Pairwise Logical Selection of Enthymeme Completions under Semantic-Link Uncertainty

Xuyao Feng; Antonis Bikakis

cs.AIAug 19, 2026

Pairwise Logical Selection of Enthymeme Completions under Semantic-Link Uncertainty

Authors: Xuyao Feng, Antonis Bikakis

Organizations: Department of Computer Science, University College London · Department of Information Studies, University College London

Abstract

Arguments often omit premises or claims, forming enthymemes. We study pairwise logical selection between two candidates for the omitted component. Existing natural language methods can identify or generate candidates but often do not expose how the selected candidate completes the inference, while logic-based approaches usually assume that the required formulae and background knowledge are available. We extend a prior neuro-symbolic pipeline from missing-premise to missing-claim selection and replace binary entailment outcomes with logical-resistance scores. Top-Link uses weighted Partial MaxSAT under a single configuration of highest-confidence semantic links. We then introduce Possible-World Atom-Link Formalization (PWAL), which keeps translated formulae fixed and marginalizes logical resistance over alternative cross-formula semantic-link configurations. We evaluate PWAL on five tasks: ARCT and a CDED-derived task for missing-premise selection, iDebate- and AAE2-derived tasks for missing-claim selection, and alphaNLI for abductive hypothesis selection. Relative to Top-Link, PWAL raises strict accuracy by 2.95-30.86 percentage points and reduces tie rates by 4.57-58.00 percentage points on all five tasks. When ties receive half credit, accuracy still increases by 0.45-6.04 percentage points. PWAL also records the translated formulae, sampled link configurations, and resistance components for every comparison, providing a transparent trace of each score.

Figures & tables

Figure 1: Constructed common-target trace. Atom identifiers precede their italic sans-serif verbalizations. Arrows show confidence and clause weight (blue: entailment; dashed red: contradiction; thick: Top-Link ). PWAL treats non-neutral links and no-link as local alternatives; dotted neutral is discarded.

Dataset ( $N$ )	Metric	FH-HardSAT	Top-Link	PWAL	PWAL w/o no-link	PWAL w/o neg. guard	Direct NLI
$\alpha$ NLI (400)	Acc	11.00	36.50	$\mathbf{49.55\pm 0.96}$	$46.15\pm 0.41$	$49.48\pm 0.89$	57.75
	Tie	55.75	27.25	$2.15\pm 0.50$	$11.98\pm 0.14$	$4.63\pm 0.38$	0.00
	EAcc	38.88	50.13	$50.63\pm 0.98$	$52.14\pm 0.41$	$51.79\pm 0.93$	57.75
ARCT (444)	Acc	7.43	20.05	$\mathbf{50.90\pm 1.12}$	$47.77\pm 1.23$	$40.59\pm 1.12$	70.05
	Tie	86.71	64.86	$6.87\pm 0.41$	$17.32\pm 0.20$	$31.89\pm 0.28$	0.00
	EAcc	50.79	52.48	$54.34\pm 1.07$	$56.43\pm 1.17$	$56.53\pm 1.10$	70.05

Table 1: Pairwise test results (%). PWAL variants report ten-seed means

\pm

sample SD (

K=100

); other methods are deterministic. Failed FH-HardSAT evaluations count as errors; bold marks the best Acc among non-ablated logical methods.

Dataset	Metric	$K=100$	$K=200$	Ref.
$\alpha$ NLI	Acc	$49.55\pm 0.96$	$49.92\pm 1.16$	$49.75$
	Tie	$2.15\pm 0.50$	$1.48\pm 0.25$	$3.25$
	EAcc	$50.63\pm 0.98$	$50.66\pm 1.17$	$51.38$
ARCT	Acc	$50.90\pm 1.12$	$52.00\pm 1.59$	$48.20$
	Tie	$6.87\pm 0.41$	$6.71\pm 0.51$	$14.41$
	EAcc	$54.34\pm 1.07$	$55.36\pm 1.47$	$55.41$

Table 2: Sampling and exact/

10^{5}

-capped reference results (%). Ref. denotes the exact/

10^{5}

-capped reference.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Candidate pair $(S_{c},T_{c})$	Candidate construction	Dev/Test; cluster
$\alpha$ NLI ( Bhagavatula et al. 2020 )	$S_{c}=(O_{1},H_{c})$ , $T_{c}=(O_{2})$	The two released hypotheses are retained in their released order.	400/400; example
ARCT ( Habernal et al. 2018 )	$S_{c}=(R,W_{c})$ , $T_{c}=(C)$	The two released warrants are retained.	400/444; example
CDED ( Rinott et al. 2015 )	$S_{c}=(H_{c})$ , $T_{c}=(C)$	Same-topic evidence annotated as evidence for another claim but not annotated as evidence for $C$ .	200/400; topic
iDebate ( Wang and Ling 2016 )	$S_{c}=(P_{1},\ldots,P_{m})$ , $T_{c}=(C_{c})$	A different central claim from the same debate.	400/400; debate
AAE2 ( Stab and Gurevych 2017 )	$S_{c}=(P_{1},\ldots,P_{m})$ , $T_{c}=(C_{c})$	An opposite-stance Claim from the same essay with a graph-disjoint support subtree.	200/350; essay

Appendix

Table A.1: Source–target mappings, split sizes, and cluster units. Dataset subsampling is without replacement. Candidate positions are balanced for the three derived tasks (CDED, iDebate, and AAE2);

\alpha

NLI and ARCT retain their released candidate order. Parentheses denote ordered statement collections, including singleton targets.

Figure A.1: AAE2 extraction. The source contains every Premise with a direct support edge to the gold Claim and no other argument component. The opposite-stance distractor is selected from the same essay but has a graph-disjoint support subtree.

Dataset	$N$	Clust.	Verb.
CDED	400	39	0
iDebate	400	128	1
AAE2	350	127	0

Appendix

Table A.2: Automatic audit of the three derived pairwise tasks. “Clust.” is the number of test clusters; “Split ov.” counts development–test cluster overlap; “Dup.” counts duplicate complete examples; “Ident.” counts within-example identical candidates; and “Verb.” counts examples in which the normalized gold-candidate text occurs as a substring of the normalized source. A complete example includes its source–target context and both ordered candidates.

Dataset	$N$	Only gold	Both	Only distractor	Neither	Gold or both
CDED	50	39	9	0	2	48 (96%)
iDebate	50	37	13	0	0	50 (100%)
AAE2	100	91	3	4	2	94 (94%)

Appendix

Table A.3: Author-confirmed outcomes of the AI-assisted validity audit. “Gold or both” counts examples for which the gold candidate alone or both candidates were plausible. No item was unclear.

Dataset	$\tau_{c}$	$\tau_{m}$	Dev Acc. (%)	Test $W/L/T$
$\alpha$ NLI	90	0.55	11.50	44/133/223
ARCT	100	0.60	5.25	33/26/385
CDED	100	0.60	35.50	138/63/199
iDebate	100	0.70	29.25	132/48/220
AAE2	100	0.60	17.50	66/67/217

Appendix

Table B.1: FH-HardSAT development selection and held-out test counts.

\tau_{m}

is the BGE cosine-similarity threshold and

\tau_{c}

the NLI contradiction-confidence threshold (in percent). Test

W/L/T

gives correct unique decisions, errors, and valid ties, respectively;

L

includes unsuccessful evaluations. The corresponding metrics appear in the main results.

Dataset	TL ties $(N)$	PWAL EAcc on TL ties $(\%)$	Repairs $(N)$	Damages $(N)$
$\alpha$ NLI	109	$49.36\pm 1.97$	$32.1\pm 2.13$	$29.4\pm 1.84$
ARCT	288	$51.82\pm 1.49$	$11.7\pm 1.16$	$8.7\pm 1.77$
CDED	54	$57.13\pm 3.55$	$39.0\pm 2.40$	$18.7\pm 0.82$
iDebate	20	$52.50\pm 2.64$	$33.9\pm 1.97$	$32.6\pm 1.35$
AAE2	26	$51.15\pm 3.17$	$26.6\pm 1.07$	$23.4\pm 1.17$

Appendix

Table C.1: Top-Link -to-PWAL transition summary. Tied-subset EAcc is PWAL EAcc on the deterministic Top-Link ties. Repairs and damages are full-test mean counts of unique-to-unique transitions from incorrect to correct and correct to incorrect, respectively. Entries are ten-seed means

\pm

sample standard deviations.

Dataset	Wrong	Tension	Witness	Both
$\alpha$ NLI	$193.20\pm 4.26$	$31.73\pm 1.72$	$15.42\pm 2.00$	$52.86\pm 2.47$
ARCT	$187.50\pm 4.72$	$33.07\pm 2.17$	$47.36\pm 2.33$	$19.57\pm 2.45$
CDED	$95.70\pm 2.75$	$16.71\pm 2.07$	$27.60\pm 2.53$	$55.69\pm 2.90$
iDebate	$145.20\pm 2.62$	$8.73\pm 1.23$	$9.91\pm 1.45$	$81.35\pm 1.54$
AAE2	$149.50\pm 1.96$	$11.24\pm 1.31$	$8.15\pm 1.06$	$80.61\pm 1.57$

Appendix

Table C.2: Score-component attribution of PWAL’s incorrect unique decisions at

K=100

. Wrong is the mean count of incorrect unique decisions; the remaining columns are mean percentages. Entries are means

\pm

sample standard deviations over seeds 2026–2035.

Figure C.1: Development-only diagnostic of the logical-resistance components under

\rho_{\alpha}=\alpha\Delta T/C_{\max}-R_{\mathrm{sat}}

. Curves contain every point in the fixed grid

\alpha=0,0.05,\ldots,5

. PWAL curves are ten-seed means and shaded bands show one sample standard deviation; Top-Link is deterministic. Filled markers identify the reported score

\rho_{\alpha}=\rho

\alpha=1

Dataset	$K$	Accuracy	Tie	EAcc
$\alpha$ NLI	10	47.83 $\pm$ 1.62	5.63 $\pm$ 0.48	50.64 $\pm$ 1.68
	20	49.08 $\pm$ 1.83	4.28 $\pm$ 0.59	51.21 $\pm$ 1.80
	50	49.30 $\pm$ 0.86	3.18 $\pm$ 0.51	50.89 $\pm$ 0.76
	100	49.55 $\pm$ 0.96	2.15 $\pm$ 0.50	50.63 $\pm$ 0.98
	200	49.92 $\pm$ 1.16	1.48 $\pm$ 0.25	50.66 $\pm$ 1.17
	Ref.	49.75	3.25	51.38

Appendix

Table D.1: PWAL stability and exact/

10^{5}

-capped reference results. Finite-

K

cells are ten-seed mean percentages

\pm

sample standard deviations in percentage points; Ref. is a fixed percentage.

Dataset	Exact pairs	Capped pairs	Median	P95
$\alpha$ NLI	357	43	294	205,920
ARCT	443	1	42	1,848
CDED	216	184	6,210	$7.47\times 10^{10}$
iDebate	146	254	37,440	$5.00\times 10^{11}$
AAE2	93	257	170,586	$5.40\times 10^{13}$

Appendix

Table D.2: Exact/

10^{5}

-capped reference coverage and world-space scale. World-count summaries are over candidate sides before capping; reference outcomes are included in Table D.1 and summarized in the main paper. A capped pair has at least one candidate with

N_{c}^{\mathrm{world}}>10^{5}

; P95 denotes the 95th percentile.

(a) Ten-seed mean-score approximation
Dataset	Score MAE	Margin MAE	Decision agr.	Tie-status agr.
$\alpha$ NLI	0.0080	0.0117	95.25	97.25
ARCT	0.0078	0.0111	88.51	91.67
CDED	0.0069	0.0105	98.75	99.25
iDebate	0.0089	0.0122	99.00	100.00
AAE2	0.0066	0.0094	99.71	100.00

Appendix

Table D.3: Reference diagnostics. The first block compares decisions formed from ten-seed

K=100

mean scores with the exact/

10^{5}

-capped reference. In panel (a), MAEs are in resistance-score units and the agreement columns are percentages. The second block compares every

K=100

seed separately with exact marginalization on fully exact pairs; entries are mean percentages

\pm

sample standard deviations over seeds 2026–2035. Its final three columns partition sampled incorrect unique decisions into those that become correct, become a tie, or remain wrong.

Dataset	Configuration	Mean	Median	P95
$\alpha$ NLI	Top-Link	1.66	1.67	2.16
	PWAL $K=10$	14.58	14.37	21.71
	PWAL $K=20$	27.31	26.19	41.30
	PWAL $K=50$	68.19	66.75	101.80
	PWAL $K=100$	135.59	133.95	203.65
	PWAL $K=200$	280.89	283.08	411.32

Appendix

Table D.4: Controlled logical-scoring runtime on 20 fully enumerable pairs per dataset (ms/example). P95 denotes the 95th percentile.

(a) Main-method contrasts
Dataset	Contrast	$\Delta$ Acc [95% CI]	$\Delta$ EAcc [95% CI]	Tie reduction [95% CI]
$\alpha$ NLI	PWAL $-$ Direct NLI	-8.20 [-14.52, -1.60] †	-7.12 [-13.46, -0.57] †	-2.15 [-3.38, -1.10] †
$\alpha$ NLI	PWAL $-$ Top-Link	+13.05 [+8.45, +17.68] †	+0.50 [-3.70, +4.69]	+25.10 [+21.00, +29.38] †
$\alpha$ NLI	PWAL $-$ FH-HardSAT	+38.55 [+33.27, +43.85] †	+11.75 [+6.41, +17.14] †	+53.60 [+48.62, +58.67] †
ARCT	PWAL $-$ Direct NLI	-19.14 [-25.18, -13.11] †	-15.71 [-21.69, -9.90] †	-6.87 [-9.14, -4.73] †
ARCT	PWAL $-$ Top-Link	+30.86 [+26.80, +34.91] †	+1.86 [-1.58, +5.24]	+58.00 [+53.33, +62.50] †

Appendix

Table E.1: Paired two-way bootstrap contrasts in percentage points. Brackets are pointwise two-sided 95% bootstrap percentile intervals;

\dagger

marks an interval that excludes zero.

AMR record	Compiler action	Formula effect
and / or with :op $i$	ordered branch traversal	$\land$ / $\lor$
multi-sentence with :snt $i$ or :rel	ordered sentence traversal	$\land$
:polarity -	local scope marker	$\lnot$
:condition	antecedent edge	$\to$
:name , :quant , :unit , and date/value fields	endpoint identity or lexical realization	no separate atom
:wiki	construction-only metadata	no separate atom

Appendix

Table F.1: Principal structural and metadata cases. Records not assigned to a structural or metadata category are semantic records.

(a) PropBank function-tag resolution
PropBank tag	Resolved relation class
PAG , CAU , SRC , EXT , PRP , TMP	agent, cause, source, extent, purpose, and time, respectively
PPT	refine to agent, property, instrument, accompaniment, path, source, location, topic, extent, attribute, active theme, or state; otherwise patient
GOL	refine to beneficiary, result, location, or instrument; otherwise goal
LOC	distinguish path, source, and goal; otherwise location
DIR	distinguish patient, source, goal, path, location, and beneficiary; otherwise direction

Appendix

Table F.2: PropBank-numbered dyadic verbalization. Panel (a) resolves PropBank function tags to relation classes; panel (b) maps those classes to surface templates. Here

C

is the predicate surface and

D

is the argument surface. Relation classes that share a row use the same template.

AMR role or atom kind	Verbalization template
purpose , beneficiary	$C\ \textit{is for}\ D$
time , location	$C\ \textit{occurs at}\ D$
direction	$C\ \textit{proceeds toward}\ D$
domain	$D\ \textit{is}\ C$
mod	$D\ C$
manner	$C\ \textit{occurs in manner}\ D$

Appendix

Table F.3: Surface templates for non-numbered dyads and unary carriers. For :prep-

q

q

is the role suffix with hyphens replaced by spaces.

Join or resolved relation class	$v(\operatorname{Tri}_{r_{a},r_{b}}(c,p,d))$
ARG0+ARG1	$C\ P\ D$
accompanier ; beneficiary / purpose	$C\ P\ \textit{with}\ D$ ; $C\ P\ \textit{for}\ D$
cause ; destination ; direction	$C\ P\ \textit{because of}\ D$ ; $C\ P\ \textit{to}\ D$ ; $C\ P\ \textit{toward}\ D$
duration ; extent	$C\ P\ \textit{for}\ D$ ; $C\ P\ \textit{by}\ D$
instrument ; location ; manner	$C\ P\ \textit{using}\ D$ ; $C\ P\ \textit{at}\ D$ ; $C\ P\ \textit{by}\ D$
medium ; path ; source	$C\ P\ \textit{via}\ D$ ; $C\ P\ \textit{along}\ D$ ; $C\ P\ \textit{from}\ D$

Appendix

Table F.4: Surface templates for same-event triple atoms. Monospaced entries are AMR roles or joins; roman entries are PropBank-resolved classes or fallbacks. Here

C

P

, and

D

are the anchor, predicate, and non-anchor surfaces. Explicit adjunct roles precede PropBank classes. Slash-separated prepositions map in order to the listed classes; no slash is emitted.

Premise	People learn a lot from comment sections.
Candidate A	too much learning hurts
Candidate B	learning never hurts
Claim	Comment sections have not failed

Appendix

Table H.1: ARCT success trace. Candidate B is gold.

Candidate	Link	Source surface	Label	$p$	$w(p)$	Soft clause
A	$a_{1}\!\to f$	comment section	Con	0.848	84	$[\neg a_{1}\lor\neg f]_{84}$
A	$a_{2}\!\to f$	person learn lot	Con	0.982	98	$[\neg a_{2}\lor\neg f]_{98}$
A	$a_{3}\!\to f$	person learn toward section	Con	0.971	97	$[\neg a_{3}\lor\neg f]_{97}$
A	$a_{5}\!\to f$	hurt much	Ent	0.421	42	$[\neg a_{5}\lor f]_{42}$
B	$b_{1}\!\to f$	comment section	Con	0.848	84	$[\neg b_{1}\lor\neg f]_{84}$
B	$b_{2}\!\to f$	person learn lot	Con	0.982	98	$[\neg b_{2}\lor\neg f]_{98}$

Appendix

Table H.2: All retained candidate-specific NLI links. Each candidate’s five local outcomes—the four displayed links and no-link—have probability

1/5

World	Link	$n_{2026}$	$(\Delta T,C_{\max})$	$(R_{\mathrm{sat}},\rho)$
A0	no-link	22	$(0,100)$	$(0,0)$
A1	$a_{1}\!\to f$	17	$(0,184)$	$(1,-1)$
A2	$a_{2}\!\to f$	17	$(0,198)$	$(1,-1)$
A3	$a_{3}\!\to f$	17	$(0,197)$	$(1,-1)$
A4	$a_{5}\!\to f$	27	$(42,142)$	$(0,0.295775)$

Appendix

Table H.3: Candidate-A worlds and seed-2026 counts. Link labels and weights are given in Table H.2 .

World	Link	$n_{2026}$	$(\Delta T,C_{\max})$	$(R_{\mathrm{sat}},\rho)$
B0	no-link	24	$(0,100)$	$(0,0)$
B1	$b_{1}\!\to f$	29	$(0,184)$	$(1,-1)$
B2	$b_{2}\!\to f$	18	$(0,198)$	$(1,-1)$
B3	$b_{3}\!\to f$	16	$(0,197)$	$(1,-1)$
B4	$b_{4}\!\to f$	13	$(0,158)$	$(0,0)$

Appendix

Table H.4: Candidate-B worlds and seed-2026 counts. Link labels and weights are given in Table H.2 .

Premise	Bill was poor.
Candidate A	His business went bankrupt and he had to take a new job.
Candidate B	His business is worth millions.
Outcome	Bill made less money as a computer scientist than he did before.

Appendix

Table I.1: Stable

\alpha

NLI failure trace. Candidate A is gold.

Candidate A source		Candidate B source		Target
$a_{1}$	Bill is poor	$b_{1}$	Bill is poor	$h_{1}$	computer scientist
$a_{2}$	business is bankrupted	$b_{2}$	multiple 1000000 dollars	$h_{2}$	Bill make money
$a_{3}$	job is new	$b_{3}$	he’s business	$h_{3}$	Bill make money
$a_{4}$	he’s business	$b_{4}$	business worth multiple	$h_{4}$	Bill make toward scientist
$a_{5}$	he take job			$h_{5}$	Bill make during before
$a_{6}$	he obligate take			$h_{6}$	money have quant by less

Appendix

Table I.2: Complete atom inventory.

h_{2}

and

h_{3}

are distinct compiled role occurrences even though their verbalizations coincide.

Cand.	Target	Source	Source surface	Label	$p$	$w(p)$
A	$h_{2},h_{3}$	$a_{1}$	Bill is poor	Con	0.989	98
A	$h_{2},h_{3}$	$a_{2}$	business is bankrupted	Con	0.973	97
A	$h_{2},h_{3}$	$a_{4}$	he’s business	Ent	0.670	67
A	$h_{2},h_{3}$	$a_{6}$	he obligate take	Ent	0.587	58
A	$h_{5}$	$a_{3}$	job is new	Con	0.983	98
A	$h_{6}$	$a_{1}$	Bill is poor	Ent	0.579	57

Appendix

Table I.3: All 27 retained NLI links, grouped only when two target occurrences have identical source, label, and confidence.

Target	Candidate A choices	Candidate B choices
$h_{1}$	no-link (probability 1)	no-link (probability 1)
$h_{2}$	no-link, $a_{1}$ Con , $a_{2}$ Con , $a_{4}$ Ent , $a_{6}$ Ent (each $1/5$ )	no-link, $b_{1}$ Con , $b_{2}$ Ent , $b_{3}$ Ent , $b_{4}$ Ent (each $1/5$ )
$h_{3}$	same five A choices as $h_{2}$ (each $1/5$ )	same five B choices as $h_{2}$ (each $1/5$ )
$h_{4}$	no-link (probability 1)	no-link (probability 1)
$h_{5}$	no-link, $a_{3}$ Con (each $1/2$ )	no-link (probability 1)
$h_{6}$	no-link, $a_{1}$ Ent , $a_{2}$ Ent (each $1/3$ )	no-link, $b_{1}$ Ent , $b_{2}$ Con , $b_{4}$ Con (each $1/4$ )

Appendix

Table I.4: Per-target categorical choices. Every listed choice within a cell has the displayed common probability.

Cand.	Draw	Active links	$\Delta T$	$C_{\max}$	$R_{\mathrm{sat}}$	$\rho$
A	86	$a_{2}\!\to h_{3}$ Con	97	97	0	1.000000
A	76	$a_{1}\!\to h_{2}$ Con ; $a_{6}\!\to h_{3}$ Ent ; $a_{3}\!\to h_{5}$ Con ; $a_{2}\!\to h_{6}$ Ent ; $a_{1}\!\to h_{7}$ Con	287	387	$2/7$	0.455888
B	12	$b_{4}\!\to h_{2}$ Ent ; $b_{3}\!\to h_{3}$ Ent ; $b_{1}\!\to h_{7}$ Con	91	252	$2/7$	0.075397
B	20	$b_{4}\!\to h_{2}$ Ent ; $b_{4}\!\to h_{3}$ Ent ; $b_{1}\!\to h_{6}$ Ent ; $b_{4}\!\to h_{7}$ Ent	0	322	$4/7$	-0.571429

Appendix

Table I.5: Representative seed-2026 candidate-specific worlds for the stable

\alpha

NLI failure trace. Con and Ent refer to Table I.3 . Draw is the index within the fixed seed-2026 candidate-specific sample.

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Making Implicit Premises Explicit in Logical Understanding of Enthymemes

Xuyao Feng, Anthony Hunter

Department of Computer Science, University College London, United Kingdom

Natural Language Inference Implicit Reasoning in Language Models

Open paper AskPDF arXiv

Real-world arguments in text and dialogues are normally enthymemes (i.e. some of their premises and/or claims are implicit). Natural language processing (NLP) methods for handling enthymemes can potentially identify enthymemes in text but they do not decode their underlying logic, whereas logic-based approaches for handling them assume a knowledgebase with sufficient formulae that can be used to decode them via abduction. There is therefore a lack of a systematic method for translating textual components of an enthymeme into a logical argument and generating the logical formulae required for their decoding, and thereby showing logical entailment. To address this, we propose a pipeline that integrates: (1) a large language model (LLM) to generate intermediate implicit premises based on the explicit premise and claim; (2) another LLM to translate the natural language into logical formulas; and (3) a neuro-symbolic reasoner based on a SAT solver to determine entailment. We evaluate our pipeline on two enthymeme datasets, demonstrating promising performance in selecting the correct implicit premise, as measured by precision, recall, F1-score, and accuracy.

Abstract

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