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Logical Embeddings for Argument Analysis
Leander Heldring, Santiago Torres
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 8/18/2026, 6:13:22 AM
Summary
The paper introduces 'logical embeddings' as a framework for argument analysis, replacing traditional contextualized word embeddings with encodings based on logical semantics. The authors define a Syntactic-Semantic Jaccard similarity measure that evaluates arguments based on premise overlap and logical consequences of conclusions. This measure is proven to be a positive semi-definite kernel, allowing the derivation of unique embeddings via Reproducing Kernel Hilbert Spaces (RKHS). The method is shown to preserve all logical information and outperform standard semantic embeddings (like BERT and GloVe) on classification tasks using the IBM-ArgQ-6.3kArgs dataset.
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Leander Heldring → affiliatedwith → Northwestern University
confidence 99% · Leander Heldring Affiliation: Kellogg School of Management Affiliation: Northwestern University
Santiago Torres → affiliatedwith → MIT
confidence 99% · Santiago Torres Affiliation: Department of Economics Affiliation: MIT
Logical Embeddings → uses → Syntactic-Semantic Jaccard
confidence 95% · The resulting similarity metric has numerous theoretical properties... We call this similarity measure the Syntactic-Semantic Jaccard measure.
Syntactic-Semantic Jaccard → induces → Reproducing Kernel Hilbert Spaces
confidence 92% · 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).
Logical Embeddings → evaluatedon → IBM-ArgQ-6.3kArgs
confidence 90% · we test the performance of logical embeddings on a standard classification task, using the IBM-ArgQ-6.3kArgs corpus
Logical Embeddings → outperforms → RoBERTa-Base
confidence 88% · Additionally, we demonstrate that logical embeddings outperform most standard embedding methods on a classification task.
Logical Embeddings → outperforms → GloVe
confidence 88% · Additionally, we demonstrate that logical embeddings outperform most standard embedding methods on a classification task.
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Abstract
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 with logical embeddings, an alternative encoding that directly exploits argumentation structures. In essence, logical embeddings encapsulate 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 embeddings can 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 that logical embeddings outperform most standard embedding methods on a classification task.
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- Source: https://arxiv.org/abs/2608.15325v1
- Canonical: https://arxiv.org/abs/2608.15325v1
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Logical Embeddings for Argument Analysis Leander Heldring Affiliation: Kellogg School of Management Affiliation: Northwestern University Affiliation: 2211 Campus Drive, Evanston, IL 60208 Email: leander.heldring@kellogg.northwestern.edu Santiago Torres Affiliation: Department of Economics Affiliation: MIT Affiliation: 50 Memorial Drive, Cambridge, MA 02142 Email: storresp@mit.edu 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 with logical embeddings, an alternative encoding that directly exploits argumentation structures. In essence, logical embeddings encapsulate 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 embeddings can 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 that logical embeddings outperform most standard embedding methods on a classification task. 1 Introduction The advent of contextualized word embeddings has revolutionized the field of Natural Language Processing (NLP) (27; 18; 11). 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, we report three arguments from the IBM-ArgQ-6.3kArgs dataset (26). For the argument at the top, we prompt gpt-4o-mini to 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. 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. 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) 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, 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). 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 embeddings provide 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 embeddings are optimal embeddings in 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; 23). 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). 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 to metricize logic. Following 5 and 10, 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), such a property defines unique embeddings of arguments via their premise-conclusion decomposition. This gives rise to logical embeddings. We further prove that, because of the desirable properties of the similarity metric that spans them, logical embeddings preserve all logical information. This stems from the fact that the Syntactic-Semantic Jaccard similarity measure is a characteristic kernel (7; 8), which guarantees an injective map from an argument to its embedding. As with other RKHS applications, logical embeddings can be used for many supervised and unsupervised tasks (see 22 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 of logical embeddings. Finally, we test the performance of logical embeddings on a standard classification task, using the IBM-ArgQ-6.3kArgs corpus (26). 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. 2 Logical 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. 2.1 Arguments as premise–conclusion pairs We work with a propositional logic (ℒ,⊢)(L, ), where ℒL is the language of formulas and ⊢ is the logical consequence relation. Appendix A.1 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 (¬)( ), conjunction (∧)( ), disjunction (∨)( ), implication (→)(→), and biconditionality (↔)( ). Thus, if p and q are atomic propositions, then p∧qp q, p→qp→ q, and ¬p∨q p q are formulas. Finally, a literal is an atomic proposition or its negation. The relation ⊢ denotes logical consequence. For Γ⊆ℒ and ϕ∈ℒφ , the expression Γ⊢ϕ φ means that every truth assignment satisfying all formulas in Γ also satisfies ϕφ. For formulas ϕ,ψ∈ℒφ,ψ , we say that ϕφ and ψ are logically equivalent, and write ϕ≡ψφ≡ψ, when they entail each other: ϕ⊢ψφ ψ and ψ⊢ϕψ φ. Here ϕ⊢ψφ ψ abbreviates ϕ⊢ψ\φ\ ψ. Equivalently, ϕ≡ψφ≡ψ if, and only if, ϕφ and ψ have the same truth conditions. We denote the equivalence class of ϕφ under ≡ by [ϕ]≡=ψ∈ℒ:ψ≡ϕ[φ]_≡=\ψ :ψ≡φ\. Definition 1 (Argument). An argument is a pair a=(Φ,ϕ),a=( ,φ), where Φ⊆ℒ is a finite set of premises and ϕ∈ℒφ is a conclusion such that Φ⊢ϕ φ. 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 deriving ϕφ. The set of all arguments is denoted Arg(ℒ)Arg(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 details the formal consistency and non-redundancy conditions. Section 3 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 formulas Φ , define [Φ]≡=[ξ]≡:ξ∈Φ,[ ]_≡=\[ξ]_≡:ξ∈ \, the set of equivalence classes represented by the formulas in Φ . Definition 2 (Argument equivalence). For arguments a=(Φ,ϕ)a=( ,φ) and b=(Ψ,ψ)b=( ,ψ), we write a≈ba≈ b if [Φ]≡=[Ψ]≡andϕ≡ψ.[ ]_≡=[ ]_≡ φ≡ψ. 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. 2.2 A 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 and 23. 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, 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→(q r) is logically equivalent to the CNF formula (¬p∨q)∧(¬p∨r).( p q) ( p r). This restriction loses no expressive power: every propositional formula is logically equivalent to some CNF formula (19). For a formula ϕφ, let CNℱ(ϕ)CN_F(φ) be its set of logical consequences in CNF form. This is the finite set of CNF formulas that are entailed by ϕφ, use only the literals relevant to ϕφ, and are listed only once up to logical equivalence.11 1 Appendix A.1 gives the formal construction of this set. For example, CNℱ(p∧q)=p,q,p∨q,p∧q.CN_F(p q)=\p,q,p q,p q\. Indeed, p∧qp q entails p, entails q, entails p∨qp q, and entails itself. In this case, the relevant literals are p and q, and the displayed set lists the distinct CNF consequences over those literals. Given two arguments a=(Φ,ϕ)a=( ,φ) and b=(Ψ,ψ)b=( ,ψ), define their premise similarity, or syntactic similarity, by the Jaccard overlap of their premise equivalence classes: ssyn(Φ,Ψ)=|[Φ]≡∩[Ψ]≡||[Φ]≡∪[Ψ]≡|,s_syn( , )= |[ ]_≡∩[ ]_≡||[ ]_≡∪[ ]_≡|, with the convention that the ratio is 11 when 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_sem(φ,ψ)= |CN_F(φ) _F(ψ)||CN_F(φ) _F(ψ)|, again with the convention that the ratio is 11 when both sets are empty. The syntactic–semantic Jaccard similarity of a and b is the convex combination simσ(a,b)=σssyn(Φ,Ψ)+(1−σ)ssem(ϕ,ψ),0<σ<1.sim^σ(a,b)=σ s_syn( , )+(1-σ)s_sem(φ,ψ), 0<σ<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 parameter σ controls the relative weight placed on shared premises versus shared conclusions. 5 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 states the full list of properties formally. The following result is a direct consequence of 1. Theorem 1 (Logical faithfulness). For any 0<σ<10<σ<1 and any arguments a,b∈Arg(ℒ)a,b (L), simσ(a,b)=1if and only ifa≈b.sim^σ(a,b)=1 and only if a≈ b. Theorem 1 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, simσsim^σ 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. 2.3 From 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 any 0<σ<10<σ<1, simσsim^σ is a positive semidefinite kernel on Arg(ℒ)Arg(L). By the Moore–Aronszajn theorem (3), the similarity measure simσsim^σ determines a reproducing kernel Hilbert space ℋσH_σ and a canonical feature map Θσ:Arg(ℒ)→ℋσ _σ:Arg(L) _σ such that, for all a,b∈Arg(ℒ)a,b (L), ⟨Θσ(a),Θσ(b)⟩ℋσ=simσ(a,b). _σ(a), _σ(b) _H_σ=sim^σ(a,b). We call Θσ(a) _σ(a) the logical embedding of the argument a. Appendix A.6 gives a brief review of reproducing kernel Hilbert spaces. This construction gives the embedding a direct interpretation. The vector Θσ(a) _σ(a) encodes the logical content of a: two arguments receive the same embedding exactly when they are equivalent under the premise–conclusion representation. This encoding is relational. The position of a in the Hilbert space is determined by how a relates to every other possible argument through the similarity function simσ(a,⋅)sim^σ(a,·). 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 arguments a1,…,ana_1,…,a_n, the formal construction defines its Gram matrix, that is, the matrix of pairwise similarities Kijσ=simσ(ai,aj)=⟨Θσ(ai),Θσ(aj)⟩ℋσ.K^σ_ij=sim^σ(a_i,a_j)= _σ(a_i), _σ(a_j) _H_σ. 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. Let K^σ K^σ denote the resulting empirical kernel matrix. We then apply kernel PCA to K^σ K^σ to obtain finite-dimensional coordinates for the observed arguments.22 2 Appendix A.7 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 ≈ receive distinct embeddings, whereas equivalent arguments receive the same embedding. The next theorem formalizes this property. Theorem 3 (No loss of logical information). For any 0<σ<10<σ<1, let ℋσH_σ be the RKHS induced by simσsim^σ, and let Θσ:Arg(ℒ)→ℋσ,Θσ(a)=simσ(⋅,a) _σ:Arg(L) _σ, _σ(a)=sim^σ(·,a) be the canonical feature map. Then, for any a,b∈Arg(ℒ)a,b (L), Θσ(a)=Θσ(b)if and only ifa≈b. _σ(a)= _σ(b) and only if a≈ b. Consequently, Θσ _σ induces an injective map Θ¯σ:Arg(ℒ)/≈→ℋσ,Θ¯σ([a]≈)=Θσ(a). _σ:Arg(L)/≈ _σ, _σ([a]_≈)= _σ(a). 3 NLP 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 using existing NLP and Machine Learning tools. In essence, we propose a three-step procedure to computing Syntactic-Semantic Jaccard similarity measures: 1. Training an entailment model. 2. Using a large language model (LLM) to decompose an argument into its constituent premises and conclusion. 3. Using an LLM to map a conclusion into its CNF. We further detail each step in detail in the upcoming sections. 3.1 Training an entailment model The key to establishing logical overlap is determining whether two propositions, p and q, are logically equivalent. This task can be further decomposed into establishing that p entails q and that q entails p. 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. 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 exceeding 0.90.9 in several studies (28; 17; 29). Consequently, logical entailment classification can be satisfactorily accomplished via zero-shot learning or fine-tuning of these models. 3.2 Argument 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; 16; 24). 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: Text Phase 2: You are a skilled logician. Given a text and its conclusion, extract its premises. Inputs: Text + Conclusion from Phase 1. 3.3 Conclusion 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 3.4 Putting everything together Given two arguments, a1a_1 and a2a_2, we can now compute their logic-based similarity using the previously explored tools. The process unfolds as follows: • Decompose a1a_1 and a2a_2 into their constituent premises and conclusions, (Φ1,ϕ1)( _1, _1) and (Φ2,ϕ2)( _2, _2). • Find the CNF representation of ϕ1 _1 and ϕ2 _2, and calculate CNℱ(ϕ1)CN_F( _1) and CNℱ(ϕ2)CN_F( _2). • Using the entailment model, calculate ssyn(Φ1,Φ2)s_syn( _1, _2) and ssem(ϕ1,ϕ2)s_sem( _1, _2) by assessing the overlap, modulo logical equivalence, between Φ1 _1 and Φ2 _2, and CNℱ(ϕ1)CN_F( _1) and CNℱ(ϕ2)CN_F( _2). • Fix a σ∈[0,1]σ∈[0,1] and aggregate into simσ(a1,a2)sim^σ(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 we provide an example of how similarity is constructed for a pair of arguments. 3.5 Performance evaluation We evaluate the performance of our embeddings on the standard IBM-ArgQ-6.3kArgs dataset (26). 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 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. 4 Conclusion In this paper we introduced and tested logical embeddings. We show that these are optimal embeddings and provide implementation details. Implementation code is available at https://github.com/lheldring/logical_embeddings. Table 1: F1 scores 100-dimensional logical embeddings Estimation: F1 Scores Logistic Lasso Ridge Neural Network Random Forest Logical emb. 0.842 0.883 0.869 0.884 0.857 GloVe 0.668 0.700 0.668 0.667 0.667 BERT 0.684 0.775 0.665 0.766 0.694 SBERT 0.674 0.761 0.670 0.781 0.760 RoBERTa 0.725 0.696 0.688 0.751 0.733 GPT 0.826 0.837 0.780 0.872 0.840 GloVe + Logical emb. 0.766 0.814 0.793 0.834 0.871 BERT + Logical emb. 0.844 0.838 0.693 0.803 0.866 SBERT + Logical emb. 0.787 0.845 0.757 0.860 0.871 RoBERTa + Logical emb. 0.836 0.868 0.752 0.825 0.870 GPT + Logical emb. 0.871 0.885 0.781 0.875 0.890 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. 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Ma Entailment as few-shot learner. External Links: 2104.14690 Cited by: §3.1. Zhang et al. (2020) Z. Zhang, Y. Wu, H. Zhao, Z. Li, S. Zhang, X. Zhou, and X. Zhou Semantics-aware bert for language understanding. External Links: 1909.02209 Cited by: §3.1. Appendix A Appendix / supplemental material A.1 Logical preliminaries This appendix gives the formal logical background used in Section 2. 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, 5, and 23. A.1.1 Sentential 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,…\ of atoms. Atoms are the basic units of the language. For example, an atom p may stand for a truth-valued statement such as “the policy reduces debt.” The language L specifies which atoms are available in the application. Definition 4 (Propositional connectives). The propositional connectives are ¬,∧,∨,→,↔, , , , →, , 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 by L, denoted ℒL, is the smallest set satisfying the following conditions: 1. if p∈Lp∈ L, then p∈ℒp ; 2. the truth constants ⊤ and ⊥ belong to ℒL; 3. if ϕ∈ℒφ , then ¬ϕ∈ℒ φ ; 4. if ϕ,ψ∈ℒφ,ψ , then (ϕ∧ψ),(ϕ∨ψ),(ϕ→ψ),(ϕ↔ψ)(φ ψ), (φ ψ), (φ→ψ), (φ ψ) are in ℒL. Thus, atoms are formulas, and formulas can be combined recursively to form more complex formulas. For instance, if p,q∈Lp,q∈ L, then p∧qp q, p→qp→ q, and ¬p∨q p q are 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\, where F denotes falsity and T denotes truth. Definition 6 (Valuation). A valuation is a map v:ℒ→F,Tv:L→\F,T\ that satisfies the usual truth-functional rules. In particular, v(⊤)=Tandv(⊥)=F,v( )=T v( )=F, v(¬ϕ)=Tif and only ifv(ϕ)=F,v( φ)=T and only if v(φ)=F, v(ϕ∧ψ)=Tif and only ifv(ϕ)=T and v(ψ)=T,v(φ ψ)=T and only if v(φ)=T and v(ψ)=T, v(ϕ∨ψ)=Tif and only ifv(ϕ)=T or v(ψ)=T,v(φ ψ)=T and only if v(φ)=T or v(ψ)=T, v(ϕ→ψ)=Tif and only ifv(ϕ)=F or v(ψ)=T,v(φ→ψ)=T and only if v(φ)=F or v(ψ)=T, and v(ϕ↔ψ)=Tif and only ifv(ϕ)=v(ψ).v(φ ψ)=T and only if v(φ)=v(ψ). 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,⊢)(L, ), where ℒL is the set of well-formed formulas generated by a sentential language L, and ⊢ is the semantic consequence relation. For Γ⊆ℒ and φ∈ℒ , we write Γ⊢φ if and only if every valuation that assigns T to all formulas in Γ also assigns T to φ . Fixing a logic (ℒ,⊢)(L, ) lets us define when two formulas have the same logical content. Definition 8 (Logical equivalence of formulas). Two formulas ϕ,ψ∈ℒφ,ψ are logically equivalent if ϕ⊢ψandψ⊢ϕ,φ ψ ψ φ, where ϕ⊢ψφ ψ abbreviates ϕ⊢ψ\φ\ ψ. In this case, we write ϕ≡ψφ≡ψ. 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 formula ϕ∈ℒφ , its equivalence class under logical equivalence is [ϕ]≡=ψ∈ℒ:ψ≡ϕ.[φ]_≡=\ψ :ψ≡φ\. 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 atom p∈Lp∈ L or its negation ¬p p. A clause is a finite disjunction of literals. A formula ϕ∈ℒφ is in conjunctive normal form (CNF) if it can be written as ϕ=⋀i=1mci,φ= _i=1^mc_i, where each clause cic_i has the form ci=⋁j=1niℓij,c_i= _j=1^n_i _ij, and each ℓij _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]. For example, p→(q∧r)≡(¬p∨q)∧(¬p∨r).p→(q r)≡( p q) ( p r). Following 2, 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). Let ℱ⊆ℒF . We say that ℱF is a finite CNF language for ℒL if, for every ϕ∈ℒφ , there exists a unique ψ∈ℱψ such that 1. ψ∈[ϕ]≡ψ∈[φ]_≡; 2. Lit(ψ)=Lit(ϕ)Lit(ψ)=Lit(φ); 3. ψ is in CNF. Thus, ℱF contains a unique CNF representative for each formula, up to logical equivalence. We write CNFℱ(ϕ)CNF_F(φ) for the unique element of ℱF that is logically equivalent to ϕφ and has the same literals as ϕφ. A.2 Logical arguments We now define the logical arguments used in the main text, following 4 and 1. 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 formulas Φ⊆ℒ is consistent if Φ⊬⊥. . It is inconsistent otherwise. Equivalently, Φ is consistent if there exists a valuation under which all formulas in Φ are true. We also need to compare premise sets up to logical equivalence. Definition 13 (Logical equivalence of sets of formulas). For Φ,Ψ⊆ℒ , , write [Φ]≡=[ϕ]≡:ϕ∈Φ.[ ]_≡=\[φ]_≡:φ∈ \. We say that Φ and Ψ are logically equivalent, written Φ≅Ψ , if [Φ]≡=[Ψ]≡.[ ]_≡=[ ]_≡. We now define the logical arguments used in the main text, following 4 and 1. 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,⊢)(L, ) is a pair a=(Φ,ϕ),a=( ,φ), where Φ⊆ℒ is finite, ϕ∈ℒφ , and: 1. Φ is consistent; 2. Φ⊢ϕ φ; 3. Φ is non-redundant, meaning that there is no Φ′⊊Φ such that Φ′⊢ϕ φ. The set Φ is the set of premises of a, and ϕφ is its conclusion. We write Prem(a)=ΦandConc(a)=ϕ.Prem(a)= (a)=φ. The set of all arguments over ℒL is denoted Arg(ℒ)Arg(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 arguments a,b∈Arg(ℒ)a,b (L) are equivalent, written a≈ba≈ b, if Prem(a)≅Prem(b)andConc(a)≡Conc(b).Prem(a) (b) (a) (b). A.3 Measuring 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 and 5. These measures compare arguments by comparing their premises and their conclusions, and they satisfy the rationality properties stated in Appendix A.5. We first need an operator for comparing sets of formulas modulo logical equivalence. Definition 16 (Common formulas operator). For sets of formulas Φ,Ψ⊆ℒ , , define Co(Φ,Ψ)=ϕ∈Φ: there exists ψ∈Ψ such that ϕ≡ψ.Co( , )=\φ∈ : there exists ψ∈ such that φ≡ψ\. The operator Co(Φ,Ψ)Co( , ) selects the formulas in Φ that also occur in Ψ , up to logical equivalence. Thus, Φ≅Ψ if and only if Co(Φ,Ψ)=ΦandCo(Ψ,Φ)=Ψ.Co( , )= ( , )= . We next define the consequence sets used to compare conclusions. Definition 17 (Logical consequences of a formula). For a formula ϕ∈ℒφ , define CN(ϕ)=ψ∈ℒ:ϕ⊢ψ.CN(φ)=\ψ :φ ψ\. This is the set of all logical consequences of ϕφ. The set CN(ϕ)CN(φ) is typically infinite. For example, p⊢p p, p⊢p∧p p p, p⊢(p∧p)∧p (p p) 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 which ϕφ depends. Let L±L^± denote the set of literals, that is, atoms and their negations. For a formula ϕφ, let Lit(ϕ)Lit(φ) denote the set of literals occurring in ϕφ after rewriting ϕφ in negation normal form. A formula ϕφ is independent of a literal ℓ∈L± ∈ L^± if there exists a formula χ∈ℒχ such that χ≡ϕχ≡φ and ℓ∉Lit(χ) (χ). Otherwise, ϕφ depends on ℓ . We write DepLit(ϕ)DepLit(φ) for the set of literals on which ϕφ depends. Fix a finite CNF representative language ℱ⊆ℒF . That is, ℱF 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 formula ϕ∈ℒφ , define CNℱ(ϕ)=ψ∈ℱ:ϕ⊢ψ and Lit(ψ)⊆DepLit(ϕ).CN_F(φ)=\ψ :φ ψ and Lit(ψ) (φ)\. Thus, CNℱ(ϕ)CN_F(φ) contains the CNF representatives of the logical consequences of ϕφ that use only literals on which ϕφ depends. The restriction to ℱF removes equivalent reformulations; the restriction to DepLit(ϕ)DepLit(φ) removes consequences involving irrelevant literals. Since the atom set is finite, CNℱ(ϕ)CN_F(φ) is finite. For example, CNℱ(p)=p,CNℱ(p∨q)=p∨q,CN_F(p)=\p\, _F(p q)=\p q\, and CNℱ(p∧q)=p,q,p∨q,p∧q.CN_F(p q)=\p,q,p q,p q\. The last equality holds because p∧qp q entails p, entails q, entails p∨qp q, and entails itself. These are the distinct representative consequences over the literals on which p∧qp q depends. 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,⊢)(L, ) is a map :Arg(ℒ)×Arg(ℒ)→[0,1].S:Arg(L)×Arg(L)→[0,1]. 5 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). Let a=(Φ,ϕ)a=( ,φ) and b=(Ψ,ψ)b=( ,ψ) be arguments in Arg(ℒ)Arg(L). For 0<σ<10<σ<1, define simσ(a,b)=σssyn(Φ,Ψ)+(1−σ)ssem(ϕ,ψ),sim^σ(a,b)=σ s_syn( , )+(1-σ)s_sem(φ,ψ), 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_syn( , )= cases |Co( , )|| |+| |-|Co( , )|&if ≠ and ≠ ,\\[5.69054pt] 1&if = = ,\\ 0&otherwise, cases and the conclusion similarity is ssem(ϕ,ψ)=|CNℱ(ϕ)∩CNℱ(ψ)||CNℱ(ϕ)∪CNℱ(ψ)|.s_sem(φ,ψ)= |CN_F(φ) _F(ψ)||CN_F(φ) _F(ψ)|. The measure is a convex combination of premise similarity and conclusion similarity. The term ssyns_syn measures how many premises the two arguments share, modulo logical equivalence. The term ssems_sem measures how many canonical consequences their conclusions share. The parameter σ controls the relative weight placed on premises and conclusions. Importantly, the value of ssems_sem is invariant to the particular choice of ℱF, as long as ℱF 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 of CNℱ(ϕ)CN_F(φ) without changing the cardinalities of the intersections and unions defining the quantity. A.4 Example 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. Let a1=(Φ1,ϕ1)a_1=( _1, _1) and a2=(Φ2,ϕ2)a_2=( _2, _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. array[]lp_1&=&``John is Susan's brother,′\\ p_2&=&``Susan is shorter than all her brothers,′\\ p_3&=&``Susan is the only girl in the household,′ array _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. array[]lq_1&=&``Susan is John's sister,′\\ q_2&=&``Susan is shorter than all her brothers,′\\ q_3&=&``No boy in the household has blond hair,′\\ q_4&=&``Susan is the only girl in the household,′\\ q_5&=&``There is at least one sibling with blond hair,′ array _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≡ q_1, p_2=q_2, p_3=q_4. Thus, Co(Φ1,Φ2)=p1,p2,p3.Co( _1, _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_syn( _1, _2)= |Co( _1, _2)|| _1|+| _2|-|Co( _1, _2)|= 33+5-3= 35. Now let r1=“Susan is the shortest sibling”andr2=“Susan is the only one with blond hair.”r_1=``Susan is the shortest sibling′ r_2=``Susan is the only one with blond hair.′ Then ϕ1=r1andϕ2=r1∧r2. _1=r_1 _2=r_1 r_2. Under the fixed CNF representative convention, CNℱ(ϕ1)=r1,CN_F( _1)=\r_1\, whereas CNℱ(ϕ2)=r1,r2,r1∨r2,r1∧r2.CN_F( _2)=\r_1,r_2,r_1 r_2,r_1 r_2\. The two conclusion consequence sets share only r1r_1. Hence, ssem(ϕ1,ϕ2)=|CNℱ(ϕ1)∩CNℱ(ϕ2)||CNℱ(ϕ1)∪CNℱ(ϕ2)|=14.s_sem( _1, _2)= |CN_F( _1) _F( _2)||CN_F( _1) _F( _2)|= 14. Combining the two terms gives, for 0<σ<10<σ<1, simσ(a1,a2)=σ(35)+(1−σ)(14)=14+720σ.sim^σ(a_1,a_2)=σ ( 35 )+(1-σ) ( 14 )= 14+ 720σ. A.5 Formal definitions of the properties satisfied by similarity measures We state the relevant properties for an arbitrary similarity measure S:Arg(ℒ)×Arg(ℒ)→[0,1]S:Arg(L)×Arg(L)→[0,1]. 5 proves that the syntactic–semantic Jaccard similarity satisfies each property below. Property 1 (Maximality). A similarity measure S satisfies maximality if, for every a∈Arg(ℒ)a (L), S(a,a)=1.S(a,a)=1. 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 value 11. Property 2 (Symmetry). A similarity measure S satisfies symmetry if, for all a,b∈Arg(ℒ)a,b (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 measure S satisfies the triangle inequality if, for all a,b,c∈Arg(ℒ)a,b,c (L), 1+S(a,c)≥S(a,b)+S(b,c).1+S(a,c)≥ S(a,b)+S(b,c). This condition is the triangle inequality written in similarity form. Equivalently, the dissimilarity d(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)≤ d(a,b)+d(b,c). Thus, if a is close to b and b is close to c, then a cannot be arbitrarily far from c. Property 4 (Substitution). A similarity measure S satisfies substitution if, for all a,b,c∈Arg(ℒ)a,b,c (L), S(a,b)=1⟹S(a,c)=S(b,c).S(a,b)=1 S(a,c)=S(b,c). Substitution says that maximally similar arguments are interchangeable for similarity comparisons with any third argument. For a formula ϕφ, let At(ϕ)At(φ) denote the set of atoms occurring in ϕφ. For a finite set of formulas Φ , define At(Φ)=⋃ϕ∈ΦAt(ϕ).At( )= _φ∈ At(φ). Property 5 (Minimality). A similarity measure S satisfies minimality if, for all a=(Φ,ϕ)a=( ,φ) and b=(Ψ,ψ)b=( ,ψ) in Arg(ℒ)Arg(L), the following conditions imply S(a,b)=0S(a,b)=0: 1. a≉ba ≈ b; 2. At(Φ)∩At(Ψ)=∅At( ) ( )= ; 3. At(ϕ)∩At(ψ)=∅At(φ) (ψ)= . 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 measure S satisfies non-zero similarity if, for all a=(Φ,ϕ)a=( ,φ) and b=(Ψ,ψ)b=( ,ψ) in Arg(ℒ)Arg(L), Co(Φ,Ψ)≠∅⟹S(a,b)>0.Co( , )≠ 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 measure S satisfies monotony if, for all a=(Φ,ϕ)a=( ,φ), b=(Ψ,ψ)b=( ,ψ), and c=(Ξ,ξ)c=( ,ξ) in Arg(ℒ)Arg(L), the following conditions imply S(a,b)≥S(a,c)S(a,b)≥ S(a,c): 1. ϕ≡ψφ≡ψ or At(ϕ)∩At(ξ)=∅At(φ) (ξ)= ; 2. Co(Φ,Ξ)⊆Co(Φ,Ψ)Co( , ) ( , ); 3. Ψ∖Co(Ψ,Φ)=Co(Ψ∖Co(Ψ,Φ),Ξ∖Co(Ξ,Φ)). ( , )=Co ( ( , ), ( , ) ). It satisfies strict monotony if, in addition, either Co(Φ,Ξ)⊊Co(Φ,Ψ),Co( , ) ( , ), or Co(Φ,Ξ)≠∅and|Ξ∖Co(Ξ,Φ)|>|Ψ∖Co(Ψ,Φ)|.Co( , )≠ | ( , )|>| ( , )|. 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 to a as it shares more of a’s premises and adds fewer unrelated premises. Property 8 (Strict dominance). A similarity measure S satisfies dominance if, for all a=(Φ,ϕ)a=( ,φ), b=(Ψ,ψ)b=( ,ψ), and c=(Ξ,ξ)c=( ,ξ) in Arg(ℒ)Arg(L), the following conditions imply S(a,b)≥S(a,c)S(a,b)≥ S(a,c): 1. Ψ≅Ξ ; 2. CNℱ(ϕ)∩CNℱ(ξ)⊆CNℱ(ϕ)∩CNℱ(ψ);CN_F(φ) _F(ξ) _F(φ) _F(ψ); 3. CNℱ(ψ)∖CNℱ(ϕ)⊆CNℱ(ξ)∖CNℱ(ϕ).CN_F(ψ) _F(φ) _F(ξ) _F(φ). It satisfies strict dominance if, in addition, either the inclusion in condition 2 is strict, or CNℱ(ϕ)∩CNℱ(ξ)≠∅CN_F(φ) _F(ξ)≠ and 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.6 More information on Reproducing Kernel Hilbert Spaces Let X be a set. A function K:X×X→ℝK:X× X is a positive semi-definite kernel if, for every finite collection x1,…,xn∈Xx_1,…,x_n∈ X, the Gram matrix =(K(xi,xj))i,j=1nK= (K(x_i,x_j) )_i,j=1^n is positive semi-definite. Equivalently, for every c1,…,cn∈ℝc_1,…,c_n , ∑i=1n∑j=1ncicjK(xi,xj)≥0. _i=1^n _j=1^nc_ic_jK(x_i,x_j)≥ 0. A reproducing kernel Hilbert space (RKHS) associated with K is a Hilbert space ℋKH_K of real-valued functions on X such that, for every x∈Xx∈ X, the function K(⋅,x):X→ℝK(·,x):X belongs to ℋKH_K, and the reproducing property holds: h(x)=⟨h,K(⋅,x)⟩ℋKfor all h∈ℋK.h(x)= h,K(·,x) _H_K all h _K. The Moore–Aronszajn theorem states that every positive semi-definite kernel K uniquely determines such an RKHS ℋKH_K [3, 15]. The associated canonical feature map is ΘK:X→ℋK,ΘK(x)=K(⋅,x). _K:X _K, _K(x)=K(·,x). Thus, each object x∈Xx∈ X is represented by the function that records its kernel similarity to every other element of X. In this sense, ΘK _K is an embedding: it represents each element of X as a point in a Hilbert space whose inner products are given by K. The key identity is ⟨ΘK(x),ΘK(y)⟩ℋK=K(x,y), _K(x), _K(y) _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 representations ΘK(x) _K(x) and ΘK(y) _K(y), even when these representations are not written explicitly or are infinite-dimensional. A.7 Finite-dimensional approximation by kernel PCA The logical embedding defined in Section 2 maps each argument a to an element Θσ(a)∈ℋσ _σ(a) _σ of the RKHS induced by the kernel simσsim^σ. 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). _σ(a)(·)=sim^σ(·,a). Thus, the embedding of a is a function that records the similarity of a to every possible argument. On a finite corpus a1,…,ana_1,…,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)). (sim^σ(a_1,a),…,sim^σ(a_n,a) )=( _σ(a)(a_1),·s, _σ(a)(a_n)). By restricting to the argument corpus, these restricted feature functions form the columns of the Gram matrix Kijσ=simσ(ai,aj).K^σ_ij=sim^σ(a_i,a_j). where each column j is then a finite-dimensional approximation to the logical embedding of argument j. When n is 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-r linear approximation to the centered logical embeddings on the observed corpus. In practice, we do not observe KσK^σ exactly. We estimate it from natural-language text, obtaining K^σ K^σ. Kernel PCA applied to K^σ K^σ therefore provides a finite-dimensional approximation to the ideal logical embeddings, with two sources of approximation: the empirical similarity estimates and the rank-r PCA truncation. We now state the finite-sample construction for this approximation. Let K:X×X→ℝK:X× X be a positive semidefinite kernel on a set X, with RKHS ℋKH_K, and let x1,…,xn∈Xx_1,…,x_n∈ X. Define the Gram matrix =(K(xi,xj))i,j=1n.K=(K(x_i,x_j))_i,j=1^n. Let =In−1n⊤andc=H=I_n- 1n11 _c=HKH be the centering matrix and the centered Gram matrix. Let ΘK(x)=K(⋅,x) _K(x)=K(·,x) be the canonical RKHS feature map, and define the centered embeddings Zi=ΘK(xi)−1n∑j=1nΘK(xj).Z_i= _K(x_i)- 1n _j=1^n _K(x_j). The centered Gram matrix records exactly the inner products among these centered embeddings: (c)ij=⟨Zi,Zj⟩ℋK.(K_c)_ij= Z_i,Z_j _H_K. Thus, kernel PCA performs ordinary PCA on the centered RKHS embeddings Z1,…,ZnZ_1,…,Z_n, using only the finite matrix cK_c. Let c=UΛU⊤K_c=U U be an eigendecomposition, with eigenvalues λ1≥λ2≥⋯≥λn≥0 _1≥ _2≥·s≥ _n≥ 0 and orthonormal eigenvectors u1,…,unu_1,…,u_n. For each λℓ>0 _ >0, the ℓ empirical principal direction in ℋKH_K is v^ℓ=1λℓ∑j=1n(uℓ)jZj. v_ = 1 _ _j=1^n(u_ )_jZ_j. The coordinate of xix_i along this direction is ⟨Zi,v^ℓ⟩ℋK=λℓ(uℓ)i. Z_i, v_ _H_K= _ (u_ )_i. Therefore, the rank-r kernel PCA coordinates are given by the rows of UrΛr1/2,U_r _r^1/2, where UrU_r contains the first r eigenvectors and Λr _r contains the corresponding eigenvalues. These coordinates have the standard PCA optimality property. Among all rank-r orthogonal projections P on ℋKH_K, the projection onto spanv^1,…,v^rspan\ v_1,…, v_r\ minimizes the average squared reconstruction error: minrank(P)≤r1n∑i=1n‖Zi−PZi‖ℋK2=∑ℓ>rλℓn. _rank(P)≤ r 1n _i=1^n\|Z_i-PZ_i\|_H_K^2= _ >r _ n. Equivalently, by the Eckart–Young–Mirsky theorem, the rank-r truncation UrΛrUr⊤U_r _rU_r is the best rank-r approximation to cK_c in Frobenius norm: minrank(M)≤r‖c−M‖F2=∑ℓ>rλℓ2. _rank(M)≤ r\|K_c-M\|_F^2= _ >r _ ^2. Applying this construction with X=Arg(ℒ)X=Arg(L) and K=simσK=sim^σ shows that kernel PCA on the ideal Gram matrix KσK^σ gives the best r-dimensional linear approximation to the centered logical embeddings of the observed arguments. In practice, we use K^σ K^σ in place of KσK^σ. When K^σ K^σ is a good approximation to KσK^σ, 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; see also 20. The best rank-r matrix approximation property follows from 6. A.8 Omitted proofs A.8.1 Auxiliary Lemmas Given a set X and an equivalence relation ∼ on X, the quotient of X by ∼ is X╱∼:=[x]∼:x∈X, 3.41666pt$ X$ -5.0mu -4.0mu -1.83437pt$ $ 3.41666pt$ X$ -5.0mu -4.0mu -1.83437pt$ $ 2.39166pt$ X$ -5.0mu -4.0mu -1.34427pt$ $ 1.70833pt$ X$ -5.0mu -4.0mu -0.99101pt$ $:=\[x]_ :x∈ X\, where [x]∼=y∈X:y∼x[x]_ =\y∈ X:y x\. For Φ⊆ℒ , write Φ╱≡:=[ξ]≡:ξ∈Φ, 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$:=\[ξ]_≡:ξ∈ \, where [ξ]≡=η∈ℒ:η≡ξ[ξ]_≡=\η :η≡ξ\ is the equivalence class of ξ in the ambient language ℒL. Lemma 1. Let a=(Φ,ϕ)∈Arg(ℒ)a=( ,φ) (L). If ξ,ζ∈Φξ,ζ∈ and ξ≡ζξ≡ζ, then ξ=ζξ=ζ. Proof. Suppose, toward a contradiction, that ξ,ζ∈Φξ,ζ∈ , ξ≠ζξ≠ζ, and ξ≡ζξ≡ζ. Let Φ′=Φ∖ξ = \ξ\. We show that Φ′⊢ϕ φ. Let v be any valuation that satisfies every formula in Φ′ . Since ζ∈Φ′ζ∈ , v satisfies ζ. Because ζ≡ξζ≡ξ, every valuation satisfying ζ also satisfies ξ. Hence v satisfies ξ. Therefore v satisfies every formula in Φ . Since Φ⊢ϕ φ, it follows that v satisfies ϕφ. Thus every valuation satisfying Φ′ also satisfies ϕφ, so Φ′⊢ϕ φ. But Φ′⊊Φ , contradicting the non-redundancy of Φ . Therefore ξ=ζξ=ζ. ∎ Lemma 2. Let a=(Φ,ϕ)a=( ,φ) and b=(Ψ,ψ)b=( ,ψ) be arguments in Arg(ℒ)Arg(L). Then the map π:Co(Φ,Ψ)→Φ╱≡∩Ψ╱≡,ξ↦[ξ]≡,π:Co( , )→ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∩ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, ξ [ξ]_≡, is a bijection. In particular, |Co(Φ,Ψ)|=|Φ╱≡∩Ψ╱≡|.|Co( , )|= | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∩ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ |. Proof. First, the map is well-defined. If ξ∈Co(Φ,Ψ)ξ ( , ), then ξ∈Φξ∈ and there exists ζ∈Ψζ∈ such that ξ≡ζξ≡ζ. Hence [ξ]≡∈Φ╱≡[ξ]_≡∈ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ and [ξ]≡=[ζ]≡∈Ψ╱≡[ξ]_≡=[ζ]_≡∈ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$. Therefore [ξ]≡∈Φ╱≡∩Ψ╱≡[ξ]_≡∈ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∩ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$. We now prove surjectivity. Let C∈Φ╱≡∩Ψ╱≡.C∈ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∩ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$. Since C∈Φ╱≡C∈ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, there exists ξ∈Φξ∈ such that C=[ξ]≡C=[ξ]_≡. Since C∈Ψ╱≡C∈ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, there exists ζ∈Ψζ∈ such that C=[ζ]≡C=[ζ]_≡. Hence ξ≡ζξ≡ζ, so ξ∈Co(Φ,Ψ)ξ ( , ). Moreover, π(ξ)=[ξ]≡=Cπ(ξ)=[ξ]_≡=C. Thus π is surjective. We now prove injectivity. Let ξ,ξ′∈Co(Φ,Ψ)ξ,ξ ( , ) and suppose that π(ξ)=π(ξ′)π(ξ)=π(ξ ). Then [ξ]≡=[ξ′]≡,[ξ]_≡=[ξ ]_≡, so ξ≡ξ′ξ≡ξ . Since ξ,ξ′∈Φξ,ξ ∈ and Φ is the premise set of an argument, Lemma 1 implies that ξ=ξ′ξ=ξ . Therefore, π is injective. Hence π is bijective. ∎ Lemma 3. Let a=(Φ,ϕ)a=( ,φ) be an argument in Arg(ℒ)Arg(L). Then |Φ|=|Φ╱≡|.| |= | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ |. Proof. Consider the map π:Φ→Φ╱≡,ξ↦[ξ]≡.π: → 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, ξ [ξ]_≡. This map is surjective by definition of Φ╱≡ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$. We now show that it is injective. Let ξ,ζ∈Φξ,ζ∈ and suppose that π(ξ)=π(ζ)π(ξ)=π(ζ). Then [ξ]≡=[ζ]≡,[ξ]_≡=[ζ]_≡, so ξ≡ζξ≡ζ. Since Φ is the premise set of an argument, Lemma 1 implies that ξ=ζξ=ζ. Hence π is injective. Therefore π is a bijection, and consequently |Φ|=|Φ╱≡|.| |= | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ |. ∎ A.8.2 Proof of Theorem 2 Proof. Let a=(Φ,ϕ)a=( ,φ) and b=(Ψ,ψ)b=( ,ψ) be arguments in Arg(ℒ)Arg(L). For any set X, write fin(X)=A⊆X:|A|<∞P_fin(X)=\A X:|A|<∞\ for the set of finite subsets of X. 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≠∅. k_T(A,B)= |A∩ B||A∪ B|, A∪ B≠ . This is the Tanimoto kernel on finite sets, also known as the Jaccard kernel [14, 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)= cases k_T(A,B),&if A∪ B≠ ,\\[11.99998pt] 1,&if A=B= . cases We first verify that this extension is still positive semi-definite. Let A1,…,AnA_1,…,A_n be finite sets. After permutation, we may assume without loss of generality that A1=⋯=Am=∅A_1=·s=A_m= and that Am+1,…,AnA_m+1,…,A_n are nonempty. If m=0m=0, all sets are nonempty, and the result follows directly from the positive semi-definiteness of the usual Tanimoto kernel. If m=nm=n, then the Gram matrix is the all-ones matrix, which is positive semi-definite. Thus assume 0<m<n0<m<n. For i≤m<ji≤ m<j, we have kT(Ai,Aj)=k~T(∅,Aj)=|∅∩Aj||∅∪Aj|=0.k_T(A_i,A_j)= k_T( ,A_j)= | ∩ A_j|| ∪ A_j|=0. Therefore the Gram matrix (kT(Ai,Aj))i,j=1n (k_T(A_i,A_j) )_i,j=1^n has the block form (mm⊤00G), pmatrix1_m1_m &0\\ 0&G pmatrix, where mm⊤1_m1_m is the m×m× m all-ones matrix and G=(k~T(Ai,Aj))i,j=m+1nG= ( k_T(A_i,A_j) )_i,j=m+1^n is the Gram matrix of the usual Tanimoto kernel on the nonempty sets Am+1,…,AnA_m+1,…,A_n. The first block is positive semi-definite because, for any u∈ℝmu ^m, u⊤(mm⊤)u=(m⊤u)2≥0.u (1_m1_m )u=(1_m u)^2≥ 0. The second block is positive semi-definite by the positive semi-definiteness of the Tanimoto kernel [14, p. 301]. Hence the block-diagonal Gram matrix is positive semi-definite. Therefore kTk_T is a positive semi-definite kernel on fin(X)P_fin(X), for any ambient set X. We now rewrite the syntactic part of the similarity. By Lemmas 2 and 3, |Co(Φ,Ψ)|=|Φ╱≡∩Ψ╱≡|,|Co( , )|= | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∩ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ |, and |Φ|=|Φ╱≡|,|Ψ|=|Ψ╱≡|.| |= | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ |, | |= | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ |. Therefore, whenever Φ╱≡∪Ψ╱≡≠∅ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∪ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$≠ , inclusion– exclusion gives ssyn(Φ,Ψ)=|Φ╱≡∩Ψ╱≡||Φ╱≡∪Ψ╱≡|=kT(Φ╱≡,Ψ╱≡).s_syn( , )= | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∩ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ | | 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$∪ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ |=k_T ( 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ ). If Φ╱≡=Ψ╱≡=∅ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$= 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$= , then Φ=Ψ=∅ = = , and both sides are equal to 11 by the empty-set convention for kTk_T. Hence, in all cases, ssyn(Φ,Ψ)=kT(Φ╱≡,Ψ╱≡).s_syn( , )=k_T ( 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$ ). Similarly, by definition of the semantic component, ssem(ϕ,ψ)=kT(CNℱ(ϕ),CNℱ(ψ)),s_sem(φ,ψ)=k_T (CN_F(φ),CN_F(ψ) ), again using the convention that the similarity of two empty consequence sets is 11. Now define Π:Arg(ℒ)→fin(ℒ╱≡),Π(Φ,ϕ)=Φ╱≡, :Arg(L) _fin ( 3.41666pt$ L$ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ L$ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ L$ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ L$ -5.0mu -4.0mu -1.287pt$ ≡$ ), ( ,φ)= 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, and ρ:Arg(ℒ)→fin(ℱ),ρ(Φ,ϕ)=CNℱ(ϕ).ρ:Arg(L) _fin(F), ρ( ,φ)=CN_F(φ). These maps are well-defined because premise sets are finite and ℱF is finite. Define KΠ(a,b)=kT(Π(a),Π(b))K_ (a,b)=k_T( (a), (b)) and Kρ(a,b)=kT(ρ(a),ρ(b)).K_ρ(a,b)=k_T(ρ(a),ρ(b)). Since kTk_T is positive semi-definite on finite subsets, KΠK_ and KρK_ρ are positive semi-definite kernels on Arg(ℒ)Arg(L) by closure of positive semi-definite kernels under pullback along arbitrary maps [3, 15]. Finally, for any 0<σ<10<σ<1, simσ(a,b)=σKΠ(a,b)+(1−σ)Kρ(a,b).sim^σ(a,b)=σ K_ (a,b)+(1-σ)K_ρ(a,b). Positive semi-definite kernels are closed under nonnegative scalar multiplication and finite sums [3, 15]. Since σ>0σ>0 and 1−σ>01-σ>0, it follows that simσsim^σ is a positive semi-definite kernel on Arg(ℒ)Arg(L). ∎ A.8.3 Proof of Theorem 3 Proof. Let a,b∈Arg(ℒ)a,b (L). Since Θσ _σ is the canonical feature map associated with simσsim^σ, we have ⟨Θσ(x),Θσ(y)⟩ℋσ=simσ(x,y) _σ(x), _σ(y) _H_σ=sim^σ(x,y) for all x,y∈Arg(ℒ)x,y (L). First suppose that Θσ(a)=Θσ(b) _σ(a)= _σ(b). Then 0=‖Θσ(a)−Θσ(b)‖ℋσ2.0=\| _σ(a)- _σ(b)\|_H_σ^2. Expanding the squared norm gives 0=simσ(a,a)+simσ(b,b)−2simσ(a,b).0=sim^σ(a,a)+sim^σ(b,b)-2sim^σ(a,b). Since each argument has maximal similarity with itself, simσ(a,a)=simσ(b,b)=1.sim^σ(a,a)=sim^σ(b,b)=1. Therefore simσ(a,b)=1.sim^σ(a,b)=1. By Theorem 1, this implies a≈ba≈ b. Conversely, suppose that a≈ba≈ b. By Theorem 1, simσ(a,b)=1.sim^σ(a,b)=1. Again using simσ(a,a)=simσ(b,b)=1sim^σ(a,a)=sim^σ(b,b)=1, we get ‖Θσ(a)−Θσ(b)‖ℋσ2=1+1−2=0.\| _σ(a)- _σ(b)\|_H_σ^2=1+1-2=0. Hence Θσ(a)=Θσ(b) _σ(a)= _σ(b). Thus Θσ(a)=Θσ(b)if and only ifa≈b. _σ(a)= _σ(b) and only if a≈ b. It remains only to check that the quotient map is well-defined. If [a]≈=[b]≈[a]_≈=[b]_≈, then a≈ba≈ b, and the result just proved implies Θσ(a)=Θσ(b). _σ(a)= _σ(b). Therefore Θ¯σ([a]≈)=Θσ(a) _σ([a]_≈)= _σ(a) is well-defined. Moreover, if Θ¯σ([a]≈)=Θ¯σ([b]≈), _σ([a]_≈)= _σ([b]_≈), then Θσ(a)=Θσ(b) _σ(a)= _σ(b), so a≈ba≈ b, and hence [a]≈=[b]≈.[a]_≈=[b]_≈. Therefore Θ¯σ _σ is injective. ∎ Appendix B Exhibits B.1 Features 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. 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 illustrates this point using only the subset of arguments under the doping topic from from the IBM-ArgQ-6.3kArgs dataset [26]. 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, with 8959889598 edges, covering 98.5%98.5\% of all possible pairs. The Logical Embeddings graph, in contrast, contains only 64086408 edges, or 7.0%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. 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 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 contributes 0.5∗1=0.50.5*1=0.5 to 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 scores 0.250.25 and 0.50.5 obtained using premises only, the number of mutual entailments is the same, but in the second case all clauses are connected, which gives a score of 0.5×1/10.5× 1/1, while in the first case the score is only 0.5×1/20.5× 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? 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 shows exactly how a similarity score of 0.750.75 is 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 contributes 0.50.5 to 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 to 12 12, that contributes 0.250.25 weighted. Adding this to the 0.50.5 of the subconclusion, the overall similarity score is 0.750.75. Figure 4: Construction of a Logical Embedding 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.2 F1 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 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 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.3 Practical implementation In the paper, Theorem 3 establishes that simσ(a,b)=σkT(Π(a),Π(b))+(1−σ)kT(ρ(a),ρ(b))sim^σ(a,b)=σ\,k_T ( (a), (b) )+(1-σ)\,k_T (ρ(a),ρ(b) ) is a positive semi-definite kernel on Arg(ℒ)Arg(L), provided that Π and ρ are 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 map Π(a)=Φ╱≡ (a)= 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ ≡$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ ≡$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ ≡$, the implementation uses a pair dependent map Πa,b(a)=Φ╱≡a,b, _a,b(a)= 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ _a,b$ 3.41666pt$ $ -5.0mu -4.0mu -2.31873pt$ _a,b$ 2.39166pt$ $ -5.0mu -4.0mu -1.72855pt$ _a,b$ 1.6994pt$ $ -5.0mu -4.0mu -1.287pt$ _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 threshold M is left unchanged (that we put to be M=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+ (i)∩ (j) , where Γ(i) (i) denotes the neighborhood of clause i in 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 same M=20M=20 size. Only after this recursive reduction do we collapse clauses into equivalence classes. That is, each final component of size at most M is treated as one global equivalence class and represented by a single node. This yields fixed global reduction maps ΠG ^G and ρGρ^G prior to any pairwise comparison. The resulting similarity can therefore be written as KσG(a,b)=σkT(ΠG(a),ΠG(b))+(1−σ)kT(ρG(a),ρG(b)).K^G_σ(a,b)=σ\,k_T ( ^G(a), ^G(b) )+(1-σ)\,k_T (ρ^G(a),ρ^G(b) ). Since ΠG ^G and ρGρ^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) Estimation: F1 Scores Logistic Lasso Ridge Neural Network Random Forest Logical emb. 0.848 0.851 0.853 0.840 0.829 GloVe 0.668 0.700 0.668 0.667 0.667 BERT 0.667 0.775 0.665 0.766 0.694 SBERT 0.674 0.761 0.670 0.781 0.760 RoBERTa 0.723 0.696 0.688 0.751 0.733 GPT 0.832 0.837 0.780 0.871 0.843 GloVe + Logical emb. 0.761 0.820 0.819 0.828 0.837 BERT + Logical emb. 0.837 0.833 0.748 0.824 0.799 SBERT + Logical emb. 0.768 0.855 0.771 0.768 0.833 RoBERTa + Logical emb. 0.838 0.851 0.728 0.806 0.811 GPT + Logical emb. 0.864 0.880 0.807 0.879 0.880 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.3.1 LLM 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. 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. ⬇ You are assessing the degree to which two arguments share the same underlying logical content. Assign a continuous score from 0 to 1 reflecting how much the logical content of these two arguments overlap, where 0 means the arguments share no logical content whatsoever and 1 means the arguments are logically equivalent. Keep it within two decimal places. Important: focus on the logical structure and inferential content, not the topic or surface wording. Consider the following two arguments: Argument 1: text1 Argument 2: text2 Please provide your response as a valid JSON object: "sentence_id_1": "sentence_id_1", "sentence_id_2": "sentence_id_2", "answer": "N/A", "score": <a float between 0 and 1>, "reasoning": "brief reasoning for your score", "comment": "N/A" Ensure the response is strictly a valid JSON object with no extra characters or formatting. 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. ⬇ You are working on assessing whether statements made in certain contexts entail one another. In natural language, logical implication makes sense in context. For example, if a statement speaks about a ruler and another one speaks about a king, they may be talking about the same figure of authority, if both statements talk about or within a monarchy. Question: Do the ideas entailed in Statement 1 imply the ideas entailed in Statement 2? Answer with YES or NO, state a brief reasoning for your answer, then give a score from 1 to 10 evaluating how confident you are with your answer (1 means ’I am absolutely not confident about my answer’ and 10 means ’I am completely sure about my assessment’), and give a brief comment on your level of certainty on your answer. Please provide your response as a valid JSON object in the following format: Please consider the following statements: Statement 1: text1 Statement 2: text2 "sentence_id_1": "sentence_id_1", "sentence_id_2": "sentence_id_2", "answer": "Your YES or NO answer goes here (as a string)", "reasoning": "The reasoning behind your answer goes here (as a string)", "score": "A score between 0 and 10 based on your confidence (as an integer)", "comment": "Additional comments on your confidence here (as a string)" Ensure the response is strictly a valid JSON object with no extra characters or formatting.