Learning First-Order Logic Rules for Argumentation Mining
Authors: Yang Sun, Guanrong Chen, Hamid Alinejad-Rokny, Jianzhu Bao, Yuqi Huang, Bin Liang, Kam-Fai Wong, Min Yang, Ruifeng Xu
Accepted at: acl25

Learning First-Order Logic Rules for Argumentation Mining

Yang Sun1,2*, Guanrong Chen1,5*, Hamid Alinejad-Rokny6, Jianzhu Bao1,2,
Yuqi Huang1, Bin Liang1,4, Kam-Fai Wong,4 Min Yang3† and Ruifeng Xu1,2,5†
1 Harbin Institute of Technology, Shenzhen, China 2 Peng Cheng Laboratory, Shenzhen, China
3 SIAT, Chinese Academy of Sciences, Shenzhen, China 4 The Chinese University of Hong Kong
5 Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies
6 School of Biomedical Engineering, UNSW Sydney
yang.sun@stu.hit.edu.cn, 23S051030@stu.hit.edu.cn

Abstract

Argumentation Mining (AM) aims to extract argumentative structures from texts by identifying argumentation components (ACs) and their argumentative relations (ARs). While previous works focus on representation learning to encode ACs and AC pairs, they fail to explicitly model the underlying reasoning patterns of AM, resulting in limited interpretability. This paper proposes a novel First-Order Logic reasoning framework for AM (FOL-AM), designed to explicitly capture logical reasoning paths within argumentative texts. By interpreting multiple AM subtasks as a unified relation query task modeled using FOL rules, FOL-AM facilitates multi-hop relational reasoning and enhances interpretability. The framework supports two flexible implementations: a fine-tuned approach to leverage task-specific learning, and a prompt-based method utilizing large language models to harness their generalization capabilities. Extensive experiments on two AM benchmarks demonstrate that FOL-AM outperforms strong baselines while significantly improving explainability.

1 Introduction

Argumentation Mining (AM) focuses on extracting argumentation structures from texts, which has garnered significant attention in recent years (Lawrence and Reed, 2019; Schaefer and Stede, 2021; Vecchi et al., 2021). AM enables automation of human argumentation logic and has been shown to provide support across various domains, such as decision-making (Vassiliades et al., 2021; Walker et al., 2018), writing support (Wambsganss and Rietsche, 2019; Wambsganss et al., 2020), and argument generation (Khatib et al., 2021; Hua et al., 2019; Slonim et al., 2021).

Given an argumentative text, an Argumentation Mining (AM) system must identify argumentation components (ACs) and their corresponding argumentative relations (ARs). AM generally consists of three key subtasks (Morio et al., 2022a; Stab and Gurevych, 2017a): (i) Argumentation Component Segmentation (ACS), which identifies and extracts the boundaries of ACs within the input text; (ii) Argumentation Component Type Classification (ACTC), which categorizes ACs into predefined types (e.g., Claim and Premise); and (iii) Argumentative Relation Classification (ARC), which determines the type of relation (e.g., Support or Attack) between pairs of ACs. Following previous works (Potash et al., 2017a; Kuribayashi et al., 2019a), we assume that the ACS step has already been performed, meaning that ACs are pre-segmented. Thus, we focus on addressing the ACTC and ARC subtasks.

Previous works (Kuribayashi et al., 2019b; Bao et al., 2021a; Morio et al., 2022b; Sun et al., 2024) primarily focus on representation learning to encode ACs and AC pairs for AM. For instance,

• ARC example: ?(AC1,AC5)

[The government should increase taxes on tobacco]_{AC1}. According to statistics, [taxes account for over 50% of commodity prices]_{AC2}. [Taxation will increase the consumption cost of tobacco]_{AC3}, [but high prices can effectively suppress the willingness to purchase tobacco]_{AC4}. [Although increasing taxes is negative for consumers]_{AC5}

Ground Truth: Support(AC1,AC5)
PITA Predication: Attack(AC1,AC5) ✗

[The government should increase taxes on tobacco]_{AC1} ... | \( r_1 = \text{Promotion} \)
[but high prices can effectively suppress the willingness to purchase tobacco]_{AC4} [Although increasing taxes is negative for consumers]_{AC5}
\( r_2 = \text{Entailment} \)

FOL-AM Predication: Support(AC1,AC5) \( \leftarrow \) Promotion(AC1,AC4) \( \wedge \) Entailment(AC4,AC5)

Figure 1: Illustration of ARC subtask, comparing the results of PITA (previous state-of-the-art model) and FOL-AM (ours).
Kuribayashi et al. (2019b) enhances AC representations using discourse markers, while Bao et al. (2021a) employs a transition-based network for AC and AC pair modeling. Morio et al. (2022b) leverages biaffine functions to encode relational representations, and Sun et al. (2024) incorporates task interactions into AC and AC pair representations. However, these methods fail to explicitly model the reasoning patterns underlying AM, potentially leading to sub-optimal performance. Moreover, they lack interpretability, as they rely on black-box neural networks.

For example, as shown in Figure 1, the state-of-the-art (SOTA) model PITA misclassifies the relation between AC1 and AC5 as Attack due to their shared contextual cue increase taxes, whereas AC5 actually expresses a negative stance toward the topic. In contrast, humans can deduce the correct relation using logical reasoning, recognizing that “if AC1 promotes AC4, and AC4 entails AC5, then AC1 and AC5 are in a Support relation.”

In this paper, we propose a novel first-order logic reasoning framework for multi-task AM (FOL-AM). As indicated in Figure 1, our method aligns with human reasoning by introducing interpretable FOL rules derived from the argumentative text for AM. For example, we may use the following logic rule to identify the relation between AC pair AC1 and AC5 as “Support”: Support(AC1, AC5) \( \leftarrow \) Promotion\( ^1 \)(AC1, AC4) \( \land \) Entailment(AC4, AC5). This logic reasoning framework has two clear benefits over previous approaches: First, it naturally captures multi-hop relations among ACs owing to the compositional structure of the reasoning chain. Second, it improves interpretability as the reasoning processes are visible.

Specifically, we uniformly transform ACTC and ARC subtasks as relation query \( q(x_1, x_{L+1}) \) associated with an AC pair \( (x_1, x_{L+1}) \) within the argumentative text. The query is modeled as the FOL rule to conduct logical deduction: \( q(x_1, x_{L+1}) \leftarrow r_1(x_1, x_2) \land \cdots \land r_L(x_L, x_{L+1}) \). The body of the rule (on the left) is formed by attentively selecting proper ACs \( x_i \) and predicates \( r_i \) upon given argumentative text. We implement the FOL-AM framework in two settings: a fine-tuned method and a prompt-based method using LLMs. Intuitively, the former method can learn the nuanced patterns specific to the AM task directly from the dataset, while the latter can leverage the remarkable performance of the LLMs, which is supposed to have better generalization in various corpora.

In summary, our contributions are as follows: (i) We introduce the first FOL-based reasoning framework for multi-task AM; (ii) We propose a unified FOL rule-based approach to model reasoning patterns in AM; (iii) We implement both fine-tuned and LLM prompt-based methods, demonstrating the flexibility of our framework; (iv) Extensive experiments on two AM datasets showcase the superiority of FOL-AM.

2 Background

Task Definition Building on previous works (Potash et al., 2017a; Morio et al., 2020a), we assume that the ACS subtask has been completed, meaning the spans of ACs are given. Given an argumentative text \( W = \{w_1, w_2, \cdots, w_n\} \) consisting of \( n \) tokens and \( m \) ACs \( X = \{x_1, x_2, \cdots, x_m\} \), with the start and end word indexes \( (b_i, e_i) \) of each AC \( x_i \), the objective is twofold: first, to classify each AC’s type \( y_i \in Y_{ACTC} \) (e.g., Claim or Premise), and second, to determine the argumentative relation (AR) type \( y_{(i,j)} \in Y_{ARC} \) (e.g., Support, Attack, or No-Relation) between two ACs \( x_i \) and \( x_j \).

FOL Rule In this work, we define a \( L \)-hop FOL rule that has the conjunctive form as follows:

\[
q(a_1, a_{L+1}) \leftarrow r_1(a_1, a_2) \land r_2(a_2, a_3) \cdots \land r_L(a_L, a_{L+1})
\]

where \( a_i \) denotes an argument, \( r_i \) denotes a predicate, i.e., a relation between \( a_i \) and \( a_{i+1} \). Each \( r_i(a_i, a_{i+1}) \) is a logic atom and an atom is composed of a predicate \( r_i \) and its arguments \( a_i \) and \( a_{i+1} \). The left side of \( \leftarrow \) represents the head atom, while the right side constitutes a rule body. \( L \) denotes the number of predicates in the rule body. In this way, the body rule suggests a reasoning path to deduce the relation query \( q(a_1, a_{L+1}) \) in the head atom.

The ACTC and ARC have different task objectives that hinder the direct incorporation of FOL rules into a multi-task AM. Concretely, ACTC classifies individual ACs, while ARC predicts the relation type between AC pairs. To bridge the gap between multi-task AM and FOL rule, we reformulate the ACTC and ARC subtasks into a unified relation query task driven by the FOL rule. For ACTC, we define the relation query \( q(x_i, x_w) \) as the head atom of a FOL rule that predicts the AC

1These phrases serve as human-interpretable descriptions of the abstract logical predicates induced via LLM prompting, as described in Section 3.3.
type \( y_{x_i} \) of the AC \( x_i \). Similarly, for ARC, the relation query \( q(x_i, x_j) \) is defined as the head atom of a FOL rule that predicts the AR type \( y_{(x_i, x_j)} \) for AC pair \( (x_i, x_j) \). Here, we introduce the argumentative text input \( W \) as a special AC \( x_W \). This is because the type of an AC is determined by its context and its relations with other ACs (Stab and Gurevych, 2014).

3 Methodologies

We propose FOL-AM, a First-Order Logic (FOL) reasoning framework designed to explicitly uncover the reasoning patterns in Argumentation Mining (AM). FOL-AM consists of two key components: (i) an *FOL Term Constructor*, which learns representations of Argumentation Components (ACs) and predicates; and (ii) an *FOL Rule Generator*, which utilizes these representations to instantiate logical atoms. The rule bodies are then constructed by selecting relevant atoms, and subsequently evaluated to infer the relation query in the head atom.

3.1 FOL Term Constructor

**AC Representation Learning** We define the ACs as the argument candidates within the FOL rules, as the relation query in the head atom operates on AC pairs. Given an argumentative text \( W \) with AC set \( X \), we employ a Longformer (Beltagy et al., 2020) to model the representations of the text \( W \) and ACs. Specifically, the text \( W \) is passed through Longformer to obtain contextual representations \( H^W \). Then the whole text representation \( h^W \) is then derived by applying mean-pooling over \( H^W \). For each AC \( x_i \), its representation \( h_i \) is computed by mean-pooling over the tokens in the span \( (b_i, e_i) \), i.e., \( h_i = \frac{1}{e_i - b_i + 1} \sum_{k=b_i}^{e_i} H_k^W \).

**Predicate Generation** We assume that there are \( M \) atomic predicates with indecomposable semantics, represented by a predicate set \( \mathcal{R} = \{ r_i \}_{i=1}^M \). Each predicate is assigned a learnable vector to represent its semantics, with \( U \in \mathbb{R}^{M \times d} \) denoting the embedding matrix for the \( M \) predicates.

3.2 FOL Rule Generator

The FOL Rule Generator aims to conduct complex FOL reasoning by first generating multiple probable rule bodies and then ranking each rule body to derive the final FOL rule.

3.2.1 Rules Generation

Given the AC set \( X \) and predicate set \( \mathcal{R} \), the \( l \)-hop FOL rule \( q(x_1, x_{l+1}) \leftarrow r_1(x_1, a_2) \land \cdots \land r_l(a_l, x_{l+1}) \) consists of atoms \( r_i(a_i, a_{i+1}) \) defined over arguments \( a_i \) and predicates \( r_i \) to derive the relation query \( q(x_1, x_{l+1}) \). Generating the FOL rule involves three steps: (1) AC autoregression learning selects relevant ACs \( \{x_2, \cdots, x_l\} \) to instantiate the bridging arguments \( \{a_2, \cdots, a_l\} \) in the rule body. (2) Predicate autoregressive learning sequentially generates predicate \( r_i \) for each atom in the rule body, ensuring logical consistency across steps. (3) Body atom generation integrates the selected ACs and predicates to form the body atoms. These steps work together to generate multiple probable rule bodies.

**AC Autoregression Learning** We first apply AC autoregression learning to select ACs from which one may be chosen as a bridging argument \( a_t (1 < t \leq l) \) of a rule body. The first and last arguments (\( a_1, a_{l+1} \)) are referred to as \( (x_1, x_{l+1}) \) based on the head atom \( q(x_1, x_{l+1}) \). Specifically, we calculate a score \( A_t^i \) for each AC using the previously instantiated argument \( x_{t-1} \). The score indicates the probability of selecting an AC as the instantiation of the argument \( a_t \) from the AC set \( X \). For example, the score for \( x_i \) is:

\[
A_t^i = \frac{\exp(W^A[\mathbf{h}_{t-1}; h_i])}{\sum_{k=1}^m \exp(W^A[\mathbf{h}_{t-1}; h_k])}
\]

where \( W^A \) is a learnable weight matrix. \( \mathbf{h}_{t-1} \) is the AC representation of instantiated argument \( x_{t-1} \). [ ; ] represents concatenation. Instead of selecting only the AC with the highest score, we choose the top-K ACs \( \{x_1^1, \ldots, x_K^1\} \subseteq X \) as the possible instantiation of the argument \( a_t \) according to \( A_t^i \). This is because the local optimum derived from autoregression learning may not be the global optimum, as demonstrated in Section 4.6.

**Predicate Autoregressive Learning** Then, we learn a predicate association that indicates which predicates are likely to be generated for constructing a rule body. To do this, we denote \( \mathbf{u}' \in \mathbb{R}^D \) as the query vector for the \( l \)-hop reasoning rule and use predicate autoregressive learning to generate a probability vector \( s_t^j \in \mathbb{R}^M \). This probability vector represents the distribution of the predicate in the \( t \)-th atom over the predicate set \( \mathcal{R} \):

\[
s_t^j = \mathrm{softmax}((W_t^j \mathbf{u}')^T (W_b^j U))
\]

2We denote the \( a_t \) as uninstantiated argument and \( x_t \) as instantiated argument with an AC.
Argumentation Text: Despite the fact that [advertisements can be exaggerated]AC1, it is also true that [it plays an important role economically]AC2. [They introduce new products]AC3. [Advertising also helps to keep prices at reasonable levels]AC4. [Otherwise the market may be monopolistic]AC5.

Relation Query: \( q(AC_1, AC_5) \)

\[
\text{Support}(AC_1, AC_5) \gets \\
r_1(AC_1, AC_2) \land r_2(AC_2, AC_5)
\]

\( \sqrt{0.9} \quad r_1(AC_1, AC_2) \land r_2(AC_2, AC_5) \)
\( \times 0.7 \quad r_2(AC_1, AC_3) \land r_3(AC_3, AC_5) \)
\( \times 0.1 \quad r_3(AC_1, AC_4) \land r_1(AC_4, AC_5) \)

Figure 2: The overview of our approach, with a running example for classifying the relation between an AC pair.

\[
u_{t+1}^l = u_t^l + s_t^l \cdot (W_b^l U)
\]
(4)

where \( u_0^l = u^l \), \( W_t^l \) and \( W_b^l \) are learnable weight matrices. Intuitively, to learn to define a l-hop rule body, Eq. 3 and 4 sequentially produce predicate \( r_t \) at each step \( t \in \{1, ..., l\} \) by attending to all the predicates with probability \( s_t^l \).

Body Atom Generation We integrate the selected ACs and predicates to generate the body atoms of a rule body. However, the AC and predicate autoregression learning ignore the correlation between predicates and ACs. This causes always generating the same predicates for different selected AC pairs in each step and has a pretty poor performance in Section 4.5. To address this, we propose a relational network that produces an AC-specific probability distribution \( p(x_t, x_{t+1}) \in \mathbb{R}^M \) over all predicates based on selected AC pairs \( x_t, x_{t+1} \). This distribution denotes the compatibility of selected AC pair \( (x_t, x_{t+1}) \) with each predicate \( r \in \mathcal{R} \) to form an atom \( r_t(x_t, x_{t+1}) \):

\[
p(x_t, x_{t+1}) = \operatorname{softmax}(W^r \tanh([h_t; h_{t+1}]))
\]
(5)

where \( W^r \) is a weight matrix.

We combine the probabilities from predicate autoregression learning with AC-specific probabilities to compute the value \( ^3 \mu(r_t(x_t, x_{t+1})) \) of an body atom \( r_t(x_t, x_{t+1}) \) being true formed by the predicate \( r_t \) and two ACs \( (x_t, x_{t+1}) \):

\[
\mu(r_t(x_t, x_{t+1})) = \max(s_t^l \odot p(x_t, x_{t+1}))
\]
(6)

\( ^3 \)For smooth optimization, we use \( \mu(\cdot) \in [0, 1] \) to denote the value of an atom or a rule body, which indicates the probability of the atom or rule body being true.

where \( \odot \) denotes element-wise multiplication and \( r_t \) is selected as the entry with the highest probability. Note that the predicate autoregression learning captures the sequential dependency among predicates and the relational network models the correlation between the predicate and ACs. By combining the two information, we can construct more robust reasoning rules.

Finally, we can combine the body atoms \( r_t(x_t, x_{t+1})(1 \leq t \leq l) \) into a complete rule body. Since the AC instantiations of bridging arguments are variable, we input each probable AC pair from AC autoregression learning into Eq. 5 and 6 for each body atom, forming multiple probable rule bodies. For example, consider the case in Figure 2 where \( l = 2 \). Given the possible AC instantiations \( (x_1, x_2, x_5) \) for all arguments, Eq. 5 and 6 first generate a body atom \( r_1(x_1, x_2) \) based the AC pair \( (x_1, x_2) \). Then, the next AC pair \( (x_2, x_5) \) is used to generate another predicate \( r_2 \), forming the second body atom \( r_2(x_2, x_5) \). As a result, the rule body \( r_1(x_1, x_2) \land r_2(x_2, x_5) \) is constructed.

3.2.2 Rules Ranking

These rule body candidates need to be evaluated and selected to induce the relation query in the head atom. We compute the value of each probable rule body being true. Given a l-hop rule body consisting of body atoms (i.e., \( r_t(x_t, x_{t+1}) \)) and conjunction operators (\( \land \)), computing its value is difficult due to its discrete nature (Qu et al., 2020). Here, we introduce T-Norm logic (Klement et al., 2013) to relax the discrete nature of the rule body evaluation, enabling an end-to-end learning process. The T-Norm logic defines the value of a
rule with two atoms and conjunction operator as \( T(\mu_1 \land \mu_2) = \min(\mu_1, \mu_2) \), where \( \mu_1, \mu_2 \in [0, 1] \) refer to the values of two atoms. Thus, the value of the \( l \)-hop rule body based on the atom values can be computed as \( \min(\{\mu(r_l(x_t, x_{t+1}))\}_{l=0}^l) \).

We rank all probable rule bodies over all possible AC instantiations and use a max operator to select the rule body with the highest value, representing the truth value \( \mu^l \) of the \( l \)-hop rule body.

\[
\mu^l = \max_{x \in \mathcal{X}} (\min(\{\mu(x_t, x_{t+1})\}_{t=1}^l))
\]

where \( \mathcal{X} = \{x|x = (x_1, x_2^k, ..., x_l^k, x_{l+1})\}_{k=1}^K \) denotes the set for all possible AC instantiations for the \( l \)-hop rule body. The selected ACs and predicates form the rule body with the maximum score. In this way, the generated \( l \)-hop FOL rule is \( q(x_1, x_{L+1}) \leftarrow r_1(x_1, x_2) \land \cdots \land r_l(x_l, x_{l+1}) \), suggesting a reasoning path for deducing the relation between \( x_1 \) and \( x_{L+1} \).

The number of reasoning hops \( l \) required to derive the query head should be flexibly decided by the model itself. We iterate through the rule generation and ranking process for different \( l \)-hop (\( 1 \leq l \leq L \)) rules and obtain the values \( \{\mu^1, \cdots, \mu^L\} \) of different \( l \)-hop FOL rules. Finally, the value \( \mu(q(x_1, x_{L+1})) \) of the head atom \( q(x_1, x_{L+1}) \) is computed by selecting the maximum score over the \( L \) rule bodies:

\[
\begin{align*}
\mu(q(x_1, x_{L+1})) &= \max_{1 \leq l \leq L} (\exp(s \log([\mu^1; \cdots; \mu^L]))) \\
s &= v_q[u^1, \ldots, u^L]
\end{align*}
\]

where \( s \) is a selective distribution over the \( L \) rule bodies with different hop and \( v_q \) denotes the learnable vector of the relation label \( q \) between the AC pair \( (x_1, x_{L+1}) \).

Training and Inference The generated FOL rules need to be tested and refined against the ground-truth label, known as learning from entailment, which maximizes the probabilities of the ground-truth label and minimizes those of other labels. We use cross-entropy loss to train the model during the training process:

\[
L = -\log \frac{\mu(\hat{q}(x_i, x_j))}{\sum_{q \in Y} \mu(q(x_i, x_j))}
\]

where \( \hat{q}(x_i, x_j) \) denotes the ground-truth label for AC pair \( (x_i, x_j) \) in ACTC (ARI) and \( Y \) denotes the label set \( Y_{ACTC} \) (\( Y_{ARC} \)) of ACTC (ARC). The choice of the label set depends on the type of relation query \( q(x_i, x_j) \). For example, if \( x_j \) denotes the specific AC \( x_w \), then \( Y_{ACTC} \) is used; otherwise, \( Y_{ARC} \) is applied. During inference, we query all labels in ACTC (ARC) and select the label with the highest truth value of the reasoning rule as the prediction result. Although our methods apply autoregressive learning to produce instantiated ACs and predicates, our use of parallel tensor operations allows it to perform as efficiently as previous methods. The time cost of FOL-AM is detailed in Appendix A.6.

3.3 Adaptiveness for LLM

Inspired by recent advancements in LLMs (Minaee et al., 2024), we extend our FOL-AM framework to leverage LLMs, prompting them to perform FOL-based reasoning for AM. The reasoning process in LLM-enhanced FOL-AM\(_{PT}\)\footnote{We distinguish two implementations: *FOL-AM\(_{FT}\)* using task-specific fine-tuning and *FOL-AM\(_{PT}\)* employing LLM prompting. Unless otherwise specified, FOL-AM refers to the *FOL-AM\(_{FT}\)*.} closely follows the fine-tuned FOL-AM\(_{FT}\) approach, including both *FOL Term Construction* and *FOL Rule Generation*.

**FOL Term Construction** We construct the fundamental FOL terms, including arguments and predicates, within the FOL rule. ACs are used to instantiate the arguments. Unlike fine-tuned FOL-AM, which employs shared parameterized predicates across all samples, the predicates in the LLM-based approach are dynamically derived for each sample. Specifically, we input the argumentative text \( W \) and each AC pair \( (x_i, x_j) \) into an LLM and instruct it to generate multiple relation phrases \( R_{i,j} \) as potential predicates for the given AC pair.

**FOL Rule Generation** To generate FOL rules, we first identify relevant ACs and predicates. Given a relation query \( q(x_i, x_j) \) for ACTC (ARC) as the head atom, we provide the argumentative text \( W \) and the AC set \( X \) as input to the LLM and instruct it to select the three most relevant ACs from \( X \) for each AC in the head atom. These selected ACs instantiate the bridging arguments in the FOL rules, while the relation phrases between them serve as potential predicates for each AC pair. Next, we input the selected ACs, along with all probable predicates for each AC pair, into the LLM and instruct it to generate several candidate rule bodies for the head atom.

Finally, we perform *FOL rule ranking* by instructing the LLM to identify the most plausible rule body from the generated rule set and use it
<table>
  <tr>
    <th>Model</th>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
    <th>Avg</th>
  </tr>
  <tr>
    <td>Joint-ILP</td>
    <td>82.6</td>
    <td>58.5</td>
    <td>-</td>
    <td>-</td>
  </tr>
  <tr>
    <td>Joint-PN</td>
    <td>84.9</td>
    <td>60.8</td>
    <td>-</td>
    <td>-</td>
  </tr>
  <tr>
    <td>BERT-Trans</td>
    <td>88.4</td>
    <td>70.6</td>
    <td>-</td>
    <td>-</td>
  </tr>
  <tr>
    <td>LSTM+dist</td>
    <td>85.7</td>
    <td>67.8</td>
    <td>54.3</td>
    <td>69.3</td>
  </tr>
  <tr>
    <td>BART</td>
    <td>82.7</td>
    <td>62.9</td>
    <td>53.4</td>
    <td>66.3</td>
  </tr>
  <tr>
    <td>ST</td>
    <td>86.8</td>
    <td>69.3</td>
    <td>57.1</td>
    <td>71.1</td>
  </tr>
  <tr>
    <td>PITA</td>
    <td>88.3</td>
    <td>73.5</td>
    <td>59.2</td>
    <td>73.7</td>
  </tr>
  <tr>
    <td>FOL-AM (our)</td>
    <td><b>89.3</b></td>
    <td><b>75.7</b></td>
    <td><b>64.0</b></td>
    <td><b>76.3 (+2.6)</b></td>
  </tr>
</table>

Table 1: Performance comparison on the PE dataset. Avg indicates the average value across all metrics. Our improvements over baselines are statistically significant with p < 0.05.

<table>
  <tr>
    <th>Model</th>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
    <th>Avg</th>
  </tr>
  <tr>
    <td>SSVM-strict</td>
    <td>73.2</td>
    <td>26.7</td>
    <td>-</td>
    <td>-</td>
  </tr>
  <tr>
    <td>TSP-PLBA</td>
    <td>78.9</td>
    <td>34.0</td>
    <td>-</td>
    <td>-</td>
  </tr>
  <tr>
    <td>BERT-Trans</td>
    <td>82.5</td>
    <td>37.3</td>
    <td>-</td>
    <td>-</td>
  </tr>
  <tr>
    <td>DR-LG</td>
    <td>65.3</td>
    <td>29.3</td>
    <td>15.0</td>
    <td>36.5</td>
  </tr>
  <tr>
    <td>BART</td>
    <td>81.4</td>
    <td>32.9</td>
    <td>16.7</td>
    <td>43.7</td>
  </tr>
  <tr>
    <td>ST</td>
    <td>82.3</td>
    <td>40.2</td>
    <td>20.4</td>
    <td>47.6</td>
  </tr>
  <tr>
    <td>PITA</td>
    <td>83.6</td>
    <td>44.9</td>
    <td>23.8</td>
    <td>50.8</td>
  </tr>
  <tr>
    <td>FOL-AM (our)</td>
    <td><b>85.3</b></td>
    <td><b>47.7</b></td>
    <td><b>28.7</b></td>
    <td><b>53.9 (+3.1)</b></td>
  </tr>
</table>

Table 2: Performance comparison on CDCP dataset. Avg indicates the average value across all metrics. Our improvements over baselines are statistically significant with p < 0.05.

for label prediction. Specifically, the LLM is directed to select the most relevant rule body and apply logical reasoning within the FOL framework to determine the final label for the AC pair in the head atom for ACTC (ARC). The details of the prompt template are provided in Appendix A.7.

4 Experiments

4.1 Experimental Settings

We evaluate our approach on two AM benchmark datasets: PE (Stab and Gurevych, 2017b) and CDCP (Park and Cardie, 2018). The statistical details of these datasets are in Appendix A.1. For evaluation, we report the macro-averaged F1 score (Macro) for ACTC. Following prior work (Morio et al., 2022b; Sun et al., 2024), we adopt a hierarchical evaluation protocol for ARC, decomposing it into two metrics: ARI, which measures the F1 score for identifying whether an argumentative relation exists between a pair of ACs; and ARTC, which measures the Macro score for correctly classifying the type of relation (e.g., Support and Attack for PE; Reason and Evidence for CDCP) for AC pairs where a relation is present.

For optimization, we use AdaW (Loshchilov and Hutter, 2017) with a learning rate of \(3e^{-5}\) and weight decay of \(1e^{-2}\) for both PE and CDCP datasets. The reasoning hops \(L\) is set to 2 for both datasets. The number of predicates \(M\) is set to 40 for PE and 30 for CDCP, while the number of instantiations \(K\) of bridging argument is set to 5 for PE and 4 for CDCP. To reduce randomness, we repeat the experiments three times with different seeds and report the average results. More implementation details are in Appendix A.3. Our code is available at https://github.com/syiswell/logic-AM.

4.2 Baselines

We select strong AM models from recent years as baselines, following previous work (Morio et al., 2022b). For the PE dataset, we compare our FOL-AM model with seven baselines: Joint-ILP (Stab and Gurevych, 2017b), Joint-PN (Potash et al., 2017b), LSTM+dist (Kuribayashi et al., 2019b), BERT-Trans (Bao et al., 2021a), BART (Lewis et al., 2019), ST (Morio et al., 2022b), and PITA (Sun et al., 2024). For the CDCP dataset, we compare with seven baselines: DR-LG (Galassi et al., 2018), SSVM-strict (Niculae et al., 2017), TSP-PLBA (Morio et al., 2020b), BERT-Trans (Bao et al., 2021a), BART (Lewis et al., 2019), ST (Morio et al., 2022b), and PITA (Sun et al., 2024).

Additionally, we apply our prompt-based FOL-AM framework to two large language models (LLMs): GPT-4o-mini (OpenAI, 2024) and Llama-3.1-8B (Dubey et al., 2024). We compare our prompt-based FOL-AM\(_{PT}\) with two baselines: (i) Standard LLMs, which predict argumentation labels directly without reasoning, and (ii) Chain-of-Thought (CoT) (Wei et al., 2022), which generates intermediate explanations before making final predictions. The experiments are conducted under both zero-shot and few-shot settings.

4.3 Overall Performance

The experimental results for both the PE and CDCP datasets are summarized in Table 1 and Table 2, respectively. We observe that FOL-AM consistently achieves superior performance across both datasets. Specifically, on the PE dataset, our FOL-AM model outperforms the state-of-the-art (SOTA) model PITA by 4.8% in ARC and 2.6% in Avg. A similar trend is observed for the CDCP dataset, where FOL-AM surpasses PITA by 4.9% in ARC and 3.1% in Avg.

Additionally, all models exhibit a consistent performance gap between ARI and ARTC scores (e.g.,
<table>
  <tr>
    <th rowspan="2">Data</th>
    <th rowspan="2">LLM</th>
    <th rowspan="2">Method</th>
    <th colspan="4">Zero-Shot</th>
    <th colspan="4">Few-Shot</th>
  </tr>
  <tr>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
    <th>Avg</th>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
    <th>Avg</th>
  </tr>
  <tr>
    <td rowspan="3">PE</td>
    <td rowspan="3">LLaMA-3.1-8B</td>
    <td>Standard</td>
    <td>28.7</td>
    <td>49.0</td>
    <td>24.9</td>
    <td>34.2</td>
    <td>35.8</td>
    <td>49.5</td>
    <td>25.1</td>
    <td>36.8</td>
  </tr>
  <tr>
    <td>CoT</td>
    <td>36.0</td>
    <td>49.3</td>
    <td>30.5</td>
    <td>38.6</td>
    <td>47.1</td>
    <td>49.4</td>
    <td>30.3</td>
    <td>42.3</td>
  </tr>
  <tr>
    <td>FOL-AM<sub>PT</sub></td>
    <td>35.6</td>
    <td>50.7</td>
    <td>31.2</td>
    <td>39.2</td>
    <td>44.4</td>
    <td>51.0</td>
    <td>32.2</td>
    <td>42.5</td>
  </tr>
  <tr>
    <td rowspan="3">CDCP</td>
    <td rowspan="3">GPT-4o-mini</td>
    <td>Standard</td>
    <td>37.4</td>
    <td>53.7</td>
    <td>31.4</td>
    <td>40.8</td>
    <td>53.9</td>
    <td>54.4</td>
    <td>38.6</td>
    <td>49.0</td>
  </tr>
  <tr>
    <td>CoT</td>
    <td>45.8</td>
    <td>51.9</td>
    <td>33.3</td>
    <td>43.7</td>
    <td>56.9</td>
    <td>52.3</td>
    <td>39.1</td>
    <td>49.4</td>
  </tr>
  <tr>
    <td>FOL-AM<sub>PT</sub></td>
    <td><b>48.6</b></td>
    <td><b>54.6</b></td>
    <td><b>36.2</b></td>
    <td><b>47.1 (+3.4)</b></td>
    <td><b>58.2</b></td>
    <td><b>56.0</b></td>
    <td><b>41.4</b></td>
    <td><b>51.9 (+2.5)</b></td>
  </tr>
  <tr>
    <td rowspan="3">CDCP</td>
    <td rowspan="3">LLaMA-3.1-8B</td>
    <td>Standard</td>
    <td>44.9</td>
    <td>13.6</td>
    <td>7.5</td>
    <td>22.0</td>
    <td>42.0</td>
    <td>13.7</td>
    <td>8.9</td>
    <td>21.3</td>
  </tr>
  <tr>
    <td>CoT</td>
    <td>50.2</td>
    <td>15.9</td>
    <td>8.2</td>
    <td>24.8</td>
    <td>50.7</td>
    <td>14.5</td>
    <td>8.0</td>
    <td>24.4</td>
  </tr>
  <tr>
    <td>FOL-AM<sub>PT</sub></td>
    <td>53.5</td>
    <td>15.4</td>
    <td>9.1</td>
    <td>26.0</td>
    <td>54.3</td>
    <td>14.4</td>
    <td>8.4</td>
    <td>25.7</td>
  </tr>
  <tr>
    <td rowspan="3">CDCP</td>
    <td rowspan="3">GPT-4o-mini</td>
    <td>Standard</td>
    <td>50.8</td>
    <td>23.9</td>
    <td>12.7</td>
    <td>29.1</td>
    <td>62.7</td>
    <td>21.2</td>
    <td>12.9</td>
    <td>32.2</td>
  </tr>
  <tr>
    <td>CoT</td>
    <td>56.0</td>
    <td>23.4</td>
    <td>13.4</td>
    <td>31.0</td>
    <td>57.2</td>
    <td>23.8</td>
    <td>13.3</td>
    <td>31.4</td>
  </tr>
  <tr>
    <td>FOL-AM<sub>PT</sub></td>
    <td><b>63.1</b></td>
    <td><b>24.8</b></td>
    <td><b>14.3</b></td>
    <td><b>34.1 (+3.1)</b></td>
    <td><b>65.2</b></td>
    <td><b>25.1</b></td>
    <td><b>14.8</b></td>
    <td><b>35.0 (+2.8)</b></td>
  </tr>
</table>

Table 3: Performance comparison of LLMs on PE and CDCP datasets under zero-shot/few-shot settings, presenting the F1 for ARI and Macro for ACTC and ARC. Avg indicates the average value across all metrics.

11.7 points for FOL-AM and 14.3 points for PITA on the PE dataset), underscoring the greater difficulty of fine-grained relation classification (i.e., ARTC) compared to binary relation identification (i.e., ARI). Notably, FOL-AM shows a smaller drop between ARI and ARTC scores than baseline models, suggesting that it more effectively preserves argumentative semantics—likely due to the explicit logical constraints imposed during relation classification. These consistent improvements highlight the effectiveness of integrating FOL rules to explicitly capture reasoning patterns in AM.

4.4 Adaptability Experiment

Table 3 shows the results of LLM-based methods for AM. We observe that CoT-based methods consistently outperform standard methods, emphasizing the crucial role of reasoning ability in AM. Our prompt-based FOL-AM<sub>PT</sub> methods significantly surpass both baselines (i.e., standard and CoT) across all metrics on PE and CDCP, demonstrating the effectiveness of our framework in enhancing model reasoning capabilities. Moreover, few-shot methods generally outperform zero-shot methods, and our FOL-AM<sub>PT</sub> methods improve performance in both settings, further underscoring its universality. Notably, our fine-tuned FOL-AM<sub>FT</sub> model in Table 1 and 2 significantly outperforms all zero-shot and few-shot methods. This improvement stems from its ability to learn logic rules directly from the dataset, thereby enhancing its reasoning capacity. In contrast, zero-shot and few-shot methods rely on the model’s inherent reasoning ability and a limited number of examples, leading to lower performance. Given the strong performance of our fine-tuned FOL-AM<sub>FT</sub> model, we conduct further experiments based on it in the following sections.

<table>
  <tr>
    <th rowspan="2">Model</th>
    <th colspan="3">PE</th>
    <th colspan="3">CDCP</th>
  </tr>
  <tr>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
  </tr>
  <tr>
    <td>FOL-AM</td>
    <td>89.3</td>
    <td>75.7</td>
    <td>64.0</td>
    <td>85.3</td>
    <td>47.7</td>
    <td>28.7</td>
  </tr>
  <tr>
    <td>w/o PG</td>
    <td>85.4</td>
    <td>71.4</td>
    <td>50.2</td>
    <td>73.1</td>
    <td>45.4</td>
    <td>23.4</td>
  </tr>
  <tr>
    <td>w/o PAL</td>
    <td>77.3</td>
    <td>60.3</td>
    <td>41.1</td>
    <td>66.2</td>
    <td>29.8</td>
    <td>14.3</td>
  </tr>
  <tr>
    <td>w/o RN</td>
    <td>79.7</td>
    <td>65.9</td>
    <td>43.8</td>
    <td>72.6</td>
    <td>44.3</td>
    <td>22.5</td>
  </tr>
  <tr>
    <td>w/o AAL</td>
    <td>88.4</td>
    <td>74.6</td>
    <td>61.1</td>
    <td>84.8</td>
    <td>47.5</td>
    <td>24.3</td>
  </tr>
</table>

Table 4: The impact of various components presenting the F1 for ARI and Macro for ACTC and ARC.

![Line graph showing the impact of the number of AC instantiations on PE and CDCP](page_1012_1342_377_246.png)

Figure 3: The impact of the number of AC instantiations on PE and CDCP.

4.5 Ablation Study

Table 4 compares our FOL-AM method with several variants where different components are removed: w/o predicate generation (w/o PG), which eliminates predicate generation and instead computes only binary relations to determine whether two ACs are related; w/o predicate autoregression learning (w/o PAL), which disregards the sequential dependencies among predicates; and w/o relational network (w/o RN), which removes the relational network and computes the value of a body atom by simply multiplying the similarity of the AC pair with the predicate probability distribution. w/o AC autoregression learning (w/o AAL), which selects all ACs as bridging arguments. The results show that predicate generation is crucial for reasoning, as its removal causes a significant performance drop. Similarly, removing predicate autoregression learn-
Example 1

Argumentation Text:
... For instance, [after studying and sitting in the classroom for the whole morning, it is better for students to do some outdoor activities, such as stretching, jogging and playing ball games, in order to improve their learning ability during the afternoon classes]_{AC1}. [Removing physical classes from school curriculum will not only be harmful to young people's health, it will also decrease their learning capability]_{AC2}.

Relation Query: ?(AC2,Text)
FOL Rule: Claim(AC2,Text) \rightarrow r_2(AC2,AC3) \wedge r_4(AC3,AC4)
Ground Truth: Claim(AC2,Text)

r_2: Further elaboration of viewpoints
r_4: Exemplification using suggestions

Example 2

Argumentation Text:
Admittedly, to some extent, [technology may make people's life more complicated]_{AC1}. In addition, [games on their phone exert a tremendous fascination on teenagers, leading to a large problem that they got addicted to their phones]_{AC2}. Consequently, [making their life more complicated and tired]_{AC5}.

Relation Query: ?(AC1,AC4)
FOL Rule: Support(AC1,AC4) \rightarrow r_3(AC1,AC5) \wedge r_4(AC5,AC4)
Ground Truth: Support(AC1,AC4)

r_3: Elaboration
r_4: Explanation for life's complexity

Figure 4: Examples of interpretability of our method for ACTC and ARC.

ing harms performance, highlighting the importance of sequential dependencies among predicates. Omitting the relational network also reduces performance by losing the correlation between predicates and ACs. Lastly, the w/o AAL variant suffers from degraded performance, confirming precise AC selection is vital for reasoning rule modeling.

4.6 Impact of AC Instantiation

Figure 3 examines the effect of varying the number of instantiations \( K \) of bridging argument in AC autoregression learning on the PE and CDCP datasets. The value of \( K \) determines the number of probable rule bodies. The results indicate that performance remains relatively stable across both datasets as \( K \) varies. However, when \( K = 1 \), performance is consistently the lowest, reinforcing our assumption that AC autoregression learning struggles to find the global optimum in this scenario. Additionally, when we randomly select \( K \) ACs as bridging arguments (denoted as Random) rather than employing the AC autoregression learning (AAL) mechanism, the performance of the Random variant is consistently worse than that of the model with AAL. This suggests that similarity-based AC selection through AAL enables the model to more effectively capture and utilize logical dependencies among ACs, ultimately leading to improved performance.

4.7 Impact of Reasoning Hops in FOL Rules

Table 5 analyzes the impact of reasoning hops in FOL rules. The Strict setting enforces an exact \( L \)-hop reasoning approach, while the Adaptive setting allows for a maximum of \( L \) reasoning hops, selecting the prediction with the highest score adaptively. Notably, the 1-hop setting remains identical in both Strict and Adaptive configurations.

The results indicate that the Strict setting leads to suboptimal performance, whereas the Adaptive setting achieves significantly better results. Furthermore, the 1-hop FOL rule performs poorly, suggesting that relation queries in AM benefit more from multi-hop reasoning patterns.

<table>
  <tr>
    <th rowspan="2">Setting</th>
    <th rowspan="2">L</th>
    <th colspan="3">PE</th>
    <th colspan="3">CDCP</th>
  </tr>
  <tr>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
    <th>ACTC</th>
    <th>ARI</th>
    <th>ARTC</th>
  </tr>
  <tr>
    <td rowspan="4">Strict</td>
    <td>One</td>
    <td>73.5</td>
    <td>70.1</td>
    <td>49.9</td>
    <td>59.3</td>
    <td>35.8</td>
    <td>17.7</td>
  </tr>
  <tr>
    <td>Two</td>
    <td>78.6</td>
    <td>66.3</td>
    <td>44.4</td>
    <td>69.5</td>
    <td>28.7</td>
    <td>15.2</td>
  </tr>
  <tr>
    <td>Three</td>
    <td>76.9</td>
    <td>63.2</td>
    <td>40.3</td>
    <td>66.9</td>
    <td>25.2</td>
    <td>13.9</td>
  </tr>
  <tr>
    <td>Four</td>
    <td>74.1</td>
    <td>61.8</td>
    <td>39.6</td>
    <td>59.9</td>
    <td>22.4</td>
    <td>12.1</td>
  </tr>
  <tr>
    <td rowspan="3">Adaptive</td>
    <td>Two</td>
    <td><b>89.3</b></td>
    <td><b>75.7</b></td>
    <td><b>64.0</b></td>
    <td><b>85.3</b></td>
    <td><b>47.7</b></td>
    <td><b>28.7</b></td>
  </tr>
  <tr>
    <td>Three</td>
    <td>86.8</td>
    <td>73.9</td>
    <td>60.6</td>
    <td>83.4</td>
    <td>44.1</td>
    <td>23.0</td>
  </tr>
  <tr>
    <td>Four</td>
    <td>83.2</td>
    <td>69.5</td>
    <td>49.2</td>
    <td>79.2</td>
    <td>39.2</td>
    <td>20.3</td>
  </tr>
</table>

Table 5: The impact of varying FOL rule's lengths in different FOL settings. The "Strict" setting refers to precise adoption with fixed hop counts, while the "Adaptive" setting allows up to \( L \)-hop, with "one hop" indicating no bridging AC.

4.8 Interpretability and Case Study

Figure 4 examines the interpretability of our method through a case study for ACTC and ARC. The results demonstrate that FOL-AM effectively generates FOL rules with selected bridging ACs and predicates to infer the relation query. In the first example, the type of AC2 is determined by leveraging bridging AC1, with \( r_2 \) interpreted as elaboration and \( r_8 \) as exemplification. The second case highlights the ability of FOL-AM to interpret multi-hop reasoning in identifying relations between distant AC pairs. Here, \( r_3 \) serves as a summary connecting two similar viewpoints, while \( r_6 \) explains why "life is complicated." Although AC1 and AC4 appear to describe different topics at the surface semantic level, a Support relation between them can be inferred through the bridging AC5.

5 Related Works

5.1 Argumentation Mining

Argumentation Mining (AM) is a computational task that aims to automatically identify and
extract argument structures from argumentative texts (Lawrence and Reed, 2020). With the advent of deep learning, Laha and Raykar (2016) introduced the first neural model for AM. Later, studies incorporated linguistic features and discourse relations for AM (Kuribayashi et al., 2019b; Chakrabarty et al., 2019). Recent models used advanced architectures like biaffine attention (Morio et al., 2020a, 2022b), transition-based network (Bao et al., 2021a) and Graph Convolutional Networks (Sun et al., 2024). Additionally, some studies explored integrating knowledge bases for logical reasoning. (Moens, 2018) enhanced AM with world knowledge, while (Paul et al., 2020) used unsupervised graph methods for multi-hop reasoning. In contrast, our approach employs FOL rules to model reasoning directly without relying on predefined rules or external knowledge bases.

5.2 FOL Rule

First-order logic (FOL) rules provide a rigorous foundation for formal reasoning, enabling precise knowledge representation and logical deduction (Hughes, 1996). In the deep learning era, several studies explored integrating FOL rules with neural networks for tasks, such as knowledge base inference (Qu et al., 2020), text entailment (Li and Srikumar, 2019), question answering (Wang and Pan, 2022), and event argument extraction (Liu et al., 2023). With the rise of LLMs, integrating FOL reasoning into LLMs has gained attention (Ye et al., 2023; Pan et al., 2023; Olausson et al., 2023; Xu et al., 2024). However, these approaches are not directly applicable to AM due to differences in task structure and the reasoning required. AM involves identifying and classifying complex arguments and relations in textual contexts, which existing methods may not capture effectively. Our work is inspired by knowledge base inference but addresses the unique challenges of AM. To the best of our knowledge, FOL-AM is the first attempt to unify multiple AM tasks into a relation query framework while incorporating FOL reasoning.

6 Conclusion

In conclusion, we introduce the FOL-AM framework, which enhances argumentative structure extraction by explicitly capturing logical reasoning paths through FOL rules. The framework, implemented with both fine-tuning and LLM prompt-based methods, achieves superiority performance and offers enhanced explainability on AM benchmarks, demonstrating its effectiveness for AM.

Limitation

To outline potential future research directions for AM, we conduct a detailed error analysis on 100 cases where FOL-AM made incorrect predictions. Our findings highlight two main challenges.

First, FOL-AM struggles with classifying instances that rely on finer-grained bridging units for ACTC. For example, expressions such as “Thus, it is apparent that...” and “I strongly believe that...” typically indicate a Claim. However, since ACs are defined as FOL terms, these expressions may not constitute an entire AC or may only form a small segment within a longer AC. As a result, the model tends to overlook these fine-grained features during classification. To address this limitation, we suggest that future research explore multi-hop reasoning in AM by incorporating finer-grained bridging units, such as elementary discourse units.

Second, despite our adaptive multi-hop reasoning mechanism, FOL-AM tends to overconstruct long reasoning paths for irrelevant AC pairs. This issue becomes particularly pronounced as the rule length increases, likely due to inaccuracies or error propagation in the intermediate steps of multi-hop reasoning. To mitigate this problem, we propose designing a loss function that penalizes excessive rule length, encouraging the model to generate more effective and concise FOL reasoning chains. Such an approach would help capture AC relationships more accurately while reducing the risk of error propagation.

By addressing these challenges, future work can further enhance the interpretability and robustness of FOL-based reasoning in AM.

Acknowledgements

We thank the anonymous reviewers for their valuable suggestions to improve the quality of this work. This work was partially supported by the National Key Research and Development Program of China (2022YFF0902100), the National Natural Science Foundation of China 62176076, Natural Science Foundation of Guang Dong (2023A1515012922, 2024A1515030166), the Shenzhen Foundational Research Funding (KQTD20190929172835662, JCYJ20220818102415032), the Major Key Project of (PCL2021A06, PCL2023A09), Guangdong Provincial Key Laboratory of Novel Secu-
rity Intelligence Technologies 2022B1212010005 and CIPSC-SMP-ZHIPU Large Model Cross-Disciplinary Fund ZPCG20241119405.

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A Appendix

<table>
  <tr>
    <th>Type</th>
    <th>PE</th>
    <th>CDCP</th>
  </tr>
  <tr>
    <td>Paragraphs</td>
    <td>1833</td>
    <td>731</td>
  </tr>
  <tr>
    <td>Train</td>
    <td>1464</td>
    <td>581</td>
  </tr>
  <tr>
    <td>Test</td>
    <td>369</td>
    <td>150</td>
  </tr>
  <tr>
    <td>Components</td>
    <td>6089</td>
    <td>4779</td>
  </tr>
  <tr>
    <td>Relations</td>
    <td>3832</td>
    <td>1353</td>
  </tr>
  <tr>
    <td>Components Per Sample</td>
    <td>3.5</td>
    <td>6.5</td>
  </tr>
</table>

Table 6: The statistics of the PE and CDCP datasets.

A.1 Data statistics

We evaluate our approach on two popular benchmarks. Table 6 provides detailed dataset statistics.

• PE (Stab and Gurevych, 2017b): Comprising 420 essays totaling 1,833 paragraphs, this corpus categorizes argumentation components (ACs) into three classes (MajorClaim, Claim, Premise) and argumentative relations (ARs) into two types (Support, Attack). The dataset’s structural constraint allows each AC to possess a maximum of one outgoing AR, forming directed tree/forest graphs. We maintain consistent data partitioning with prior works (Kuribayashi et al., 2019a; Bao et al., 2021b): 1,464 ACs for training (10% as validation) and 369 ACs for testing.

• CDCP (Park and Cardie, 2018): This collection contains 731 paragraphs with finer-grained AC categorization (Value, Policy, Testimony, Fact, Reference) and AR types (Reason, Evidence). Unlike PE’s tree structure, CDCP permits multiple AR connections from a single AC, enabling non-tree graph configurations. Following recent implementations (Bao et al., 2021a; Morio et al., 2022b), we adopt the standard split: 581 ACs for training (10% as validation) and 150 ACs for testing.
A.2 Baselines
PE Dataset: Our experimental evaluation compares against seven competitive baselines following previous works (Morio et al., 2022b):

• Joint-ILP (Stab and Gurevych, 2017b): Utilizes Integer Linear Programming for simultaneous optimization of argumentation component type classification (ACTC) and argumentative relation identification (ARI).

• Joint-PN (Potash et al., 2017b): Implements a Pointer Network architecture with attention mechanisms for joint ACTC-ARI learning.

• BERT-Trans (Bao et al., 2021a): Neural transition-based approach employing action sequence generation for argument analysis.

• LSTM+dist (Kuribayashi et al., 2019b): Pioneers LSTM-minus span representations combined with ELMo embeddings for argument mining.

• BART (Lewis et al., 2019): Sequence-to-sequence formulation using BART with target structures.

• ST (Morio et al., 2022b): Longformer-based architecture with biaffine scoring for structural prediction.

• PITA (Sun et al., 2024): A generation framework with dynamic prompt templates and heterogeneous graph modeling is proposed to explicitly coordinate the interaction of different subtasks.

CDCP Dataset: We evaluate against seven competitive baselines:

• SSVM-strict (Niculae et al., 2017): Structured SVM variant with constrained factor graphs for dual-task learning.

• TSP-PLBA (Morio et al., 2020b): Features task-specific parameterization and proposition-level biaffine attention.

• BERT-Trans (Bao et al., 2021a): Neural transition-based approach employing action sequence generation for argument analysis.

• DR-LG (Galassi et al., 2018): Residual networks with link-guided training for joint ACTC, ARI, and ARTC.

• BART (Lewis et al., 2019): Sequence-to-sequence formulation using BART with target structures.

• ST (Morio et al., 2022b): Longformer-based architecture with biaffine scoring for structural prediction.

• PITA (Sun et al., 2024): A generation framework with dynamic prompt templates and heterogeneous graph modeling is proposed to explicitly coordinate the interaction of different subtasks.

A.3 More Implementation Details
We implement the proposed framework using the Transformers library (Wolf et al., 2020) in PyTorch and conduct experiments on a GPU. The model is optimized with AdaW (Loshchilov and Hutter, 2017), utilizing a learning rate of \(3e^{-5}\) and a weight decay of \(1e^{-2}\) for both the PE and CDCP datasets. The batch size is set to 4 for both datasets. For both datasets, we adopt dropout with a dropout rate of 0.1 to avoid overfitting. We set the reasoning hop of \(L\) to 2 for both datasets. We set the number of predicates \(M\) to 40 for PE and 30 for CDCP, while the number of instantiations \(K\) for bridging argument is set to 5 and 4, respectively. To mitigate randomness, we repeat our approach three times with different random seeds and report the average results.

A.4 Overall Performance Supplement
The experimental results for both the PE and CDCP datasets are summarized in Table 1 and Table 2, respectively. We observe that FOL-AM consistently achieves superior performance across both datasets. Specifically, on the PE dataset, our FOL-AM model outperforms the state-of-the-art (SOTA) model PITA by 4.8% in ARC and 2.6% in Avg. A similar trend is observed for the CDCP dataset, where FOL-AM surpasses PITA by 4.9% in ARC and 3.1% in Avg. These consistent improvements highlight the effectiveness of integrating FOL rules to explicitly capture reasoning patterns in AM.

Further analysis reveals three key trends: (i) Traditional feature-engineered models (Joint-ILP, SSVM-strict) exhibit limited competitiveness due to constraints imposed by manual feature engineering; (ii) Pretrained language model-based approaches (BERT-Trans, ST, and PITA) significantly outperform sequential architectures (LSTM+dist,
Figure 5: The impact of the number of predicates. Avg indicates the average value across all metrics.

TSP-PLBA), suggesting that PLMs’ ability to leverage extensive linguistic knowledge benefits argument structure modeling; (iii) Our logic-enhanced framework consistently surpasses all baselines. This improvement aligns with our design objective, demonstrating that explicit integration of logical rules aids in disentangling complex argumentative dependencies that pure representation-learning-based methods may overlook.

A.5 Impact of different number of predicates

Figure 5 analyzes the effect of the number of predicates on the performance of the PE and CDCP datasets. The results indicate that performance improves initially but declines as the number of predicates increases, with the optimal performance observed at 40 predicates for PE and 30 for CDCP. This finding suggests that a moderate number of predicates achieves a balance between capturing multi-hop reasoning paths and mitigating overfitting. An excessive number of predicates may introduce redundant rules, leading to conflicts with essential reasoning steps and diminishing overall model performance.

<table>
  <tr>
    <th>Data</th>
    <th>Model</th>
    <th>TT (min)</th>
    <th>IT (sec)</th>
  </tr>
  <tr>
    <td rowspan="4">PE</td>
    <td>BART</td>
    <td>1.17</td>
    <td>15.00</td>
  </tr>
  <tr>
    <td>ST</td>
    <td>1.23</td>
    <td>4.08</td>
  </tr>
  <tr>
    <td>PITA</td>
    <td>1.27</td>
    <td>2.54</td>
  </tr>
  <tr>
    <td>FOL-AM</td>
    <td>1.42</td>
    <td>9.16</td>
  </tr>
  <tr>
    <td rowspan="4">CDCP</td>
    <td>BART</td>
    <td>0.63</td>
    <td>18.63</td>
  </tr>
  <tr>
    <td>ST</td>
    <td>0.81</td>
    <td>1.75</td>
  </tr>
  <tr>
    <td>PITA</td>
    <td>0.73</td>
    <td>1.26</td>
  </tr>
  <tr>
    <td>FOL-AM</td>
    <td>0.68</td>
    <td>4.83</td>
  </tr>
</table>

Table 7: Computational cost in terms of Training Time (TT) per epoch (minutes) and Inference Time (IT) in the test set (second).

A.6 Computational Cost

We evaluate the computational cost of baseline methods and our FOL-AM model during both training and inference. For a fair comparison, all models use a batch size of 4 for both training and inference.

Table 7 presents the training and inference times on the PE and CDCP datasets. FOL-AM demonstrates competitive efficiency compared to baselines in training and inference. For instance, FOL-AM increases inference time by 6.62 seconds and 3.57 seconds for all samples in PE and CDCP, respectively, compared to PITA. These datasets contain 369 and 150 samples, respectively. The average increase in time per sample is 0.018 seconds for PE and 0.023 seconds for CDCP. Given these small overheads, the computational cost remains acceptable for practical applications.

A.7 Prompt Templates for LLM

The prompt templates of LLM-based FOL-AM are shown in Figure 6, 7, 8 and 9.
Prompt 1: FOL Term Construction.

Given an argumentative text, a pair of argumentative components within the paragraph, please identify the predicates between the argumentative component pair based on the definition of labels. Several words or phrases can represent predicates.

<Definition of Each Label>
[label definition]
</Definition of Each Label >

<Examples>
{examples}
</Examples>

Based on the above definitions, you can extract according to the following steps: Observe the definition of each label in turn, and then identify the predicates in the argumentative component pair that have commonalities with these definitions.

Use the following JSON format to output the extracted properties:
{
    "The identified predicates": [ no more than 10 predicates ]
}

< Argumentative Text >
{argumentative text}
</Argumentative Text >

<The Argumentative Component Pair>
{argumentative component pair}
</The Argumentative Component Pair>

Let's think step by step.

Figure 6: The format of natural language prompts for FOL term construction.

Prompt 2: Generation of Bridging Arguments.

Given an argumentative text, all argumentative components within the paragraph, for a specific argumentative component, please identify the three other argumentative components that are most relevant to it within that paragraph.

<Examples>
{examples}
</Examples>

Use <Answer> and </Answer> to enclose your answer, now please identify the three relevant argumentative components:

<Argumentative Text >
{argumentative text}
</Argumentative Text>

<All Argumentation Components>
{all argumentation components}
</All Argumentation Components>

<The Specific Argumentation Component>
{argumentation component}
</The Specific Argumentation Component>

Let's think step by step.

Figure 7: The format of natural language prompts for the generation of bridging arguments.
Prompt 3: FOL Rule Generation.
Given an argumentative text, a pair of argumentative components within the paragraph, and a relation graph between the argumentative components, please generate several FOL reasoning chains based on the relation graph.

<Examples>
{examples}
</Examples>

Use the following JSON format to output the inference chains:
{
    “Chain1”: “….”,
    “Chain2”: “….”,
    …
}

<Argumentative Text >
{argumentative text}
</Argumentative Text>

<All Argumentation Components>
{all argumentation components}
</All Argumentation Components>

<The Specific Argumentation Component>
{argumentation component}
</The Specific Argumentation Component>

< Relation Graph>
…
</Relation Graph>

Let’s think step by step.

Figure 8: The format of natural language prompts for FOL rule generation.

Prompt 4: FOL Rule Ranking.
Given a argumentation text, a pair of argumentative components within the paragraph, and several FOL reasoning chains, please choose the correct reasoning chain to determine the relation label of the argumentative component pair.

<Definiton of Each Label>
{label definition}
</Definiton of Each Label >

<Examples>
{examples}
</Examples>

Using <Answer> and </Answer> to enclose your answer, now please determine the relation label of the argumentative component pair:

<Argumentative Text >
{argumentative text}
</Argumentative Text>

<All Argumentation Components>
{all argumentation components}
</All Argumentation Components>

<The Specific Argumentation Component>
{argumentation component}
</The Specific Argumentation Component>

< Reasoning Chains>
…
</Reasoning Chains >

Let’s think step by step.

Figure 9: The format of natural language prompts for FOL rule ranking.

Reviewer paper summaries: The paper presents a novel framework for representing reasoning patterns in text using First-Order Logic, FOL-AM. The framework can be incorporated within an argument mining pipeline to address such tasks as argument component and argument relation classification. The conducted experiments demonstrate the effectiveness of the approach compared to previous SOTA when used as a fine-tuning technique as well as the improvement of the LLM performance when incorporating FOL-AM into prompting compared to standard and CoT prompting.
Summary of strengths: 1. The methodology for FOL term construction, rule generation and rule ranking is well-designed. 
2. The experiments include a large number of baselines for comparison; the results are tested for statistical significance.
3. Interesting follow-up experiments as well as an ablation study are conducted.
Summary of weaknesses: 1. It is not clear why the authors choose specifically FOL for the representation of the argumentative relations. There exist other formal argumentation frameworks, for instance, Dung's framework from 1995, therefore the choice of FOL needs to be properly motivated given previous research.
2. The details of the rule generation are underspecified, specifically in terms of the potential relations that make up the rules. In the examples (figures 1 and 4) such relations as *promotion, entailment, elaboration, summary, exemplification*, and *explanation* are mentioned. How are these specific relations derived and what are potential other relations for each type of the rule? The generation of the FOL rules is one of the main features of the FOL-AM framework, however, it does not seem to be grounded in the existing theoretical research on what elements constitute argumentative components or relations. 
3. I have concerns regarding how well the generated rules reflect the actual argumentative structures in texts. 
- According to example 1 in figure 4, the rule for claim consists of a "further elaboration of viewpoints" and "exemplification using suggestions", however, from my understanding of what a claim is, it does not require an elaboration or exemplifications to be considered a claim, furthermore, elaboration and exemplification in relation to AC1 seem to be redundant to me as they mean almost the same thing (you could say that it is an elaboration by means of exemplification). 
- In example 2 a relation of "supportive summary" is identified between AC1 "technology may make people's life more complicated" and AC5 "making their life more complicated and tired". This is not a summary but rather a reformulation (potentially an elaboration because of the use of "tired" in addition to "complicated") and is unnecessary for defining the relation of support between AC1 "technology may make people's life more complicated" and AC4 "games on their phone exert a tremendous fascination on teenagers, leading to a large problem that they got addicted to their phones". The AC4 here can support AC1 on its own, while a relation of reformulation between AC1 and AC5 can be viewed as a separate element of the analysis. It is quite frequent that an argument would require only one premise, so I would like to know if there are any reasons as to why the authors set the number of reasoning hops to 2 in their experimental setting. 
- Based on this, I would say a thorough examination of the generated FOL rules would be a good idea in order to establish whether the generated rules do in fact reflect the argumentative structures in the text. 
3. The authors claim two benefits of their approach over previous work (lines 094-097), one being the fact that FOL-AM captures a compositional structure of the reasoning chain, however, in terms of elementary units of the argumentation as well as relations between them, the structures underlying them are not always as complex as to require multiple reasoning hops (as discussed on the basis of the examples above). And if the analysis of these structures generated by the model is incorrect, then the interpretability (which is claimed to be the second benefit), will also be lacking. 
3.  In the background section that introduces the FOL-AM, the task definition describes an experimental set-up with argumentative components such as *premises* and *claims* and relations of *support*, *attack* or *no-relation*. However, the CDCP corpus (as described in Appendix A.1), contains very different components and relations (*value, policy, fact, testimony, reference* for components and *reason, evidence* for relations). From this, I can only assume that the current framework can be generalised to these types of elements as well. However, given how different these elements are, I believe an explanation of how the framework accounts for these differences is warranted. 
4. In tables 1, 2, 3 and 4 the results are reported for three tasks, ACTC, ARI, and ARC. However, earlier the authors state that they are only "addressing ACTC and ARC subtasks" (line 056-057), as far as I can tell ARI is first introduced in the experiments section as "score to measure the F1 score for identifying argumentative relations between ACs as defined in Morio et al., 2022b" (lines 421-423). What is the difference between ARI and ARC (defined in lines 051-053 as a task that "determines the type of relation (e.g. Support or Attack) between pairs of ACs")? The results indicate that the ARC is a more challenging task than ARI, however, the exact nature of the difference between the tasks is not made clear, furthermore, the ARI results are not discussed in the paper.
5. It is unclear what model is used in the fine-tuning experiments.
Comments, suggestions and typos: Typos:
- Round brackets around in-text citations such as "For instance, (Kuribayashietal.,2019b) enhances AC representations..." (line 062) in the following lines: 063, 065, 067, 423, 580, 581.
- Lack of empty spaces between text (baseline names) and round brackets (citations) in lines 440-450. Also a missing space in line 570.

**Comments after Author Discussion**

I would like to thank the authors for their detailed response. I updated both the soundness and overall assessment scores to 3.0 (from 2.0 and 2.5, respectively) as the authors cleared up several of the main concerns that I had.
Overall assessment scores: 3
Metareview score: 3.5
Metareview: The paper addresses the tasks of Argument Component Type Classification (ACTC) and argument relation classification (ARC): the input is an argumentative text with argument components (ACs) already marked within. The output is the type of each AC (e.g. Claim or Premise), and a prediction of argument pairs that are related, with the type of the relation (e.g. Support, Attack, No-relation). There is no specific assumption on the predefined AC types or relation types.

To address the tasks, the paper suggests formulating both as a relation query task: each prediction is formulated as $rel(x_i, x_j)$, where $rel$ is an AC type or a relation type, and $x_i$ is either an argument or the entire text. For example, *Claim(AC1, text)* indicates that AC1 is predicted to be a claim, and *Attack(AC1, AC2)* means that AC2 attacks AC1.

With this joint formulation, the paper introduces a framework based on First Order Logic (FOL): it constructs rules of the form *p <- r1 ^ r2 ^ r3 ^ …*, where $p$ is a prediction induced from relations $r_i$ (the rules maximal length is defined by a parameter). These rules are then ranked. Two approaches are taken for the construction of the FOL rules and their ranking: one is based on fine-tuning, and the other on zero-shot or few-shot with an LLM.

The suggested FOL approaches are evaluated on two datasets. Results with the fine-tuning methods significantly outperform the LLM-based approach, and several baselines. The LLM-based approach itself outperforms a CoT baseline and a prompt based approach which does not involve FOL rules.
Summary of reasons to publish: - For the addressed tasks, the FOL based framework is novel, and offers explainability
- The empirical evaluation is sound, it involves multiple datasets and baselines, an ablation study, and an analysis of various aspects. The results themselves are promising.
- Having a fine-tuned approach as well as an LLM-based approach allows for an interesting comparison of their performance.