Putting everything together

Given two arguments,a1a\_\{1\}anda2a\_\{2\}, we can now compute their logic\-based similarity using the previously explored tools\. The process unfolds as follows:

- •Decomposea1a\_\{1\}anda2a\_\{2\}into their constituent premises and conclusions,\(Φ1,ϕ1\)\(\\Phi\_\{1\},\\phi\_\{1\}\)and\(Φ2,ϕ2\)\(\\Phi\_\{2\},\\phi\_\{2\}\)\.
- •Find the CNF representation ofϕ1\\phi\_\{1\}andϕ2\\phi\_\{2\}, and calculateCNℱ​\(ϕ1\)\\text\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{1\}\)andCNℱ​\(ϕ2\)\\text\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{2\}\)\.
- •Using the entailment model, calculatessyn​\(Φ1,Φ2\)s\_\{\\text\{syn\}\}\(\\Phi\_\{1\},\\Phi\_\{2\}\)andssem​\(ϕ1,ϕ2\)s\_\{\\text\{sem\}\}\(\\phi\_\{1\},\\phi\_\{2\}\)by assessing the overlap, modulo logical equivalence, betweenΦ1\\Phi\_\{1\}andΦ2\\Phi\_\{2\}, andCNℱ​\(ϕ1\)\\text\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{1\}\)andCNℱ​\(ϕ2\)\\text\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{2\}\)\.
- •Fix aσ∈\[0,1\]\\sigma\\in\[0,1\]and aggregate intosimσ​\(a1,a2\)\\text\{sim\}^\{\\sigma\}\(a\_\{1\},a\_\{2\}\)\.

Logical embeddings can then be produced and exploited via the reproducibility property of the similarity measure\. In the Appendix, Figure[4](https://arxiv.org/html/2608.15325#A2.F4)we provide an example of how similarity is constructed for a pair of arguments\.

### 3\.5Performance evaluation

We evaluate the performance of our embeddings on the standard IBM\-ArgQ\-6\.3kArgs dataset\([26](https://arxiv.org/html/2608.15325#bib.bib7)\)\. This dataset contains arguments in favor or against several topics\. To economize on implementation costs we focus on the topics of doping, vaccins, and cryptocurrencies \. For each embedding\-classifier pair, we use a fixed 70/30 train\-test split and report binary F1 on the test set\. The neural\-network classifier uses five\-fold cross\-validation within the training set for hyperparameter selection, while the other classifiers are fit directly on the training split\. We report test set F1\-scores for different embedding methods as rows and different models as columns\. We compare our logical embeddings in the first row against several pretrained alternatives such as GloVe, BERT, SBERT and OpenAI text\-embedding\-3\-small embeddings\. We also evaluate concatenated representations that combine each pretrained embedding with the logical embedding\. Classifiers include L1\-penalized logistic regression, Lasso and Ridge linear models with thresholded predictions, a one\-hidden\-layer feed\-forward neural network, and a random forest\. The neural network has a ReLU hidden layer and sigmoid output; its learning rate, weight decay, and hidden dimension are selected by five\-fold cross\-validation over the training set\. The random forest uses 200 trees\. Table[1](https://arxiv.org/html/2608.15325#S4.T1)shows that the Logical Embeddings outperform every model when compared side by side\. In combination, Open AI text\-embedding\-3\-small and logical embeddings outperform all other combinations\.

## 4Conclusion

In this paper we introduced and testedlogical embeddings\. We show that these are optimal embeddings and provide implementation details\. Implementation code is available athttps://github\.com/lheldring/logical\_embeddings\.

Table 1:F1 scores 100\-dimensional logical embeddingsNotes: This Table presents the main F1 scores of the paper\. All classifiers are estimated on a 70/30 train\-test split of 1137 arguments\. Logistic, Lasso and Ridge are linear models with L1, L1 and L2 penalties respectively\. Random Forest uses 200 trees\. The Neural Network is a single hidden layer network whose hyperparameters are selected via 5\-fold validation grid search with early stopping\. Logical emb\. refers to the clause\-level logical embedding using the top 100 eigenvectors of the pairwise entailment score matrix\. GloVe uses 100\-dimensional averaged word vectors\. BERT and RoBERTa use 768\-dimensional CLS token embeddings\. SBERT uses 384\-dimensional sentence embeddings\. GPT uses OpenAI text\-embedding\-3\-small \(1536 dimensions\)\. All embeddings are standardised before estimation\. Bold indicates the highest F1 per column\.
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## Appendix AAppendix / supplemental material

### A\.1Logical preliminaries

This appendix gives the formal logical background used in Section[2](https://arxiv.org/html/2608.15325#S2)\. The main text introduces only the notation needed to define arguments, logical equivalence, and the syntactic–semantic similarity\. Here we state the underlying propositional language, consequence relation, CNF convention, and canonical consequence sets more explicitly\. The presentation follows[2](https://arxiv.org/html/2608.15325#bib.bib18),[5](https://arxiv.org/html/2608.15325#bib.bib14), and[23](https://arxiv.org/html/2608.15325#bib.bib15)\.

#### A\.1\.1Sentential logic

Sentential logic starts from atomic propositions and builds more complex statements from them using Boolean connectives\.

###### Definition 3\(Atoms and sentential language\)\.

An atomic proposition, or atom, is a statement that can be assigned a truth value\. A sentential language is a finite set

L=\{p,q,r,s,…\}L=\\\{p,q,r,s,\\ldots\\\}of atoms\.

Atoms are the basic units of the language\. For example, an atomppmay stand for a truth\-valued statement such as “the policy reduces debt\.” The languageLLspecifies which atoms are available in the application\.

###### Definition 4\(Propositional connectives\)\.

The propositional connectives are

¬,∧,∨,→,↔,\\neg,\\quad\\land,\\quad\\lor,\\quad\\rightarrow,\\quad\\leftrightarrow,denoting negation, conjunction, disjunction, implication, and biconditionality, respectively\.

Connectives combine atoms into more complex truth\-valued expressions\. These expressions are called formulas\.

###### Definition 5\(Formulas\)\.

The set of well\-formed formulas generated byLL, denotedℒ\\mathcal\{L\}, is the smallest set satisfying the following conditions:

1. 1\.ifp∈Lp\\in L, thenp∈ℒp\\in\\mathcal\{L\};
2. 2\.the truth constants⊤\\topand⊥\\botbelong toℒ\\mathcal\{L\};
3. 3\.ifϕ∈ℒ\\phi\\in\\mathcal\{L\}, then¬ϕ∈ℒ\\neg\\phi\\in\\mathcal\{L\};
4. 4\.ifϕ,ψ∈ℒ\\phi,\\psi\\in\\mathcal\{L\}, then \(ϕ∧ψ\),\(ϕ∨ψ\),\(ϕ→ψ\),\(ϕ↔ψ\)\(\\phi\\land\\psi\),\\quad\(\\phi\\lor\\psi\),\\quad\(\\phi\\rightarrow\\psi\),\\quad\(\\phi\\leftrightarrow\\psi\)are inℒ\\mathcal\{L\}\.

Thus, atoms are formulas, and formulas can be combined recursively to form more complex formulas\. For instance, ifp,q∈Lp,q\\in L, thenp∧qp\\land q,p→qp\\rightarrow q, and¬p∨q\\neg p\\lor qare formulas\. A literal is an atom or the negation of an atom\.

Next, we define truth assignments\. A truth assignment determines which formulas are true and which are false, subject to the usual truth tables for the Boolean connectives\. Fix the set of truth values\{F,T\}\\\{F,T\\\}, whereFFdenotes falsity andTTdenotes truth\.

###### Definition 6\(Valuation\)\.

A valuation is a map

v:ℒ→\{F,T\}v:\\mathcal\{L\}\\to\\\{F,T\\\}that satisfies the usual truth\-functional rules\. In particular,

v⁡\(⊤\)=Tandv⁡\(⊥\)=F,v\(\\top\)=T\\qquad\\text\{and\}\\qquad v\(\\bot\)=F,v⁡\(¬ϕ\)=Tif and only ifv⁡\(ϕ\)=F,v\(\\neg\\phi\)=T\\quad\\text\{if and only if\}\\quad v\(\\phi\)=F,v⁡\(ϕ∧ψ\)=Tif and only ifv⁡\(ϕ\)=T​and​v​\(ψ\)=T,v\(\\phi\\land\\psi\)=T\\quad\\text\{if and only if\}\\quad v\(\\phi\)=T\\text\{ and \}v\(\\psi\)=T,v⁡\(ϕ∨ψ\)=Tif and only ifv⁡\(ϕ\)=T​or​v​\(ψ\)=T,v\(\\phi\\lor\\psi\)=T\\quad\\text\{if and only if\}\\quad v\(\\phi\)=T\\text\{ or \}v\(\\psi\)=T,v⁡\(ϕ→ψ\)=Tif and only ifv⁡\(ϕ\)=F​or​v​\(ψ\)=T,v\(\\phi\\rightarrow\\psi\)=T\\quad\\text\{if and only if\}\\quad v\(\\phi\)=F\\text\{ or \}v\(\\psi\)=T,and

v⁡\(ϕ↔ψ\)=Tif and only ifv⁡\(ϕ\)=v⁡\(ψ\)\.v\(\\phi\\leftrightarrow\\psi\)=T\\quad\\text\{if and only if\}\\quad v\(\\phi\)=v\(\\psi\)\.

The preceding definitions specify the syntax of the language and the truth conditions for its formulas\. We now package these objects into the propositional logic used throughout the paper\.

###### Definition 7\(Sentential logic\)\.

A sentential, or propositional, logic is a pair\(ℒ,⊢\)\(\\mathcal\{L\},\\vdash\), whereℒ\\mathcal\{L\}is the set of well\-formed formulas generated by a sentential languageLL, and⊢\\vdashis the semantic consequence relation\. ForΓ⊆ℒ\\Gamma\\subseteq\\mathcal\{L\}andφ∈ℒ\\varphi\\in\\mathcal\{L\}, we write

if and only if every valuation that assignsTTto all formulas inΓ\\Gammaalso assignsTTtoφ\\varphi\.

Fixing a logic\(ℒ,⊢\)\(\\mathcal\{L\},\\vdash\)lets us define when two formulas have the same logical content\.

###### Definition 8\(Logical equivalence of formulas\)\.

Two formulasϕ,ψ∈ℒ\\phi,\\psi\\in\\mathcal\{L\}are logically equivalent if

ϕ⊢ψandψ⊢ϕ,\\phi\\vdash\\psi\\qquad\\text\{and\}\\qquad\\psi\\vdash\\phi,whereϕ⊢ψ\\phi\\vdash\\psiabbreviates\{ϕ\}⊢ψ\\\{\\phi\\\}\\vdash\\psi\. In this case, we writeϕ≡ψ\\phi\\equiv\\psi\.

Logical equivalence means that two formulas entail each other\. Equivalently, they have the same truth conditions\.

Moroever, logical equivalence is an equivalence relation: it is reflexive, symmetric, and transitive\. We can therefore group formulas into equivalence classes\.

###### Definition 9\(Equivalence class of a formula\)\.

For a formulaϕ∈ℒ\\phi\\in\\mathcal\{L\}, its equivalence class under logical equivalence is

\[ϕ\]≡=\{ψ∈ℒ:ψ≡ϕ\}\.\[\\phi\]\_\{\\equiv\}=\\\{\\psi\\in\\mathcal\{L\}:\\psi\\equiv\\phi\\\}\.

Equivalence classes let us identify formulas that differ syntactically but have the same logical content\. To choose representatives of these classes, we use conjunctive normal form\.

###### Definition 10\(Conjunctive normal form\)\.

A literal is an atomp∈Lp\\in Lor its negation¬p\\neg p\. A clause is a finite disjunction of literals\. A formulaϕ∈ℒ\\phi\\in\\mathcal\{L\}is in conjunctive normal form \(CNF\) if it can be written as

ϕ=⋀i=1mci,\\phi=\\bigwedge\_\{i=1\}^\{m\}c\_\{i\},where each clausecic\_\{i\}has the form

ci=⋁j=1niℓi​j,c\_\{i\}=\\bigvee\_\{j=1\}^\{n\_\{i\}\}\\ell\_\{ij\},and eachℓi​j\\ell\_\{ij\}is a literal\. Equivalently, a CNF formula is a conjunction of clauses, each of which is a disjunction of literals\.

Every propositional formula is logically equivalent to some CNF formula\[[19](https://arxiv.org/html/2608.15325#bib.bib17)\]\. For example,

p→\(q∧r\)≡\(¬p∨q\)∧\(¬p∨r\)\.p\\to\(q\\land r\)\\equiv\(\\neg p\\lor q\)\\land\(\\neg p\\lor r\)\.
Following[2](https://arxiv.org/html/2608.15325#bib.bib18), we use CNF representatives to avoid redundant reformulations\. The goal is to keep one representative formula for each relevant equivalence class while preserving the literals of the original formula\.

###### Definition 11\(Finite CNF language\)\.

Letℱ⊆ℒ\\mathcal\{F\}\\subseteq\\mathcal\{L\}\. We say thatℱ\\mathcal\{F\}is a finite CNF language forℒ\\mathcal\{L\}if, for everyϕ∈ℒ\\phi\\in\\mathcal\{L\}, there exists a uniqueψ∈ℱ\\psi\\in\\mathcal\{F\}such that

1. 1\.ψ∈\[ϕ\]≡\\psi\\in\[\\phi\]\_\{\\equiv\};
2. 2\.Lit⁡\(ψ\)=Lit⁡\(ϕ\)\\operatorname\{Lit\}\(\\psi\)=\\operatorname\{Lit\}\(\\phi\);
3. 3\.ψ\\psiis in CNF\.

Thus,ℱ\\mathcal\{F\}contains a unique CNF representative for each formula, up to logical equivalence\. We writeCNFℱ⁡\(ϕ\)\\operatorname\{CNF\}\_\{\\mathcal\{F\}\}\(\\phi\)for the unique element ofℱ\\mathcal\{F\}that is logically equivalent toϕ\\phiand has the same literals asϕ\\phi\.

### A\.2Logical arguments

We now define the logical arguments used in the main text, following[4](https://arxiv.org/html/2608.15325#bib.bib16)and[1](https://arxiv.org/html/2608.15325#bib.bib13)\. A logical argument consists of a set of premises and a conclusion that follows from those premises\. Before giving the definition, we record two auxiliary notions for sets of formulas\.

###### Definition 12\(Consistency\)\.

A set of formulasΦ⊆ℒ\\Phi\\subseteq\\mathcal\{L\}is consistent if

It is inconsistent otherwise\.

Equivalently,Φ\\Phiis consistent if there exists a valuation under which all formulas inΦ\\Phiare true\.

We also need to compare premise sets up to logical equivalence\.

###### Definition 13\(Logical equivalence of sets of formulas\)\.

ForΦ,Ψ⊆ℒ\\Phi,\\Psi\\subseteq\\mathcal\{L\}, write

\[Φ\]≡=\{\[ϕ\]≡:ϕ∈Φ\}\.\[\\Phi\]\_\{\\equiv\}=\\\{\[\\phi\]\_\{\\equiv\}:\\phi\\in\\Phi\\\}\.We say thatΦ\\PhiandΨ\\Psiare logically equivalent, writtenΦ≅Ψ\\Phi\\cong\\Psi, if

\[Φ\]≡=\[Ψ\]≡\.\[\\Phi\]\_\{\\equiv\}=\[\\Psi\]\_\{\\equiv\}\.

We now define the logical arguments used in the main text, following[4](https://arxiv.org/html/2608.15325#bib.bib16)and[1](https://arxiv.org/html/2608.15325#bib.bib13)\. A logical argument consists of a finite set of premises and a conclusion that follows from those premises\. The premise set is required to be consistent and non\-redundant\.

###### Definition 14\(Logical argument\)\.

An argument in the logic\(ℒ,⊢\)\(\\mathcal\{L\},\\vdash\)is a pair

whereΦ⊆ℒ\\Phi\\subseteq\\mathcal\{L\}is finite,ϕ∈ℒ\\phi\\in\\mathcal\{L\}, and:

1. 1\.Φ\\Phiis consistent;
2. 2\.Φ⊢ϕ\\Phi\\vdash\\phi;
3. 3\.Φ\\Phiis non\-redundant, meaning that there is noΦ′⊊Φ\\Phi^\{\\prime\}\\subsetneq\\Phisuch thatΦ′⊢ϕ\\Phi^\{\\prime\}\\vdash\\phi\.

The setΦ\\Phiis the set of premises ofaa, andϕ\\phiis its conclusion\. We write

Prem⁡\(a\)=ΦandConc⁡\(a\)=ϕ\.\\operatorname\{Prem\}\(a\)=\\Phi\\qquad\\text\{and\}\\qquad\\operatorname\{Conc\}\(a\)=\\phi\.The set of all arguments overℒ\\mathcal\{L\}is denotedArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\.

This representation lets us define when two arguments have the same logical content\. The definition compares premise sets up to logical equivalence and compares conclusions by logical equivalence\.

###### Definition 15\(Argument equivalence\)\.

Two argumentsa,b∈Arg⁡\(ℒ\)a,b\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\)are equivalent, writtena≈ba\\approx b, if

Prem⁡\(a\)≅Prem⁡\(b\)andConc⁡\(a\)≡Conc⁡\(b\)\.\\operatorname\{Prem\}\(a\)\\cong\\operatorname\{Prem\}\(b\)\\qquad\\text\{and\}\\qquad\\operatorname\{Conc\}\(a\)\\equiv\\operatorname\{Conc\}\(b\)\.

### A\.3Measuring similarity

The logical representation of arguments makes it possible to compare their constituent parts separately\. This is the basis of the similarity measures developed by[1](https://arxiv.org/html/2608.15325#bib.bib13)and[5](https://arxiv.org/html/2608.15325#bib.bib14)\. These measures compare arguments by comparing their premises and their conclusions, and they satisfy the rationality properties stated in Appendix[A\.5](https://arxiv.org/html/2608.15325#A1.SS5)\.

We first need an operator for comparing sets of formulas modulo logical equivalence\.

###### Definition 16\(Common formulas operator\)\.

For sets of formulasΦ,Ψ⊆ℒ\\Phi,\\Psi\\subseteq\\mathcal\{L\}, define

Co⁡\(Φ,Ψ\)=\{ϕ∈Φ:there exists​ψ∈Ψ​such that​ϕ≡ψ\}\.\\operatorname\{Co\}\(\\Phi,\\Psi\)=\\\{\\phi\\in\\Phi:\\text\{ there exists \}\\psi\\in\\Psi\\text\{ such that \}\\phi\\equiv\\psi\\\}\.

The operatorCo⁡\(Φ,Ψ\)\\operatorname\{Co\}\(\\Phi,\\Psi\)selects the formulas inΦ\\Phithat also occur inΨ\\Psi, up to logical equivalence\. Thus,Φ≅Ψ\\Phi\\cong\\Psiif and only if

Co⁡\(Φ,Ψ\)=ΦandCo⁡\(Ψ,Φ\)=Ψ\.\\operatorname\{Co\}\(\\Phi,\\Psi\)=\\Phi\\qquad\\text\{and\}\\qquad\\operatorname\{Co\}\(\\Psi,\\Phi\)=\\Psi\.
We next define the consequence sets used to compare conclusions\.

###### Definition 17\(Logical consequences of a formula\)\.

For a formulaϕ∈ℒ\\phi\\in\\mathcal\{L\}, define

CN\(ϕ\)=\{ψ∈ℒ:ϕ⊢ψ\}\.\\operatorname\{CN\}\(\\phi\)=\\\{\\psi\\in\\mathcal\{L\}:\\phi\\vdash\\psi\\\}\.This is the set of all logical consequences ofϕ\\phi\.

The setCN⁡\(ϕ\)\\operatorname\{CN\}\(\\phi\)is typically infinite\. For example,p⊢pp\\vdash p,p⊢p∧pp\\vdash p\\land p,p⊢\(p∧p\)∧pp\\vdash\(p\\land p\)\\land p, and so on\. Moreover, many such consequences are redundant up to logical equivalence\. To obtain a finite set for comparison, we restrict attention to CNF representatives that use only literals on whichϕ\\phidepends\.

LetL±L^\{\\pm\}denote the set of literals, that is, atoms and their negations\. For a formulaϕ\\phi, letLit⁡\(ϕ\)\\operatorname\{Lit\}\(\\phi\)denote the set of literals occurring inϕ\\phiafter rewritingϕ\\phiin negation normal form\.

A formulaϕ\\phiis independent of a literalℓ∈L±\\ell\\in L^\{\\pm\}if there exists a formulaχ∈ℒ\\chi\\in\\mathcal\{L\}such thatχ≡ϕ\\chi\\equiv\\phiandℓ∉Lit⁡\(χ\)\\ell\\notin\\operatorname\{Lit\}\(\\chi\)\. Otherwise,ϕ\\phidepends onℓ\\ell\. We writeDepLit⁡\(ϕ\)\\operatorname\{DepLit\}\(\\phi\)for the set of literals on whichϕ\\phidepends\.

Fix a finite CNF representative languageℱ⊆ℒ\\mathcal\{F\}\\subseteq\\mathcal\{L\}\. That is,ℱ\\mathcal\{F\}contains one CNF representative from each logical equivalence class, and each representative is written with no literals beyond those on which it depends\.

###### Definition 18\(Finite CNF consequence set\)\.

For a formulaϕ∈ℒ\\phi\\in\\mathcal\{L\}, define

CNℱ\(ϕ\)=\{ψ∈ℱ:ϕ⊢ψandLit\(ψ\)⊆DepLit\(ϕ\)\}\.\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)=\\\{\\psi\\in\\mathcal\{F\}:\\phi\\vdash\\psi\\text\{ and \}\\operatorname\{Lit\}\(\\psi\)\\subseteq\\operatorname\{DepLit\}\(\\phi\)\\\}\.

Thus,CNℱ⁡\(ϕ\)\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)contains the CNF representatives of the logical consequences ofϕ\\phithat use only literals on whichϕ\\phidepends\. The restriction toℱ\\mathcal\{F\}removes equivalent reformulations; the restriction toDepLit⁡\(ϕ\)\\operatorname\{DepLit\}\(\\phi\)removes consequences involving irrelevant literals\. Since the atom set is finite,CNℱ⁡\(ϕ\)\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)is finite\.

For example,

CNℱ⁡\(p\)=\{p\},CNℱ⁡\(p∨q\)=\{p∨q\},\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(p\)=\\\{p\\\},\\qquad\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(p\\lor q\)=\\\{p\\lor q\\\},and

CNℱ⁡\(p∧q\)=\{p,q,p∨q,p∧q\}\.\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(p\\land q\)=\\\{p,q,p\\lor q,p\\land q\\\}\.The last equality holds becausep∧qp\\land qentailspp, entailsqq, entailsp∨qp\\lor q, and entails itself\. These are the distinct representative consequences over the literals on whichp∧qp\\land qdepends\.

Having introduced logical arguments, we can now define similarity measures over them\. The logical representation lets us compare arguments along two dimensions: their premises and their conclusions\.

###### Definition 19\(Similarity measure\)\.

A similarity measure on a sentential logic\(ℒ,⊢\)\(\\mathcal\{L\},\\vdash\)is a map

𝒮:Arg⁡\(ℒ\)×Arg⁡\(ℒ\)→\[0,1\]\.\\mathcal\{S\}:\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\times\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\to\[0,1\]\.

[5](https://arxiv.org/html/2608.15325#bib.bib14)studies several similarity measures for logical arguments and their properties\. The central idea is that arguments can be compared syntactically, by the overlap in their premises, and semantically, by the overlap in the logical consequences of their conclusions\. Measures based on only one of these dimensions fail to satisfy all the desired properties\. The syntactic–semantic Jaccard measure combines both dimensions\.

###### Definition 20\(Syntactic–semantic Jaccard similarity\)\.

Leta=\(Φ,ϕ\)a=\(\\Phi,\\phi\)andb=\(Ψ,ψ\)b=\(\\Psi,\\psi\)be arguments inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\. For0<σ<10<\\sigma<1, define

simσ⁡\(a,b\)=σ​ssyn​\(Φ,Ψ\)\+\(1−σ\)​ssem​\(ϕ,ψ\),\\operatorname\{sim\}^\{\\sigma\}\(a,b\)=\\sigma s\_\{\\mathrm\{syn\}\}\(\\Phi,\\Psi\)\+\(1\-\\sigma\)s\_\{\\mathrm\{sem\}\}\(\\phi,\\psi\),where the premise similarity is

ssyn​\(Φ,Ψ\)=\{\|Co⁡\(Φ,Ψ\)\|\|Φ\|\+\|Ψ\|−\|Co⁡\(Φ,Ψ\)\|if​Φ≠∅​and​Ψ≠∅,1if​Φ=Ψ=∅,0otherwise,s\_\{\\mathrm\{syn\}\}\(\\Phi,\\Psi\)=\\begin\{cases\}\\dfrac\{\|\\operatorname\{Co\}\(\\Phi,\\Psi\)\|\}\{\|\\Phi\|\+\|\\Psi\|\-\|\\operatorname\{Co\}\(\\Phi,\\Psi\)\|\}&\\text\{if \}\\Phi\\neq\\emptyset\\text\{ and \}\\Psi\\neq\\emptyset,\\\\\[5\.69054pt\] 1&\\text\{if \}\\Phi=\\Psi=\\emptyset,\\\\ 0&\\text\{otherwise,\}\\end\{cases\}and the conclusion similarity is

ssem​\(ϕ,ψ\)=\|CNℱ⁡\(ϕ\)∩CNℱ⁡\(ψ\)\|\|CNℱ⁡\(ϕ\)∪CNℱ⁡\(ψ\)\|\.s\_\{\\mathrm\{sem\}\}\(\\phi,\\psi\)=\\frac\{\|\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\cap\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\psi\)\|\}\{\|\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\cup\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\psi\)\|\}\.

The measure is a convex combination of premise similarity and conclusion similarity\. The termssyns\_\{\\mathrm\{syn\}\}measures how many premises the two arguments share, modulo logical equivalence\. The termssems\_\{\\mathrm\{sem\}\}measures how many canonical consequences their conclusions share\. The parameterσ\\sigmacontrols the relative weight placed on premises and conclusions\.

Importantly, the value ofssems\_\{\\mathrm\{sem\}\}is invariant to the particular choice ofℱ\\mathcal\{F\}, as long asℱ\\mathcal\{F\}is a valid finite CNF representative language\. Different valid choices merely select different formulas from the same logical\-equivalence classes\. They therefore relabel the elements ofCNℱ⁡\(ϕ\)\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)without changing the cardinalities of the intersections and unions defining the quantity\.

### A\.4Example of computation

We illustrate the computation with two simple arguments\.

> Argument 1\.John is Susan’s brother\. Susan is shorter than all her brothers\. Susan is the only girl in the household\. Therefore, Susan is the shortest sibling\.

> Argument 2\.Susan is John’s sister\. Susan is shorter than all her brothers\. No boy in the household has blond hair\. Susan is the only girl in the household\. There is at least one sibling with blond hair\. Therefore, Susan is the shortest sibling and is the only one with blond hair\.

Leta1=\(Φ1,ϕ1\)a\_\{1\}=\(\\Phi\_\{1\},\\phi\_\{1\}\)anda2=\(Φ2,ϕ2\)a\_\{2\}=\(\\Phi\_\{2\},\\phi\_\{2\}\)denote the corresponding logical arguments\. We encode the premises as follows:

p1=“John is Susan’s brother,”p2=“Susan is shorter than all her brothers,”p3=“Susan is the only girl in the household,”Φ1=\{p1,p2,p3\}\.\\begin\{array\}\[\]\{lll\}p\_\{1\}&=&\\text\{\`\`John is Susan's brother,''\}\\\\ p\_\{2\}&=&\\text\{\`\`Susan is shorter than all her brothers,''\}\\\\ p\_\{3\}&=&\\text\{\`\`Susan is the only girl in the household,''\}\\end\{array\}\\qquad\\Phi\_\{1\}=\\\{p\_\{1\},p\_\{2\},p\_\{3\}\\\}\.
For the second argument, write

q1=“Susan is John’s sister,”q2=“Susan is shorter than all her brothers,”q3=“No boy in the household has blond hair,”q4=“Susan is the only girl in the household,”q5=“There is at least one sibling with blond hair,”Φ2=\{q1,q2,q3,q4,q5\}\.\\begin\{array\}\[\]\{lll\}q\_\{1\}&=&\\text\{\`\`Susan is John's sister,''\}\\\\ q\_\{2\}&=&\\text\{\`\`Susan is shorter than all her brothers,''\}\\\\ q\_\{3\}&=&\\text\{\`\`No boy in the household has blond hair,''\}\\\\ q\_\{4\}&=&\\text\{\`\`Susan is the only girl in the household,''\}\\\\ q\_\{5\}&=&\\text\{\`\`There is at least one sibling with blond hair,''\}\\end\{array\}\\qquad\\Phi\_\{2\}=\\\{q\_\{1\},q\_\{2\},q\_\{3\},q\_\{4\},q\_\{5\}\\\}\.
The shared premises, modulo logical equivalence, are

p1≡q1,p2=q2,p3=q4\.p\_\{1\}\\equiv q\_\{1\},\\qquad p\_\{2\}=q\_\{2\},\\qquad p\_\{3\}=q\_\{4\}\.
Thus,

Co⁡\(Φ1,Φ2\)=\{p1,p2,p3\}\.\\operatorname\{Co\}\(\\Phi\_\{1\},\\Phi\_\{2\}\)=\\\{p\_\{1\},p\_\{2\},p\_\{3\}\\\}\.
The syntactic similarity is therefore

ssyn​\(Φ1,Φ2\)=\|Co⁡\(Φ1,Φ2\)\|\|Φ1\|\+\|Φ2\|−\|Co⁡\(Φ1,Φ2\)\|=33\+5−3=35\.s\_\{\\mathrm\{syn\}\}\(\\Phi\_\{1\},\\Phi\_\{2\}\)=\\frac\{\|\\operatorname\{Co\}\(\\Phi\_\{1\},\\Phi\_\{2\}\)\|\}\{\|\\Phi\_\{1\}\|\+\|\\Phi\_\{2\}\|\-\|\\operatorname\{Co\}\(\\Phi\_\{1\},\\Phi\_\{2\}\)\|\}=\\frac\{3\}\{3\+5\-3\}=\\frac\{3\}\{5\}\.
Now let

r1=“Susan is the shortest sibling”andr2=“Susan is the only one with blond hair\.”r\_\{1\}=\\text\{\`\`Susan is the shortest sibling''\}\\qquad\\text\{and\}\\qquad r\_\{2\}=\\text\{\`\`Susan is the only one with blond hair\.''\}
Then

ϕ1=r1andϕ2=r1∧r2\.\\phi\_\{1\}=r\_\{1\}\\qquad\\text\{and\}\\qquad\\phi\_\{2\}=r\_\{1\}\\land r\_\{2\}\.
Under the fixed CNF representative convention,

CNℱ⁡\(ϕ1\)=\{r1\},\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{1\}\)=\\\{r\_\{1\}\\\},
whereas

CNℱ⁡\(ϕ2\)=\{r1,r2,r1∨r2,r1∧r2\}\.\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{2\}\)=\\\{r\_\{1\},r\_\{2\},r\_\{1\}\\lor r\_\{2\},r\_\{1\}\\land r\_\{2\}\\\}\.
The two conclusion consequence sets share onlyr1r\_\{1\}\. Hence,

ssem​\(ϕ1,ϕ2\)=\|CNℱ⁡\(ϕ1\)∩CNℱ⁡\(ϕ2\)\|\|CNℱ⁡\(ϕ1\)∪CNℱ⁡\(ϕ2\)\|=14\.s\_\{\\mathrm\{sem\}\}\(\\phi\_\{1\},\\phi\_\{2\}\)=\\frac\{\|\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{1\}\)\\cap\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{2\}\)\|\}\{\|\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{1\}\)\\cup\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\_\{2\}\)\|\}=\\frac\{1\}\{4\}\.
Combining the two terms gives, for0<σ<10<\\sigma<1,

simσ⁡\(a1,a2\)=σ⁡\(35\)\+\(1−σ\)​\(14\)=14\+720​σ\.\\operatorname\{sim\}^\{\\sigma\}\(a\_\{1\},a\_\{2\}\)=\\sigma\\left\(\\frac\{3\}\{5\}\\right\)\+\(1\-\\sigma\)\\left\(\\frac\{1\}\{4\}\\right\)=\\frac\{1\}\{4\}\+\\frac\{7\}\{20\}\\sigma\.

### A\.5Formal definitions of the properties satisfied by similarity measures

We state the relevant properties for an arbitrary similarity measureS:Arg⁡\(ℒ\)×Arg⁡\(ℒ\)→\[0,1\]S:\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\times\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\to\[0,1\]\.[5](https://arxiv.org/html/2608.15325#bib.bib14)proves that the syntactic–semantic Jaccard similarity satisfies each property below\.

###### Property 1\(Maximality\)\.

A similarity measureSSsatisfies maximality if, for everya∈Arg⁡\(ℒ\)a\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\),

Maximality says that each argument is maximally similar to itself\. Since similarity values lie in\[0,1\]\[0,1\], maximal similarity is represented by the value11\.

###### Property 2\(Symmetry\)\.

A similarity measureSSsatisfies symmetry if, for alla,b∈Arg⁡\(ℒ\)a,b\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\),

S⁡\(a,b\)=S⁡\(b,a\)\.S\(a,b\)=S\(b,a\)\.

Symmetry says that similarity does not depend on the order in which the two arguments are compared\.

###### Property 3\(Triangle inequality\)\.

A similarity measureSSsatisfies the triangle inequality if, for alla,b,c∈Arg⁡\(ℒ\)a,b,c\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\),

1\+S⁡\(a,c\)≥S⁡\(a,b\)\+S⁡\(b,c\)\.1\+S\(a,c\)\\geq S\(a,b\)\+S\(b,c\)\.

This condition is the triangle inequality written in similarity form\. Equivalently, the dissimilarityd⁡\(a,b\)=1−S⁡\(a,b\)d\(a,b\)=1\-S\(a,b\)satisfies

d⁡\(a,c\)≤d⁡\(a,b\)\+d⁡\(b,c\)\.d\(a,c\)\\leq d\(a,b\)\+d\(b,c\)\.Thus, ifaais close tobbandbbis close tocc, thenaacannot be arbitrarily far fromcc\.

###### Property 4\(Substitution\)\.

A similarity measureSSsatisfies substitution if, for alla,b,c∈Arg⁡\(ℒ\)a,b,c\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\),

S⁡\(a,b\)=1⟹S⁡\(a,c\)=S⁡\(b,c\)\.S\(a,b\)=1\\quad\\Longrightarrow\\quad S\(a,c\)=S\(b,c\)\.

Substitution says that maximally similar arguments are interchangeable for similarity comparisons with any third argument\.

For a formulaϕ\\phi, letAt⁡\(ϕ\)\\operatorname\{At\}\(\\phi\)denote the set of atoms occurring inϕ\\phi\. For a finite set of formulasΦ\\Phi, define

At⁡\(Φ\)=⋃ϕ∈ΦAt⁡\(ϕ\)\.\\operatorname\{At\}\(\\Phi\)=\\bigcup\_\{\\phi\\in\\Phi\}\\operatorname\{At\}\(\\phi\)\.
###### Property 5\(Minimality\)\.

A similarity measureSSsatisfies minimality if, for alla=\(Φ,ϕ\)a=\(\\Phi,\\phi\)andb=\(Ψ,ψ\)b=\(\\Psi,\\psi\)inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\), the following conditions implyS⁡\(a,b\)=0S\(a,b\)=0:

1. 1\.a≉ba\\not\\approx b;
2. 2\.At⁡\(Φ\)∩At⁡\(Ψ\)=∅\\operatorname\{At\}\(\\Phi\)\\cap\\operatorname\{At\}\(\\Psi\)=\\emptyset;
3. 3\.At⁡\(ϕ\)∩At⁡\(ψ\)=∅\\operatorname\{At\}\(\\phi\)\\cap\\operatorname\{At\}\(\\psi\)=\\emptyset\.

Minimality says that arguments with no shared content in both their premises and their conclusions must have zero similarity\.

###### Property 6\(Non\-zero\)\.

A similarity measureSSsatisfies non\-zero similarity if, for alla=\(Φ,ϕ\)a=\(\\Phi,\\phi\)andb=\(Ψ,ψ\)b=\(\\Psi,\\psi\)inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\),

Co⁡\(Φ,Ψ\)≠∅⟹S⁡\(a,b\)\>0\.\\operatorname\{Co\}\(\\Phi,\\Psi\)\\neq\\emptyset\\quad\\Longrightarrow\\quad S\(a,b\)\>0\.

The non\-zero property says that shared premise content is enough to induce positive similarity\.

###### Property 7\(Strict monotony\)\.

A similarity measureSSsatisfies monotony if, for alla=\(Φ,ϕ\)a=\(\\Phi,\\phi\),b=\(Ψ,ψ\)b=\(\\Psi,\\psi\), andc=\(Ξ,ξ\)c=\(\\Xi,\\xi\)inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\), the following conditions implyS⁡\(a,b\)≥S⁡\(a,c\)S\(a,b\)\\geq S\(a,c\):

1. 1\.ϕ≡ψ\\phi\\equiv\\psiorAt⁡\(ϕ\)∩At⁡\(ξ\)=∅\\operatorname\{At\}\(\\phi\)\\cap\\operatorname\{At\}\(\\xi\)=\\emptyset;
2. 2\.Co⁡\(Φ,Ξ\)⊆Co⁡\(Φ,Ψ\)\\operatorname\{Co\}\(\\Phi,\\Xi\)\\subseteq\\operatorname\{Co\}\(\\Phi,\\Psi\);
3. 3\.Ψ∖Co⁡\(Ψ,Φ\)=Co⁡\(Ψ∖Co⁡\(Ψ,Φ\),Ξ∖Co⁡\(Ξ,Φ\)\)\.\\Psi\\setminus\\operatorname\{Co\}\(\\Psi,\\Phi\)=\\operatorname\{Co\}\\bigl\(\\Psi\\setminus\\operatorname\{Co\}\(\\Psi,\\Phi\),\\Xi\\setminus\\operatorname\{Co\}\(\\Xi,\\Phi\)\\bigr\)\.

It satisfies strict monotony if, in addition, either

Co⁡\(Φ,Ξ\)⊊Co⁡\(Φ,Ψ\),\\operatorname\{Co\}\(\\Phi,\\Xi\)\\subsetneq\\operatorname\{Co\}\(\\Phi,\\Psi\),or

Co⁡\(Φ,Ξ\)≠∅and\|Ξ∖Co⁡\(Ξ,Φ\)\|\>\|Ψ∖Co⁡\(Ψ,Φ\)\|\.\\operatorname\{Co\}\(\\Phi,\\Xi\)\\neq\\emptyset\\quad\\text\{and\}\\quad\|\\Xi\\setminus\\operatorname\{Co\}\(\\Xi,\\Phi\)\|\>\|\\Psi\\setminus\\operatorname\{Co\}\(\\Psi,\\Phi\)\|\.In either strict case,

S⁡\(a,b\)\>S⁡\(a,c\)\.S\(a,b\)\>S\(a,c\)\.

Strict monotony says that, holding the conclusion comparison fixed or irrelevant, an argument becomes more similar toaaas it shares more ofaa’s premises and adds fewer unrelated premises\.

###### Property 8\(Strict dominance\)\.

A similarity measureSSsatisfies dominance if, for alla=\(Φ,ϕ\)a=\(\\Phi,\\phi\),b=\(Ψ,ψ\)b=\(\\Psi,\\psi\), andc=\(Ξ,ξ\)c=\(\\Xi,\\xi\)inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\), the following conditions implyS⁡\(a,b\)≥S⁡\(a,c\)S\(a,b\)\\geq S\(a,c\):

1. 1\.Ψ≅Ξ\\Psi\\cong\\Xi;
2. 2\.CNℱ⁡\(ϕ\)∩CNℱ⁡\(ξ\)⊆CNℱ⁡\(ϕ\)∩CNℱ⁡\(ψ\);\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\cap\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\xi\)\\subseteq\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\cap\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\psi\);
3. 3\.CNℱ⁡\(ψ\)∖CNℱ⁡\(ϕ\)⊆CNℱ⁡\(ξ\)∖CNℱ⁡\(ϕ\)\.\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\psi\)\\setminus\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\subseteq\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\xi\)\\setminus\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\.

It satisfies strict dominance if, in addition, either the inclusion in condition 2 is strict, or

CNℱ⁡\(ϕ\)∩CNℱ⁡\(ξ\)≠∅\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\\cap\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\xi\)\\neq\\emptysetand the inclusion in condition 3 is strict\. In either strict case,

S⁡\(a,b\)\>S⁡\(a,c\)\.S\(a,b\)\>S\(a,c\)\.

Strict dominance says that, when two candidate arguments have equivalent premise sets, the one whose conclusion shares more finite logical consequences with the reference conclusion is more similar\.

### A\.6More information on Reproducing Kernel Hilbert Spaces

LetXXbe a set\. A function

K:X×X→ℝK:X\\times X\\to\\mathbb\{R\}is a positive semi\-definite kernel if, for every finite collectionx1,…,xn∈Xx\_\{1\},\\ldots,x\_\{n\}\\in X, the Gram matrix

𝐊=\(K⁡\(xi,xj\)\)i,j=1n\\mathbf\{K\}=\\big\(K\(x\_\{i\},x\_\{j\}\)\\big\)\_\{i,j=1\}^\{n\}is positive semi\-definite\. Equivalently, for everyc1,…,cn∈ℝc\_\{1\},\\ldots,c\_\{n\}\\in\\mathbb\{R\},

∑i=1n∑j=1nci​cj​K​\(xi,xj\)≥0\.\\sum\_\{i=1\}^\{n\}\\sum\_\{j=1\}^\{n\}c\_\{i\}c\_\{j\}K\(x\_\{i\},x\_\{j\}\)\\geq 0\.
A reproducing kernel Hilbert space \(RKHS\) associated withKKis a Hilbert spaceℋK\\mathcal\{H\}\_\{K\}of real\-valued functions onXXsuch that, for everyx∈Xx\\in X, the function

K⁡\(⋅,x\):X→ℝK\(\\cdot,x\):X\\to\\mathbb\{R\}belongs toℋK\\mathcal\{H\}\_\{K\}, and the reproducing property holds:

h⁡\(x\)=⟨h,K⁡\(⋅,x\)⟩ℋKfor all​h∈ℋK\.h\(x\)=\\langle h,K\(\\cdot,x\)\\rangle\_\{\\mathcal\{H\}\_\{K\}\}\\qquad\\text\{for all \}h\\in\\mathcal\{H\}\_\{K\}\.The Moore–Aronszajn theorem states that every positive semi\-definite kernelKKuniquely determines such an RKHSℋK\\mathcal\{H\}\_\{K\}\[[3](https://arxiv.org/html/2608.15325#bib.bib4),[15](https://arxiv.org/html/2608.15325#bib.bib5)\]\.

The associated canonical feature map is

ΘK:X→ℋK,ΘK​\(x\)=K⁡\(⋅,x\)\.\\Theta\_\{K\}:X\\to\\mathcal\{H\}\_\{K\},\\qquad\\Theta\_\{K\}\(x\)=K\(\\cdot,x\)\.Thus, each objectx∈Xx\\in Xis represented by the function that records its kernel similarity to every other element ofXX\. In this sense,ΘK\\Theta\_\{K\}is an embedding: it represents each element ofXXas a point in a Hilbert space whose inner products are given byKK\. The key identity is

⟨ΘK​\(x\),ΘK​\(y\)⟩ℋK=K⁡\(x,y\),\\langle\\Theta\_\{K\}\(x\),\\Theta\_\{K\}\(y\)\\rangle\_\{\\mathcal\{H\}\_\{K\}\}=K\(x,y\),which follows directly from the reproducing property\. Hence the kernel can be interpreted as an inner product between the feature representationsΘK​\(x\)\\Theta\_\{K\}\(x\)andΘK​\(y\)\\Theta\_\{K\}\(y\), even when these representations are not written explicitly or are infinite\-dimensional\.

### A\.7Finite\-dimensional approximation by kernel PCA

The logical embedding defined in Section[2](https://arxiv.org/html/2608.15325#S2)maps each argumentaato an element

Θσ​\(a\)∈ℋσ\\Theta\_\{\\sigma\}\(a\)\\in\\mathcal\{H\}\_\{\\sigma\}of the RKHS induced by the kernelsimσ\\operatorname\{sim\}^\{\\sigma\}\. This space may be infinite\-dimensional, so the embedding is not directly a finite vector representation\.

The canonical RKHS feature map has the form

Θσ​\(a\)​\(⋅\)=simσ⁡\(⋅,a\)\.\\Theta\_\{\\sigma\}\(a\)\(\\cdot\)=\\operatorname\{sim\}^\{\\sigma\}\(\\cdot,a\)\.Thus, the embedding ofaais a function that records the similarity ofaato every possible argument\. On a finite corpusa1,…,ana\_\{1\},\\ldots,a\_\{n\}, we observe only the restriction of this function to the sample:

\(simσ⁡\(a1,a\),…,simσ⁡\(an,a\)\)=\(Θσ​\(a\)​\(a1\),⋯,Θσ​\(a\)​\(an\)\)\.\\bigl\(\\operatorname\{sim\}^\{\\sigma\}\(a\_\{1\},a\),\\ldots,\\operatorname\{sim\}^\{\\sigma\}\(a\_\{n\},a\)\\bigr\)=\(\\Theta\_\{\\sigma\}\(a\)\(a\_\{1\}\),\\cdots,\\Theta\_\{\\sigma\}\(a\)\(a\_\{n\}\)\)\.
By restricting to the argument corpus, these restricted feature functions form the columns of the Gram matrix

Ki​jσ=simσ⁡\(ai,aj\)\.K^\{\\sigma\}\_\{ij\}=\\operatorname\{sim\}^\{\\sigma\}\(a\_\{i\},a\_\{j\}\)\.
where each columnjjis then a finite\-dimensional approximation to the logical embedding of argumentjj\. Whennnis large, however, representing each argument by its similarities to all observed arguments is usually too high\-dimensional\. Kernel PCA compresses this information by projecting the centered RKHS embeddings onto their leading empirical principal directions\. The resulting coordinates give the best rank\-rrlinear approximation to the centered logical embeddings on the observed corpus\.

In practice, we do not observeKσK^\{\\sigma\}exactly\. We estimate it from natural\-language text, obtainingK^σ\\widehat\{K\}^\{\\sigma\}\. Kernel PCA applied toK^σ\\widehat\{K\}^\{\\sigma\}therefore provides a finite\-dimensional approximation to the ideal logical embeddings, with two sources of approximation: the empirical similarity estimates and the rank\-rrPCA truncation\.

We now state the finite\-sample construction for this approximation\. LetK:X×X→ℝK:X\\times X\\to\\mathbb\{R\}be a positive semidefinite kernel on a setXX, with RKHSℋK\\mathcal\{H\}\_\{K\}, and letx1,…,xn∈Xx\_\{1\},\\ldots,x\_\{n\}\\in X\. Define the Gram matrix

𝐊=\(K⁡\(xi,xj\)\)i,j=1n\.\\mathbf\{K\}=\(K\(x\_\{i\},x\_\{j\}\)\)\_\{i,j=1\}^\{n\}\.Let

𝐇=In−1n​𝟏𝟏⊤and𝐊c=𝐇𝐊𝐇\\mathbf\{H\}=I\_\{n\}\-\\frac\{1\}\{n\}\\mathbf\{1\}\\mathbf\{1\}^\{\\top\}\\qquad\\text\{and\}\\qquad\\mathbf\{K\}\_\{c\}=\\mathbf\{H\}\\mathbf\{K\}\\mathbf\{H\}be the centering matrix and the centered Gram matrix\.

Let

ΘK​\(x\)=K⁡\(⋅,x\)\\Theta\_\{K\}\(x\)=K\(\\cdot,x\)be the canonical RKHS feature map, and define the centered embeddings

Zi=ΘK​\(xi\)−1n​∑j=1nΘK​\(xj\)\.Z\_\{i\}=\\Theta\_\{K\}\(x\_\{i\}\)\-\\frac\{1\}\{n\}\\sum\_\{j=1\}^\{n\}\\Theta\_\{K\}\(x\_\{j\}\)\.The centered Gram matrix records exactly the inner products among these centered embeddings:

\(𝐊c\)i​j=⟨Zi,Zj⟩ℋK\.\(\\mathbf\{K\}\_\{c\}\)\_\{ij\}=\\langle Z\_\{i\},Z\_\{j\}\\rangle\_\{\\mathcal\{H\}\_\{K\}\}\.Thus, kernel PCA performs ordinary PCA on the centered RKHS embeddingsZ1,…,ZnZ\_\{1\},\\ldots,Z\_\{n\}, using only the finite matrix𝐊c\\mathbf\{K\}\_\{c\}\.

Let

𝐊c=U​Λ​U⊤\\mathbf\{K\}\_\{c\}=U\\Lambda U^\{\\top\}be an eigendecomposition, with eigenvalues

λ1≥λ2≥⋯≥λn≥0\\lambda\_\{1\}\\geq\\lambda\_\{2\}\\geq\\cdots\\geq\\lambda\_\{n\}\\geq 0and orthonormal eigenvectorsu1,…,unu\_\{1\},\\ldots,u\_\{n\}\. For eachλℓ\>0\\lambda\_\{\\ell\}\>0, theℓ\\ellth empirical principal direction inℋK\\mathcal\{H\}\_\{K\}is

v^ℓ=1λℓ​∑j=1n\(uℓ\)j​Zj\.\\hat\{v\}\_\{\\ell\}=\\frac\{1\}\{\\sqrt\{\\lambda\_\{\\ell\}\}\}\\sum\_\{j=1\}^\{n\}\(u\_\{\\ell\}\)\_\{j\}Z\_\{j\}\.The coordinate ofxix\_\{i\}along this direction is

⟨Zi,v^ℓ⟩ℋK=λℓ​\(uℓ\)i\.\\langle Z\_\{i\},\\hat\{v\}\_\{\\ell\}\\rangle\_\{\\mathcal\{H\}\_\{K\}\}=\\sqrt\{\\lambda\_\{\\ell\}\}\(u\_\{\\ell\}\)\_\{i\}\.Therefore, the rank\-rrkernel PCA coordinates are given by the rows of

Ur​Λr1/2,U\_\{r\}\\Lambda\_\{r\}^\{1/2\},whereUrU\_\{r\}contains the firstrreigenvectors andΛr\\Lambda\_\{r\}contains the corresponding eigenvalues\.

These coordinates have the standard PCA optimality property\. Among all rank\-rrorthogonal projectionsPPonℋK\\mathcal\{H\}\_\{K\}, the projection onto

span⁡\{v^1,…,v^r\}\\operatorname\{span\}\\\{\\hat\{v\}\_\{1\},\\ldots,\\hat\{v\}\_\{r\}\\\}minimizes the average squared reconstruction error:

minrank⁡\(P\)≤r⁡1n​∑i=1n‖Zi−P​Zi‖ℋK2=∑ℓ\>rλℓn\.\\min\_\{\\operatorname\{rank\}\(P\)\\leq r\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\\|Z\_\{i\}\-PZ\_\{i\}\\\|\_\{\\mathcal\{H\}\_\{K\}\}^\{2\}=\\sum\_\{\\ell\>r\}\\frac\{\\lambda\_\{\\ell\}\}\{n\}\.Equivalently, by the Eckart–Young–Mirsky theorem, the rank\-rrtruncation

Ur​Λr​Ur⊤U\_\{r\}\\Lambda\_\{r\}U\_\{r\}^\{\\top\}is the best rank\-rrapproximation to𝐊c\\mathbf\{K\}\_\{c\}in Frobenius norm:

minrank⁡\(M\)≤r⁡‖𝐊c−M‖F2=∑ℓ\>rλℓ2\.\\min\_\{\\operatorname\{rank\}\(M\)\\leq r\}\\\|\\mathbf\{K\}\_\{c\}\-M\\\|\_\{F\}^\{2\}=\\sum\_\{\\ell\>r\}\\lambda\_\{\\ell\}^\{2\}\.
Applying this construction withX=Arg⁡\(ℒ\)X=\\operatorname\{Arg\}\(\\mathcal\{L\}\)andK=simσK=\\operatorname\{sim\}^\{\\sigma\}shows that kernel PCA on the ideal Gram matrixKσK^\{\\sigma\}gives the bestrr\-dimensional linear approximation to the centered logical embeddings of the observed arguments\. In practice, we useK^σ\\widehat\{K\}^\{\\sigma\}in place ofKσK^\{\\sigma\}\. WhenK^σ\\widehat\{K\}^\{\\sigma\}is a good approximation toKσK^\{\\sigma\}, the resulting coordinates approximate the finite\-sample kernel PCA coordinates of the ideal logical embeddings\. Thus, the empirical coordinates used in the paper are low\-dimensional approximations to the logical embeddings, restricted to the observed corpus\.

The reduction of kernel PCA to the Gram matrix eigenproblem follows[21](https://arxiv.org/html/2608.15325#bib.bib28); see also[20](https://arxiv.org/html/2608.15325#bib.bib29)\. The best rank\-rrmatrix approximation property follows from[6](https://arxiv.org/html/2608.15325#bib.bib27)\.

### A\.8Omitted proofs

#### A\.8\.1Auxiliary Lemmas

Given a setXXand an equivalence relation∼\\simonXX, the quotient ofXXby∼\\simis

X​╱∼:=\{\[x\]∼:x∈X\},\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{X\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.83437pt\}\{$\\displaystyle\{\\sim\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{X\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.83437pt\}\{$\\textstyle\{\\sim\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{X\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.34427pt\}\{$\\scriptstyle\{\\sim\}$\}\}\{\\raisebox\{1\.70833pt\}\{$\\scriptscriptstyle\{X\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-0\.99101pt\}\{$\\scriptscriptstyle\{\\sim\}$\}\}\}:=\\\{\[x\]\_\{\\sim\}:x\\in X\\\},where\[x\]∼=\{y∈X:y∼x\}\[x\]\_\{\\sim\}=\\\{y\\in X:y\\sim x\\\}\. ForΦ⊆ℒ\\Phi\\subseteq\\mathcal\{L\}, write

Φ​╱≡:=\{\[ξ\]≡:ξ∈Φ\},\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}:=\\\{\[\\xi\]\_\{\\equiv\}:\\xi\\in\\Phi\\\},where\[ξ\]≡=\{η∈ℒ:η≡ξ\}\[\\xi\]\_\{\\equiv\}=\\\{\\eta\\in\\mathcal\{L\}:\\eta\\equiv\\xi\\\}is the equivalence class ofξ\\xiin the ambient languageℒ\\mathcal\{L\}\.

###### Lemma 1\.

Leta=\(Φ,ϕ\)∈Arg⁡\(ℒ\)a=\(\\Phi,\\phi\)\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\)\. Ifξ,ζ∈Φ\\xi,\\zeta\\in\\Phiandξ≡ζ\\xi\\equiv\\zeta, thenξ=ζ\\xi=\\zeta\.

###### Proof\.

Suppose, toward a contradiction, thatξ,ζ∈Φ\\xi,\\zeta\\in\\Phi,ξ≠ζ\\xi\\neq\\zeta, andξ≡ζ\\xi\\equiv\\zeta\. LetΦ′=Φ∖\{ξ\}\\Phi^\{\\prime\}=\\Phi\\setminus\\\{\\xi\\\}\. We show thatΦ′⊢ϕ\\Phi^\{\\prime\}\\vdash\\phi\.

Letvvbe any valuation that satisfies every formula inΦ′\\Phi^\{\\prime\}\. Sinceζ∈Φ′\\zeta\\in\\Phi^\{\\prime\},vvsatisfiesζ\\zeta\. Becauseζ≡ξ\\zeta\\equiv\\xi, every valuation satisfyingζ\\zetaalso satisfiesξ\\xi\. Hencevvsatisfiesξ\\xi\. Thereforevvsatisfies every formula inΦ\\Phi\. SinceΦ⊢ϕ\\Phi\\vdash\\phi, it follows thatvvsatisfiesϕ\\phi\.

Thus every valuation satisfyingΦ′\\Phi^\{\\prime\}also satisfiesϕ\\phi, soΦ′⊢ϕ\\Phi^\{\\prime\}\\vdash\\phi\. ButΦ′⊊Φ\\Phi^\{\\prime\}\\subsetneq\\Phi, contradicting the non\-redundancy ofΦ\\Phi\. Thereforeξ=ζ\\xi=\\zeta\. ∎

###### Lemma 2\.

Leta=\(Φ,ϕ\)a=\(\\Phi,\\phi\)andb=\(Ψ,ψ\)b=\(\\Psi,\\psi\)be arguments inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\. Then the map

π:Co⁡\(Φ,Ψ\)→Φ​╱≡∩Ψ​╱≡,ξ↦\[ξ\]≡,\\pi:\\operatorname\{Co\}\(\\Phi,\\Psi\)\\to\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cap\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\},\\qquad\\xi\\mapsto\[\\xi\]\_\{\\equiv\},is a bijection\. In particular,

\|Co\(Φ,Ψ\)\|=\|Φ╱≡∩Ψ╱≡\|\.\|\\operatorname\{Co\}\(\\Phi,\\Psi\)\|=\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cap\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|\.

###### Proof\.

First, the map is well\-defined\. Ifξ∈Co⁡\(Φ,Ψ\)\\xi\\in\\operatorname\{Co\}\(\\Phi,\\Psi\), thenξ∈Φ\\xi\\in\\Phiand there existsζ∈Ψ\\zeta\\in\\Psisuch thatξ≡ζ\\xi\\equiv\\zeta\. Hence\[ξ\]≡∈Φ​╱≡\[\\xi\]\_\{\\equiv\}\\in\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}and\[ξ\]≡=\[ζ\]≡∈Ψ​╱≡\[\\xi\]\_\{\\equiv\}=\[\\zeta\]\_\{\\equiv\}\\in\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\. Therefore\[ξ\]≡∈Φ​╱≡∩Ψ​╱≡\[\\xi\]\_\{\\equiv\}\\in\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cap\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\.

We now prove surjectivity\. Let

C∈Φ​╱≡∩Ψ​╱≡\.C\\in\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cap\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\.SinceC∈Φ​╱≡C\\in\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}, there existsξ∈Φ\\xi\\in\\Phisuch thatC=\[ξ\]≡C=\[\\xi\]\_\{\\equiv\}\. SinceC∈Ψ​╱≡C\\in\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}, there existsζ∈Ψ\\zeta\\in\\Psisuch thatC=\[ζ\]≡C=\[\\zeta\]\_\{\\equiv\}\. Henceξ≡ζ\\xi\\equiv\\zeta, soξ∈Co⁡\(Φ,Ψ\)\\xi\\in\\operatorname\{Co\}\(\\Phi,\\Psi\)\. Moreover,π⁡\(ξ\)=\[ξ\]≡=C\\pi\(\\xi\)=\[\\xi\]\_\{\\equiv\}=C\. Thusπ\\piis surjective\.

We now prove injectivity\. Letξ,ξ′∈Co⁡\(Φ,Ψ\)\\xi,\\xi^\{\\prime\}\\in\\operatorname\{Co\}\(\\Phi,\\Psi\)and suppose thatπ⁡\(ξ\)=π⁡\(ξ′\)\\pi\(\\xi\)=\\pi\(\\xi^\{\\prime\}\)\. Then

\[ξ\]≡=\[ξ′\]≡,\[\\xi\]\_\{\\equiv\}=\[\\xi^\{\\prime\}\]\_\{\\equiv\},soξ≡ξ′\\xi\\equiv\\xi^\{\\prime\}\. Sinceξ,ξ′∈Φ\\xi,\\xi^\{\\prime\}\\in\\PhiandΦ\\Phiis the premise set of an argument, Lemma[1](https://arxiv.org/html/2608.15325#Thmlemma1)implies thatξ=ξ′\\xi=\\xi^\{\\prime\}\. Therefore,π\\piis injective\.

Henceπ\\piis bijective\. ∎

###### Lemma 3\.

Leta=\(Φ,ϕ\)a=\(\\Phi,\\phi\)be an argument inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\. Then

\|Φ\|=\|Φ╱≡\|\.\|\\Phi\|=\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|\.

###### Proof\.

Consider the map

π:Φ→Φ​╱≡,ξ↦\[ξ\]≡\.\\pi:\\Phi\\to\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\},\\qquad\\xi\\mapsto\[\\xi\]\_\{\\equiv\}\.This map is surjective by definition ofΦ​╱≡\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\.

We now show that it is injective\. Letξ,ζ∈Φ\\xi,\\zeta\\in\\Phiand suppose thatπ⁡\(ξ\)=π⁡\(ζ\)\\pi\(\\xi\)=\\pi\(\\zeta\)\. Then

\[ξ\]≡=\[ζ\]≡,\[\\xi\]\_\{\\equiv\}=\[\\zeta\]\_\{\\equiv\},soξ≡ζ\\xi\\equiv\\zeta\. SinceΦ\\Phiis the premise set of an argument, Lemma[1](https://arxiv.org/html/2608.15325#Thmlemma1)implies thatξ=ζ\\xi=\\zeta\. Henceπ\\piis injective\. Thereforeπ\\piis a bijection, and consequently

\|Φ\|=\|Φ╱≡\|\.\|\\Phi\|=\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|\.∎

#### A\.8\.2Proof of Theorem[2](https://arxiv.org/html/2608.15325#Thmtheorem2)

###### Proof\.

Leta=\(Φ,ϕ\)a=\(\\Phi,\\phi\)andb=\(Ψ,ψ\)b=\(\\Psi,\\psi\)be arguments inArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\.

For any setXX, write

𝒫fin​\(X\)=\{A⊆X:\|A\|<∞\}\\mathcal\{P\}\_\{\\mathrm\{fin\}\}\(X\)=\\\{A\\subseteq X:\|A\|<\\infty\\\}for the set of finite subsets ofXX\.

First define the usual Tanimoto coefficient on pairs of finite sets with nonempty union by

k~T​\(A,B\)=\|A∩B\|\|A∪B\|,A∪B≠∅\.\\tilde\{k\}\_\{T\}\(A,B\)=\\frac\{\|A\\cap B\|\}\{\|A\\cup B\|\},\\qquad A\\cup B\\neq\\varnothing\.This is the Tanimoto kernel on finite sets, also known as the Jaccard kernel\[[14](https://arxiv.org/html/2608.15325#bib.bib3), p\. 301\]\. To handle the only case in which this expression is undefined, we extend it by setting

kT​\(A,B\)=\{k~T​\(A,B\),if​A∪B≠∅,1,if​A=B=∅\.k\_\{T\}\(A,B\)=\\begin\{cases\}\\tilde\{k\}\_\{T\}\(A,B\),&\\text\{if \}A\\cup B\\neq\\varnothing,\\\\\[11\.99998pt\] 1,&\\text\{if \}A=B=\\varnothing\.\\end\{cases\}
We first verify that this extension is still positive semi\-definite\. LetA1,…,AnA\_\{1\},\\ldots,A\_\{n\}be finite sets\. After permutation, we may assume without loss of generality that

A1=⋯=Am=∅A\_\{1\}=\\cdots=A\_\{m\}=\\varnothingand thatAm\+1,…,AnA\_\{m\+1\},\\ldots,A\_\{n\}are nonempty\. Ifm=0m=0, all sets are nonempty, and the result follows directly from the positive semi\-definiteness of the usual Tanimoto kernel\. Ifm=nm=n, then the Gram matrix is the all\-ones matrix, which is positive semi\-definite\. Thus assume0<m<n0<m<n\.

Fori≤m<ji\\leq m<j, we have

kT​\(Ai,Aj\)=k~T​\(∅,Aj\)=\|∅∩Aj\|\|∅∪Aj\|=0\.k\_\{T\}\(A\_\{i\},A\_\{j\}\)=\\tilde\{k\}\_\{T\}\(\\varnothing,A\_\{j\}\)=\\frac\{\|\\varnothing\\cap A\_\{j\}\|\}\{\|\\varnothing\\cup A\_\{j\}\|\}=0\.Therefore the Gram matrix\(kT​\(Ai,Aj\)\)i,j=1n\\big\(k\_\{T\}\(A\_\{i\},A\_\{j\}\)\\big\)\_\{i,j=1\}^\{n\}has the block form

\(𝟏m​𝟏m⊤00G\),\\begin\{pmatrix\}\\mathbf\{1\}\_\{m\}\\mathbf\{1\}\_\{m\}^\{\\top\}&0\\\\ 0&G\\end\{pmatrix\},where𝟏m​𝟏m⊤\\mathbf\{1\}\_\{m\}\\mathbf\{1\}\_\{m\}^\{\\top\}is them×mm\\times mall\-ones matrix and

G=\(k~T​\(Ai,Aj\)\)i,j=m\+1nG=\\big\(\\tilde\{k\}\_\{T\}\(A\_\{i\},A\_\{j\}\)\\big\)\_\{i,j=m\+1\}^\{n\}is the Gram matrix of the usual Tanimoto kernel on the nonempty setsAm\+1,…,AnA\_\{m\+1\},\\ldots,A\_\{n\}\.

The first block is positive semi\-definite because, for anyu∈ℝmu\\in\\mathbb\{R\}^\{m\},

u⊤​\(𝟏m​𝟏m⊤\)​u=\(𝟏m⊤​u\)2≥0\.u^\{\\top\}\(\\mathbf\{1\}\_\{m\}\\mathbf\{1\}\_\{m\}^\{\\top\}\)u=\(\\mathbf\{1\}\_\{m\}^\{\\top\}u\)^\{2\}\\geq 0\.The second block is positive semi\-definite by the positive semi\-definiteness of the Tanimoto kernel\[[14](https://arxiv.org/html/2608.15325#bib.bib3), p\. 301\]\. Hence the block\-diagonal Gram matrix is positive semi\-definite\. ThereforekTk\_\{T\}is a positive semi\-definite kernel on𝒫fin​\(X\)\\mathcal\{P\}\_\{\\mathrm\{fin\}\}\(X\), for any ambient setXX\.

We now rewrite the syntactic part of the similarity\. By Lemmas[2](https://arxiv.org/html/2608.15325#Thmlemma2)and[3](https://arxiv.org/html/2608.15325#Thmlemma3),

\|Co\(Φ,Ψ\)\|=\|Φ╱≡∩Ψ╱≡\|,\|\\operatorname\{Co\}\(\\Phi,\\Psi\)\|=\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cap\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|,and

\|Φ\|=\|Φ╱≡\|,\|Ψ\|=\|Ψ╱≡\|\.\|\\Phi\|=\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|,\\qquad\|\\Psi\|=\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|\.Therefore, wheneverΦ​╱≡∪Ψ​╱≡≠∅\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cup\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\neq\\varnothing, inclusion– exclusion gives

ssyn\(Φ,Ψ\)=\|Φ╱≡∩Ψ╱≡\|\|Φ╱≡∪Ψ╱≡\|=kT\(Φ╱≡,Ψ╱≡\)\.s\_\{\\mathrm\{syn\}\}\(\\Phi,\\Psi\)=\\frac\{\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cap\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|\}\{\\left\|\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\cup\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\|\}=k\_\{T\}\\left\(\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\},\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\)\.IfΦ​╱≡=Ψ​╱≡=∅\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}=\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}=\\varnothing, thenΦ=Ψ=∅\\Phi=\\Psi=\\varnothing, and both sides are equal to11by the empty\-set convention forkTk\_\{T\}\. Hence, in all cases,

ssyn\(Φ,Ψ\)=kT\(Φ╱≡,Ψ╱≡\)\.s\_\{\\mathrm\{syn\}\}\(\\Phi,\\Psi\)=k\_\{T\}\\left\(\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\},\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Psi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\)\.
Similarly, by definition of the semantic component,

ssem​\(ϕ,ψ\)=kT​\(CNℱ⁡\(ϕ\),CNℱ⁡\(ψ\)\),s\_\{\\mathrm\{sem\}\}\(\\phi,\\psi\)=k\_\{T\}\\left\(\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\),\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\psi\)\\right\),again using the convention that the similarity of two empty consequence sets is11\.

Now define

Π:Arg⁡\(ℒ\)→𝒫fin​\(ℒ​╱≡\),Π⁡\(Φ,ϕ\)=Φ​╱≡,\\Pi:\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\to\\mathcal\{P\}\_\{\\mathrm\{fin\}\}\\left\(\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\mathcal\{L\}\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\mathcal\{L\}\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\mathcal\{L\}\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\mathcal\{L\}\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}\\right\),\\qquad\\Pi\(\\Phi,\\phi\)=\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\},and

ρ:Arg⁡\(ℒ\)→𝒫fin​\(ℱ\),ρ⁡\(Φ,ϕ\)=CNℱ⁡\(ϕ\)\.\\rho:\\operatorname\{Arg\}\(\\mathcal\{L\}\)\\to\\mathcal\{P\}\_\{\\mathrm\{fin\}\}\(\\mathcal\{F\}\),\\qquad\\rho\(\\Phi,\\phi\)=\\operatorname\{CN\}\_\{\\mathcal\{F\}\}\(\\phi\)\.These maps are well\-defined because premise sets are finite andℱ\\mathcal\{F\}is finite\.

Define

KΠ​\(a,b\)=kT​\(Π⁡\(a\),Π⁡\(b\)\)K\_\{\\Pi\}\(a,b\)=k\_\{T\}\(\\Pi\(a\),\\Pi\(b\)\)and

Kρ​\(a,b\)=kT​\(ρ⁡\(a\),ρ⁡\(b\)\)\.K\_\{\\rho\}\(a,b\)=k\_\{T\}\(\\rho\(a\),\\rho\(b\)\)\.SincekTk\_\{T\}is positive semi\-definite on finite subsets,KΠK\_\{\\Pi\}andKρK\_\{\\rho\}are positive semi\-definite kernels onArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)by closure of positive semi\-definite kernels under pullback along arbitrary maps\[[3](https://arxiv.org/html/2608.15325#bib.bib4),[15](https://arxiv.org/html/2608.15325#bib.bib5)\]\.

Finally, for any0<σ<10<\\sigma<1,

simσ⁡\(a,b\)=σ​KΠ​\(a,b\)\+\(1−σ\)​Kρ​\(a,b\)\.\\operatorname\{sim\}^\{\\sigma\}\(a,b\)=\\sigma K\_\{\\Pi\}\(a,b\)\+\(1\-\\sigma\)K\_\{\\rho\}\(a,b\)\.Positive semi\-definite kernels are closed under nonnegative scalar multiplication and finite sums\[[3](https://arxiv.org/html/2608.15325#bib.bib4),[15](https://arxiv.org/html/2608.15325#bib.bib5)\]\. Sinceσ\>0\\sigma\>0and1−σ\>01\-\\sigma\>0, it follows thatsimσ\\operatorname\{sim\}^\{\\sigma\}is a positive semi\-definite kernel onArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\)\. ∎

#### A\.8\.3Proof of Theorem[3](https://arxiv.org/html/2608.15325#Thmtheorem3)

###### Proof\.

Leta,b∈Arg⁡\(ℒ\)a,b\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\)\. SinceΘσ\\Theta\_\{\\sigma\}is the canonical feature map associated withsimσ\\operatorname\{sim\}^\{\\sigma\}, we have

⟨Θσ​\(x\),Θσ​\(y\)⟩ℋσ=simσ⁡\(x,y\)\\langle\\Theta\_\{\\sigma\}\(x\),\\Theta\_\{\\sigma\}\(y\)\\rangle\_\{\\mathcal\{H\}\_\{\\sigma\}\}=\\operatorname\{sim\}^\{\\sigma\}\(x,y\)for allx,y∈Arg⁡\(ℒ\)x,y\\in\\operatorname\{Arg\}\(\\mathcal\{L\}\)\.

First suppose thatΘσ​\(a\)=Θσ​\(b\)\\Theta\_\{\\sigma\}\(a\)=\\Theta\_\{\\sigma\}\(b\)\. Then

0=‖Θσ​\(a\)−Θσ​\(b\)‖ℋσ2\.0=\\\|\\Theta\_\{\\sigma\}\(a\)\-\\Theta\_\{\\sigma\}\(b\)\\\|\_\{\\mathcal\{H\}\_\{\\sigma\}\}^\{2\}\.Expanding the squared norm gives

0=simσ⁡\(a,a\)\+simσ⁡\(b,b\)−2​simσ⁡\(a,b\)\.0=\\operatorname\{sim\}^\{\\sigma\}\(a,a\)\+\\operatorname\{sim\}^\{\\sigma\}\(b,b\)\-2\\operatorname\{sim\}^\{\\sigma\}\(a,b\)\.Since each argument has maximal similarity with itself,

simσ⁡\(a,a\)=simσ⁡\(b,b\)=1\.\\operatorname\{sim\}^\{\\sigma\}\(a,a\)=\\operatorname\{sim\}^\{\\sigma\}\(b,b\)=1\.Therefore

simσ⁡\(a,b\)=1\.\\operatorname\{sim\}^\{\\sigma\}\(a,b\)=1\.By Theorem[1](https://arxiv.org/html/2608.15325#Thmtheorem1), this impliesa≈ba\\approx b\.

Conversely, suppose thata≈ba\\approx b\. By Theorem[1](https://arxiv.org/html/2608.15325#Thmtheorem1),

simσ⁡\(a,b\)=1\.\\operatorname\{sim\}^\{\\sigma\}\(a,b\)=1\.Again usingsimσ⁡\(a,a\)=simσ⁡\(b,b\)=1\\operatorname\{sim\}^\{\\sigma\}\(a,a\)=\\operatorname\{sim\}^\{\\sigma\}\(b,b\)=1, we get

‖Θσ​\(a\)−Θσ​\(b\)‖ℋσ2=1\+1−2=0\.\\\|\\Theta\_\{\\sigma\}\(a\)\-\\Theta\_\{\\sigma\}\(b\)\\\|\_\{\\mathcal\{H\}\_\{\\sigma\}\}^\{2\}=1\+1\-2=0\.HenceΘσ​\(a\)=Θσ​\(b\)\\Theta\_\{\\sigma\}\(a\)=\\Theta\_\{\\sigma\}\(b\)\.

Thus

Θσ​\(a\)=Θσ​\(b\)if and only ifa≈b\.\\Theta\_\{\\sigma\}\(a\)=\\Theta\_\{\\sigma\}\(b\)\\quad\\text\{if and only if\}\\quad a\\approx b\.
It remains only to check that the quotient map is well\-defined\. If\[a\]≈=\[b\]≈\[a\]\_\{\\approx\}=\[b\]\_\{\\approx\}, thena≈ba\\approx b, and the result just proved implies

Θσ​\(a\)=Θσ​\(b\)\.\\Theta\_\{\\sigma\}\(a\)=\\Theta\_\{\\sigma\}\(b\)\.Therefore

Θ¯σ​\(\[a\]≈\)=Θσ​\(a\)\\overline\{\\Theta\}\_\{\\sigma\}\(\[a\]\_\{\\approx\}\)=\\Theta\_\{\\sigma\}\(a\)is well\-defined\. Moreover, if

Θ¯σ​\(\[a\]≈\)=Θ¯σ​\(\[b\]≈\),\\overline\{\\Theta\}\_\{\\sigma\}\(\[a\]\_\{\\approx\}\)=\\overline\{\\Theta\}\_\{\\sigma\}\(\[b\]\_\{\\approx\}\),thenΘσ​\(a\)=Θσ​\(b\)\\Theta\_\{\\sigma\}\(a\)=\\Theta\_\{\\sigma\}\(b\), soa≈ba\\approx b, and hence

\[a\]≈=\[b\]≈\.\[a\]\_\{\\approx\}=\[b\]\_\{\\approx\}\.ThereforeΘ¯σ\\overline\{\\Theta\}\_\{\\sigma\}is injective\. ∎

## Appendix BExhibits

### B\.1Features of the Logical Embedding

In this section, we present a series of exhibits that illustrate more clearly the functioning and advantages of the Logical Embeddings method\. In the introduction we provided an example of how regular methods may fail to recover logical similarity\.

A useful comparison is to compare the Gram matrix generated by the Semantic\-Syntactic Jaccard similarity metric to a Gram matrix generated by LLM based comparisons\. Our prompt is in Appendix[B\.3\.1](https://arxiv.org/html/2608.15325#A2.SS3.SSS1)\. When we generate this matrix we find that LLMs rarely assign zero logical similarity\. Instead, it almost always detects some vague theme or rhetorical overlap\. The Logical Embeddings method is significantly stricter\. It relies on clause\-level decomposition and mutual entailment and assigns a positive similarity only when there is a clear logical relation between the components of two arguments\. The result is a much sparser network, but where the connections are on average stronger and more interpretable\.

Figure[2](https://arxiv.org/html/2608.15325#A2.F2)illustrates this point using only the subset of arguments under the doping topic from from the IBM\-ArgQ\-6\.3kArgs dataset\[[26](https://arxiv.org/html/2608.15325#bib.bib7)\]\. Although the two graphs contain the same number of nodes, corresponding to the same set of arguments, their edge structures are radically different\. The full\-argument graph is nearly complete, with8959889598edges, covering98\.5%98\.5\\%of all possible pairs\. The Logical Embeddings graph, in contrast, contains only64086408edges, or7\.0%7\.0\\%of all possible pairs\. It reveals that the full\-argument approach struggles to separate arguments based on their relation, and produce an almost fully connected graph where there is no possibility to see meaningful distinctions\.

This has direct implications for classification\. In the full\-argument graph, pro and con arguments are mixed, which suggests that the similarity measure does not recover the latent argumentative structure of the topic\. The Logical Embeddings graph instead displays a much clearer separation between the two sides\. This is because it is a measure designed to capture logical rather than rhetorical or thematic proximity\. Being sparse it is not an issue, it represents the selectivity of the model to connect two arguments\.

Figure 2:Logical Embeddings and Full\-argument LLM\.![Refer to caption](https://arxiv.org/html/2608.15325v1/Figures/fig_4_network_doping_comparison.png)Notes: The Figure reports, for the subset of arguments on doping, the network of pairwise connections generated by each method\. Each node represents an argument and each edge represents a non\-zero connection between two arguments\. Blue nodes denote pro\-doping arguments, while red nodes denote con\-doping arguments\.This leads to a final advantage of the Logical Embeddings model: traceability\. Since the score is produced from the number of clauses covered by mutual entailment, it is possible to trace back exactly what is connected and why the LLM judged those clauses as entailed\. Figure[3](https://arxiv.org/html/2608.15325#A2.F3)illustrates schematically how the score is obtained in the simplest, and also most recurrent, case in our dataset: two premises and one subconclusion\. First of all, the two families of clauses composing the text, premises and subconclusions, are treated separately and given equal weight\. Each component is obtained by dividing the number of shared clauses by the total number of distinct clauses in that family\. In our example, if the only subconclusion is entailed, then it contributes0\.5∗1=0\.50\.5\*1=0\.5to the final score\. The figure also makes clear another main feature of the model: it is not the raw number of mutual entailments that determines the score, but rather the number of clauses covered by those entailments\. Looking at the scores0\.250\.25and0\.50\.5obtained using premises only, the number of mutual entailments is the same, but in the second case all clauses are connected, which gives a score of0\.5×1/10\.5\\times 1/1, while in the first case the score is only0\.5×1/20\.5\\times 1/2\. This has clear implications, especially for higher scores, and helps explain why Logical Embeddings can display a higher average score than full\-argument similarity\. While mutual entailment and clause level comparison impose a stricter rule for connection, once a pair is judged entailed it has greater contributive power\.

Figure 3:How Logical Embeddings scores are computed?![Refer to caption](https://arxiv.org/html/2608.15325v1/Figures/fig_2_gram_configurations.png)Notes: The Figure shows how the Logical Embeddings score is computed for each possible entailment configuration in the case of an argument composed of two premises and one subconclusion\. Green nodes denote premises and purple nodes denote subconclusions\. A full node indicates a covered clause, a clause for which at least one mutual entailment is identified\. Only mutual entailments are represented by arrows\.To further illustrate the mechanics of the method, Figure[4](https://arxiv.org/html/2608.15325#A2.F4)shows exactly how a similarity score of0\.750\.75is obtained\. First, both texts are decomposed into premises and subconclusions\. First of all the subconclusions are judged mutually entailed\. Not making a vaccine mandatory implies that it is illegitimate to impose vaccination on individuals who oppose it for religious reasons and viceversa\. This already contributes0\.50\.5to the final score\.

Moving to the premises, the second premise of the first argument,forcing vaccination violates personal convictions, is judged to be entailed by both premises of the second argument, namely that mandatory vaccination infringes personal rights and that respecting bodily autonomy is fundamental for personal freedom\. By contrast, the first premise of the first argument,some people have strong religious beliefs against vaccines, is not entailed by either premise of the second argument, because it does not by itself imply bodily autonomy, nor does it imply a right to reject a vaccine mandate\. We therefore end up with two distinct premises overall, of which one is effectively shared across the two arguments through mutual entailment\. What matters here is that entailment creates a bridge between clauses: they are treated as similar when one logically entails the other and viceversa\. In this example, this means that one premise is shared, while the other remains unmatched\. The premise component is therefore equal to12\\tfrac\{1\}\{2\}, that contributes0\.250\.25weighted\. Adding this to the0\.50\.5of the subconclusion, the overall similarity score is0\.750\.75\.

Figure 4:Construction of a Logical Embedding![Refer to caption](https://arxiv.org/html/2608.15325v1/Figures/fig_3_exhibit_gram_77_vs_4612.png)

Notes:This figure illustrates how the Clause GRAM score is constructed for a pair of arguments\. Arguments 77 and 4612 are decomposed into clause\-level components \(two premises and one conclusion\)\. Green \>denote positive entailment judgments, while red ×denote negative ones\. In this example, the first premise of argument 77 is not bidirectionally entailed by either premise of argument 4612, whereas the second premise is bidirectionally entailed with both premises of argument 4612, producing a score of 0\.5\. The two conclusions are instead bidirectionally entailed, yielding a conclusion level score of 1\. With equal weight assigned to premise and conclusion components, the resulting Clause GRAM score is 0\.75\. The bottom panel reports the LLM reasoning used to determine whether each pair of clauses is entailed or not\.

### B\.2F1 Scores

In the main body of the paper we reported F1\-scores for the performance of logical embeddings on a standard classification task\. Here we provide additional results\. The logical embeddings used in the evaluation in the paper retain the top 100 eigenvectors of Gram matrix\. When all eigenvectors are retained, the neural network model used as one of the evaluation models suffers from severe overfitting due to the high effective dimensionality of the embedding space\. Figure[5](https://arxiv.org/html/2608.15325#A2.F5)illustrates this: test F1 decreases as more eigenvectors are included, while the train\-test gap widens\. 100 eigenvectors are chosen as the value that maximizes the F1\.

Figure 5:Neural network F1 score by number of eigenvectors![Refer to caption](https://arxiv.org/html/2608.15325v1/Figures/fig_5_eigenvalue_sweep_f1.png)

Notes:The Figure show the behaviour of the Neural Network model under changes of eigenvector retained\. The solid line reports the test F1 score of the neural network across different numbers of eigenvectors\. The shaded area represents the gap in F1 between the training and the test scores\.

### B\.3Practical implementation

In the paper, Theorem 3 establishes that

simσ⁡\(a,b\)=σ​kT​\(Π⁡\(a\),Π⁡\(b\)\)\+\(1−σ\)​kT​\(ρ⁡\(a\),ρ⁡\(b\)\)\\operatorname\{sim\}^\{\\sigma\}\(a,b\)=\\sigma\\,k\_\{T\}\\bigl\(\\Pi\(a\),\\Pi\(b\)\\bigr\)\+\(1\-\\sigma\)\\,k\_\{T\}\\bigl\(\\rho\(a\),\\rho\(b\)\\bigr\)is a positive semi\-definite kernel onArg⁡\(ℒ\)\\operatorname\{Arg\}\(\\mathcal\{L\}\), provided thatΠ\\Piandρ\\rhoare*fixed*maps from arguments to reduced representations\. The essential requirement is that the same reduction rule be applied to every argument independently of the particular pair under comparison\.

In the main empirical analysis, however, the implementation is pairwise\. For a given pair of arguments\(a,b\)\(a,b\), equivalence classes are constructed using only the clauses appearing in those two arguments\. Thus, instead of a single global reduction mapΠ⁡\(a\)=Φ​╱≡\\Pi\(a\)=\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\}$\}\}\}, the implementation uses a pair dependent map

Πa,b​\(a\)=Φ​╱​≡a,b,\\Pi\_\{a,b\}\(a\)=\{\\mathchoice\{\\raisebox\{3\.41666pt\}\{$\\displaystyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\displaystyle\{\\equiv\_\{a,b\}\}$\}\}\{\\raisebox\{3\.41666pt\}\{$\\textstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-2\.31873pt\}\{$\\textstyle\{\\equiv\_\{a,b\}\}$\}\}\{\\raisebox\{2\.39166pt\}\{$\\scriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.72855pt\}\{$\\scriptstyle\{\\equiv\_\{a,b\}\}$\}\}\{\\raisebox\{1\.6994pt\}\{$\\scriptscriptstyle\{\\Phi\}$\}\\mkern\-5\.0mu\\diagup\\mkern\-4\.0mu\\raisebox\{\-1\.287pt\}\{$\\scriptscriptstyle\{\\equiv\_\{a,b\}\}$\}\}\},\. Different entries of the Gram matrix are computed under different equivalence relations\. The fixed\-map argument underlying Theorem 3 therefore no longer applies, and positive semi\-definiteness is not guaranteed\.

However the pairwise estimator remains substantively the best way to compute the true similarity scores\. It is closer to the way a human evaluator would compare arguments, because it focuses only on the local logical relation between the two arguments at hand and it is also easier to trace since when two arguments are matched, one can directly inspect which premises and subconclusions were identified as equivalent within that particular comparison\. Empirically, this pairwise implementation yields the strongest performance\.

The main difficulty in constructing a PSD alternative is therefore not simply restoring a fixed map, but to do so without generating an excessively general global transitivity\. A naive global equivalence relation based on connected components in the bidirectional entailment graph creates a small number of extremely large components\. Due to the large number of arguments, and for LLM fluctuation in evaluations in these clusters, clauses may be reduced not because they are directly close in meaning, but because they are connected through long chains of bidirectional entailment\. The practical problem is therefore to reduce these giant global components into smaller and more coherent equivalence classes while keeping the map fixed \(and so all the properties\)\.

This global robustness implementation addresses this issue through a structural graph partition procedure\. First, we build the clause\-level graph induced by bidirectional entailment relations\. Each node is a clause, and an edge is present whenever two clauses are judged to be in bidirectional entailment\. We then compute the connected components of this graph\. Any component whose size already falls below a fixed thresholdMMis left unchanged \(that we put to beM=20M=20\)\.

However the big majority of connections falls into 4 big clusters that exceeds this threshold and where we don’t collapse it directly into one equivalence class\. Instead, we recursively partition it by a balanced spectral split\. To do so, we assign each edge a structural weight

w⁡\(i,j\)=1\+\|Γ⁡\(i\)∩Γ⁡\(j\)\|,w\(i,j\)=1\+\\lvert\\Gamma\(i\)\\cap\\Gamma\(j\)\\rvert,whereΓ⁡\(i\)\\Gamma\(i\)denotes the neighborhood of clauseiiin the entailment graph\. Hence, edges whose endpoints share many neighbors are treated as stronger and more internally supported, while edges with little local support are easier to cut\. On the weighted subgraph induced by the oversized component, we use the Fiedler vector of the Laplacian matrix to produce a bipartition\. The split is applied recursively on the median, chosen to be the one that creates the more balanced partition, were all components have the sameM=20M=20size\.

Only after this recursive reduction do we collapse clauses into equivalence classes\. That is, each final component of size at mostMMis treated as one global equivalence class and represented by a single node\. This yields fixed global reduction mapsΠG\\Pi^\{G\}andρG\\rho^\{G\}prior to any pairwise comparison\. The resulting similarity can therefore be written as

KσG​\(a,b\)=σ​kT​\(ΠG​\(a\),ΠG​\(b\)\)\+\(1−σ\)​kT​\(ρG​\(a\),ρG​\(b\)\)\.K^\{G\}\_\{\\sigma\}\(a,b\)=\\sigma\\,k\_\{T\}\\bigl\(\\Pi^\{G\}\(a\),\\Pi^\{G\}\(b\)\\bigr\)\+\(1\-\\sigma\)\\,k\_\{T\}\\bigl\(\\rho^\{G\}\(a\),\\rho^\{G\}\(b\)\\bigr\)\.SinceΠG\\Pi^\{G\}andρG\\rho^\{G\}are defined once and for all, independently of the particular pair\(a,b\)\(a,b\), Theorem 3 applies directly and the corresponding Gram matrix is positive semi\-definite by construction\.

While so this sacrifices local flexibility, since every clause must ultimately belong to a single global equivalence class, it imposes a much more disciplined reduction, because giant transitive components are recursively broken into bounded\-size subcomponents by a structural partition rule rather than being collapsed wholesale\.

Table 2:F1 scores \(100 eigenvectors\)Notes: This Table presents the main F1 scores of the paper\. All classifiers are estimated on a 70/30 train\-test split of 1137 arguments\. Logistic, Lasso and Ridge are linear models with L1, L1 and L2 penalties respectively\. Random Forest uses 200 trees\. The Neural Network is a single hidden layer network whose hyperparameters are selected via 5\-fold validation grid search with early stopping\. Logical emb\. refers to the clause\-level logical embedding using the top 100 eigenvectors of the pairwise entailment score matrix\. GloVe uses 100\-dimensional averaged word vectors\. BERT and RoBERTa use 768\-dimensional CLS token embeddings\. SBERT uses 384\-dimensional sentence embeddings\. GPT uses OpenAI text\-embedding\-3\-small \(1536 dimensions\)\. All embeddings are standardised before estimation\. Bold indicates the highest F1 per column\.#### B\.3\.1LLM prompts

This section reports the prompts used for the LLM\-based components of the analysis\. We use two distinct prompts\. The first prompt is a direct full\-argument similarity score between two arguments\. This prompt is used to construct the LLM\-based comparison Gram matrix and to generate the Figure[1](https://arxiv.org/html/2608.15325#S1.F1)\. The second prompt is the main one used in the construction of Logical Embeddings\. It asks the model to evaluate directed entailment between clause\-level statements\. Logical equivalence is then defined by mutual entailment\.

##### Full\-argument similarity prompt\.

The following prompt is used to obtain direct LLM\-based argument similarity scores\. The model receives two complete arguments and returns a continuous score between 0 and 1, together with a short explanation\.

Youareassessingthedegreetowhichtwoargumentssharethesameunderlyinglogicalcontent\.

Assignacontinuousscorefrom0to1reflectinghowmuchthelogicalcontentofthesetwoargumentsoverlap,where0meanstheargumentssharenologicalcontentwhatsoeverand1meanstheargumentsarelogicallyequivalent\.Keepitwithintwodecimalplaces\.

Important:focusonthelogicalstructureandinferentialcontent,notthetopicorsurfacewording\.

Considerthefollowingtwoarguments:

Argument1:\{text1\}

Argument2:\{text2\}

PleaseprovideyourresponseasavalidJSONobject:

\{

"sentence\_id\_1":"\{sentence\_id\_1\}",

"sentence\_id\_2":"\{sentence\_id\_2\}",

"answer":"N/A",

"score":<afloatbetween0and1\>,

"reasoning":"briefreasoningforyourscore",

"comment":"N/A"

\}

EnsuretheresponseisstrictlyavalidJSONobjectwithnoextracharactersorformatting\.

##### Clause\-level entailment prompt\.

The following prompt is used to evaluate directed entailment between clause\-level statements\. For each ordered pair of clauses, the model returns a YES/NO entailment judgment, a short reasoning, and a confidence score\. We run the prompt in both directions\. Two clauses are treated as logically equivalent only when both directed entailment judgments are positive\.

Youareworkingonassessingwhetherstatementsmadeincertaincontextsentailoneanother\.Innaturallanguage,logicalimplicationmakessenseincontext\.Forexample,ifastatementspeaksaboutarulerandanotheronespeaksaboutaking,theymaybetalkingaboutthesamefigureofauthority,ifbothstatementstalkaboutorwithinamonarchy\.

Question:DotheideasentailedinStatement1implytheideasentailedinStatement2?AnswerwithYESorNO,stateabriefreasoningforyouranswer,thengiveascorefrom1to10evaluatinghowconfidentyouarewithyouranswer\(1means’Iamabsolutelynotconfidentaboutmyanswer’and10means’Iamcompletelysureaboutmyassessment’\),andgiveabriefcommentonyourlevelofcertaintyonyouranswer\.

PleaseprovideyourresponseasavalidJSONobjectinthefollowingformat:

Pleaseconsiderthefollowingstatements:

Statement1:\{text1\}

Statement2:\{text2\}

\{

"sentence\_id\_1":"\{sentence\_id\_1\}",

"sentence\_id\_2":"\{sentence\_id\_2\}",

"answer":"YourYESorNOanswergoeshere\(asastring\)",

"reasoning":"Thereasoningbehindyouranswergoeshere\(asastring\)",

"score":"Ascorebetween0and10basedonyourconfidence\(asaninteger\)",

"comment":"Additionalcommentsonyourconfidencehere\(asastring\)"

\}

EnsuretheresponseisstrictlyavalidJSONobjectwithnoextracharactersorformatting\.