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Conditional and Modal Reasoning in Large Language Models

Wesley H. Holliday, Matthew Mandelkern, Cedegao E. Zhang

arXiv:2401.17169v4cs.CLcs.AIcs.LO

TL;DR

Whether LLMs distinguish valid from fallacious conditional and modal inferences matters because reasoning about distal possibilities is a distinctive human cognitive ability. The paper probes LLM reasoning on these patterns and finds inconsistent judgments even among the best models, with almost all models differing from reported human judgments on some complex conditional inferences.

  • Problem

    The paper asks whether LLMs have mastered conditional and modal reasoning, which concerns distal possibilities central to human cognition.

    Source: §1 · p. 1

  • Method

    The paper applies philosophical, logical, and linguistic methodologies in a focused benchmark of LLM reasoning with conditionals and modals.

    Source: §2 · p. 2, §1 · p. 2

  • Results

    Even the best models show inconsistent judgments, and almost all give answers to some complex conditional inferences that differ from reported human judgments.

    Source: §1 · p. 2, §5 · p. 9

  • Takeaways & Limitations

    LLM judgments may be unreliable on novel inference patterns, even when those patterns are natural and intuitive for humans.

    Source: §5 · p. 9

    The study focuses on conditionals and “must” and “might,” leaving other operators, connectives, and quantifiers for future study.

    Source: §7 · p. 10

Abstract

from arXiv · show

The reasoning abilities of large language models (LLMs) are the topic of a growing body of research in AI and cognitive science. In this paper, we probe the extent to which twenty-nine LLMs are able to distinguish logically correct inferences from logically fallacious ones. We focus on inference patterns involving conditionals (e.g., 'If Ann has a queen, then Bob has a jack') and epistemic modals (e.g., 'Ann might have an ace', 'Bob must have a king'). These inferences have been of special interest to logicians, philosophers, and linguists, since they play a central role in the fundamental human ability to reason about distal possibilities. Assessing LLMs on these inferences is thus highly relevant to the question of how much the reasoning abilities of LLMs match those of humans. All the LLMs we tested make some basic mistakes with conditionals or modals, though zero-shot chain-of-thought prompting helps them make fewer mistakes. Even the best performing LLMs make basic errors in modal reasoning, display logically inconsistent judgments across inference patterns involving epistemic modals and conditionals, and give answers about complex conditional inferences that do not match reported human judgments. These results highlight gaps in basic logical reasoning in today's LLMs.

1 Introduction

The paper probes whether LLMs master logical inference patterns involving conditionals and epistemic modals, reasoning capacities linked to planning, causal reasoning, and other human abilities. It reports persistent fallacies, inconsistent modal judgments, and errors on complex conditional inferences.

1 Introduction

  • Conditional and modal reasoning about distal possibilities supports planning, causal reasoning, and retroactive evaluation, motivating its study as a distinctive human cognitive ability.The introduction frames the study around whether LLMs have mastered these forms of reasoning.Source: §1 · p. 1
  • The paper situates conditional and modal reasoning in established philosophical, linguistic, logical, and computational research, emphasizing its importance and nuance.Its contribution draws on up-to-date evidence and theories from the relevant literature.Source: §1 · pp. 1–2
  • LLMs commit basic fallacies in modal and conditional reasoning; even the best-performing models show inconsistent judgments, and almost all diverge from reported human judgments on some complex conditional inferences.The paper also reports that performance on these tasks is highly correlated with Chatbot Arena Elo ratings, MMLU, and GSM8K.Source: §1 · p. 2
  • The authors propose a focused benchmark of conditional and modal logical inference, assessing accuracy across patterns and comparing zero-shot, few-shot, and chain-of-thought prompting.They report performance for a large set of LLMs across different prompting settings.Source: §1 · p. 2

2 Background and related work

The paper applies philosophical, logical, and linguistic approaches to assess LLMs’ single-step logical reasoning about conditionals and modals. It distinguishes this target from broader common-sense reasoning and considers how the models’ judgments compare with human inferential dispositions reported in the literature.

2.1 Logical inference

  • Logical validity depends only on logical-word meanings, unlike broader “logical reasoning” that includes content-based leaps; this study targets fundamental single-step patterns, including modal–conditional interactions.The authors treat single-step inference as a fundamental target that may help explain failures on multi-step reasoning problems.Source: §2.1 · pp. 2–3

2.2 Modals and conditionals

  • Natural-language “if” is poorly modeled as the material conditional, yet existing tests often assume it; conformity may be logically mistaken, while nonconformity may be correct.The material analysis can yield implausible consequences, such as a conditional becoming more probable as its antecedent becomes less probable.Source: §2.2 · p. 3
  • The paper treats modals as quantifiers over possible worlds and conditionals as restricted modal operators, testing reported human inferential dispositions without adjudicating disputed logics.The authors compare LLMs with expert claims from philosophy, logic, and semantics, while leaving direct comparison with human subjects to future work.Source: §2.2 · pp. 3–4

2.3 Natural language inference

  • Against NLI-style commonsense entailment, the authors evaluate austere one-step logical entailment and bring modal and conditional theories to bear on how closely LLMs match human reasoning.Their approach extends beyond propositional and predicate logic to assess conditionals and modals.Source: §2.3 · p. 4

3 Experiments

Experiments tested 29 open- and closed-weight LLMs on inference-pattern questions using varied stimuli and prompt conditions. Responses were broadly consistent across predicate and wording changes, though explicit validity framing affected some judgments.

3.1 Models

  • The experiments tested 29 open- and closed-weight LLMs, running models from Anthropic, Google, Mistral, and OpenAI through their APIs and open-weight models through cloud providers.The reported API-call cost was approximately $3,000.Source: §3.1 · p. 4

3.2 Data

  • For each inference pattern, researchers created 20 instances, including nonsense-predicate versions to reduce reliance on world knowledge and premise-order variants for two-premise inferences.They reviewed and manually adjusted all LLM-generated stimuli.Source: §3.2 · p. 4

3.3 Evaluation

  • The evaluation tested each instance with temperature 0 and 1 under zero-shot, few-shot, and zero-shot chain-of-thought prompting; at temperature 1, questions were repeated to estimate yes/no response frequencies.The study also assessed whether 20 instances per inference pattern were sufficient by comparing response frequencies for sensible and nonsensical predicates.Source: §3.3 · p. 4
  • The overall correlation between yes-frequencies for sensible and nonsensical instances across Table 1 inferences was .85, suggesting 20 instances captured responses to logical form rather than only particular content.The authors judged the correlations sufficient to assess each inference pattern despite the possibility of running more instances.Source: §3.3 · p. 5
  • Substituting “deduce,” “conclude,” or their “logically” variants for “infer” produced qualitatively similar results, with correlations over .9 between yes-frequencies.The study tested these prompt alternatives on the AC inference.Source: §3.3 · p. 5
  • Explicitly asking whether an argument form was logically valid decreased some LLMs’ acceptance of fallacious inferences.The authors noted this differed from the phrase substitutions, but the passage does not provide a quantified effect.Source: §3.3 · p. 5

4 Results

Across baseline and targeted tests, LLM performance varied and remained error-prone: chain-of-thought prompting improved aggregate accuracy, but modal and conditional judgments showed overgeneralization and inconsistency.

4 Results

  • No model exceeded 90% accuracy on uncontroversial inferences; Llama 3.1 405B was best overall, and performance roughly correlated with model size.Llama 3.1 405B and 70B were the only open-weights models above 80% accuracy.Source: §4 · p. 5, Table 2 · p. 6
  • Few-shot performance was similar to zero-shot, while zero-shot CoT significantly improved accuracy, bringing the best models near 90% but leaving modal and conditional errors.The paper reports the few-shot difference as not statistically significant and the CoT improvement as statistically significant.Source: §4 · pp. 5–6

4.1 Divergences from the material analysis

  • LLMs largely rejected CT and AS, matching human judgments and showing that they do not interpret natural-language conditionals as material conditionals.Treating conditionals as material would wrongly classify these rejections as mistakes.Source: §4.1 · p. 6

4.2 Inconsistency and overgeneralization

  • Many models accepted MTmu while rejecting MTmi, and similarly accepted DSmu while rejecting DSmi; the paper identifies these judgments as logically inconsistent.The MT pattern also conflicts with models’ acceptance of the equivalence between “might not” and “not must.”Source: Figure 2 · p. 7, §4.2 · p. 7, Figure 3 · p. 7
  • Leading models often remained logically inconsistent when asked about DSmu, MiN, and DSmi in the same context window, with responses strongly sensitive to question order.The inconsistency persisted despite testing the three patterns within one context window.Source: Figure 4 · p. 8, §4.2 · p. 8
  • The strongest models uniformly accepted CMP with zero-shot prompts; CoT improved performance, but most still accepted it at high rates, contrary to human judgment.This pattern, alongside the MT and DS findings, supports overgeneralization from Boolean inferences to unrestricted substitutions.Source: Figure 5 · p. 8, §4.2 · p. 8

4.3 Modal fallacies and free choice

  • Almost all models accepted the invalid MuDistOr and MiAg patterns; many accepted NSFC but rejected WSFC, potentially conflicting with their acceptance of modal disjunction equivalence.Models generally reasoned correctly about modal duality, but made basic errors in other modal inferences.Source: §4.3 · p. 8

4.4 Relationship to some popular benchmarks

  • Our benchmark correlated with LMSYS Elo, MMLU, and GSM8k at 0.81, 0.85, and 0.75, respectively, supporting a relationship between logical reasoning and broader capabilities.The reported correlations each had p-values below 0.01.Source: Figure 6 · p. 9, §4.4 · p. 9

5 Discussion

The results reflect familiar benefits of larger models and chain-of-thought prompting, yet even the best models show inconsistent, counterintuitive reasoning. The discussion highlights risks in novel inference patterns and challenges for automated natural-language reasoning systems.

5 Discussion

  • Larger models often perform better and chain-of-thought prompting often helps, but even the best models show inconsistent and counterintuitive reasoning with or without it.The inference patterns studied draw on state-of-the-art philosophy and logic research and may be less represented in LLM training and fine-tuning data.Source: §5 · p. 9
  • LLM judgments may be unreliable on novel, natural-to-humans inference patterns, suggesting these patterns can inform out-of-distribution evaluation.The authors note such patterns may be less present, though likely not absent, in training and fine-tuning data.Source: §5 · p. 9
  • A neurosymbolic semantic-parsing approach such as Olausson et al. 2023 will not automatically solve these tasks, since theorem provers lack nuanced representations and modal semantics remain controversial.The authors characterize automated natural-language reasoning that sensibly handles these tasks as challenging regardless of the model or approach.Source: §5 · p. 9

6 Conclusion

The paper uses philosophical, logical, and linguistic analyses to study LLM reasoning about conditionals and modals, and outlines several directions for future research.

6 Conclusion

  • The study applies philosophical, logical, and linguistic analyses of inference to examine current LLMs’ reasoning about conditionals and modals.The authors hope this work will begin a broader research strand on LLMs.Source: §6 · p. 9
  • Future work could compare LLMs with experimental human subjects across the tested inferences, beyond the expert judgments reported from the literature.Such experiments could reveal mistakes similar to or different from those found in LLMs.Source: §6 · p. 9
  • Further research could test more inference patterns, including quantifiers, attitude predicates, and degree constructions, and connect modal and conditional reasoning to probability, causal models, and mental simulations.The authors call more generally for studying LLMs through philosophical, linguistic, and logical insights into reasoning and meaning.Source: §6 · p. 9

7 Limitations … C.1.6 Denying the Antecedent (DA)

This section describes the evaluation’s scope and limitations, the prompts and models used, and appendix material on individual inferences, including example conversion and denying-the-antecedent prompts and a temperature setting for AC. The provided passages report no results for these tests.

7 Limitations

  • The study does not systematically test operators such as ‘and’, ‘or’, quantifiers, ‘probably’, or ‘certainly’, their interactions, more complex phrasing, or languages beyond English.The authors identify these areas as directions for future work.Source: §7 · p. 10

B Prompts used in the experiments

  • Prompts were generally shared across models, with extra step-by-step reminders added for some small models; the appendices show zero-shot, few-shot, and zero-shot chain-of-thought prompt formats.The chain-of-thought prompt asks models to explain their reasoning before giving a yes-or-no answer.Source: B Prompts used in the experiments · p. 14, §B.1 · p. 14, §B.2 · p. 14

C.1 Individual inferences

  • Appendix C.1 presents zero-shot performance on the Table 1 inferences at temperatures 0 and 1.The supplied passages introduce examples for disjunctive syllogism, modus ponens, modus tollens, and affirming the consequent, but provide no performance values.Source: §C.1 · p. 14, §C.1.1 · p. 14, §C.1.2 · p. 15

C.1.4 Affirming the Consequent (AC)

  • AC is shown with a zero-shot, temperature-1 condition, but the supplied fragment gives no readable AC prompt or outcome.Source: §C.1.4 · p. 15

C.1.5 Conversion CONV)

  • The conversion prompt asks whether “If Mary was at the wedding, then Sue was at the wedding” licenses the converse conditional.Source: §C.1.5 · p. 15

C.1.6 Denying the Antecedent (DA)

  • The DA prompt asks whether Mary’s absence, given that Mary’s presence implies Sue’s presence, licenses the conclusion that Sue was absent.The supplied DA fragments specify zero-shot testing at temperatures 0 and 1, but provide no readable outcomes.Source: §C.1.6 · p. 16

C.1.7 Inversion (INV)

This section examines inversion: whether a conditional supports a new conditional that negates both its antecedent and its consequent.

C.1.7 Inversion (INV)

  • The inversion example asks whether “If Mary was at the wedding, then Sue was at the wedding” licenses “If Mary was not at the wedding, then Sue was not at the wedding.”The supplied passages state the inference question but do not report the models’ judgments.Source: §C.1.7 · p. 16

C.1.8 Might Not (MiN) … C.1.13 DS with ‘might’ (DSmi)

This section presents example inference tests covering modal negation, conditional reasoning, and disjunctive syllogism involving epistemic modals. It also shows response-frequency plots for disjunctive-syllogism items under zero-shot conditions at temperatures 0 and 1.

C.1.8 Might Not (MiN)

  • MiN asks whether “Mary might not have been at the wedding” supports “It’s not the case that Mary must have been at the wedding.”The supplied setup labels this a MiN example; it does not provide response outcomes.Source: §C.1.8 · p. 16

C.1.9 Not Must (NMu)

  • NMu asks whether “It’s not the case that Mary must have been at the wedding” supports “Mary might not have been at the wedding.”The supplied setup labels this an NMu example; it does not provide response outcomes.Source: §C.1.9 · p. 16

C.1.10 Antecedent Strengthening (AS)

  • AS tests whether adding a condition that undermines the consequent preserves an inference, using a soaked-match example designed to expose invalid antecedent strengthening.The graph label “ASd” denotes deviant instances designed to bring out the invalidity of antecedent strengthening.Source: §C.1.10 · p. 17

C.1.11 Contraposition (CT)

  • CT tests contraposition with negation in the premise (p →¬q ⊢ q →¬p), using deviant instances intended to expose invalidity of CT.The example begins with “If it's raining, then it's not raining hard.”Source: §C.1.11 · p. 17

C.1.12 DS with ‘must’ (DSmu)

  • DSmu tests whether “Fido is inside or Fido must be in the garden” plus the denial that Fido must be in the garden supports “Fido is inside.”The passages provide the inference example but no model response outcomes.Source: §C.1.12 · p. 18

C.1.14 MT with ‘must’ (MTmu)

This section examines MT with “must” (MTmu), presenting model responses to a conditional inference and reporting yes/no frequencies under zero-shot conditions.

C.1.14 MT with ‘must’ (MTmu)

  • The displays specify zero-shot conditions at temperature 0 and temperature 1.Source: §C.1.14 · p. 19
  • The MTmu example asks whether “Mary was not at the wedding” follows from a conditional to Sue’s must-attendance and the negation that Sue must have attended.The example uses “If Mary was at the wedding, then Sue must have been at the wedding” and “It’s not the case that Sue must have been at the wedding.”Source: §C.1.14 · p. 19
  • The result display reports “Yes” and “No” response frequencies across the listed models, but the supplied passages do not provide readable values to establish a response pattern.The display labels include models such as Gemma 2 27B, Llama 2 Chat 13B, Mistral 7B, and Llama 3.1 Instruct 405B.Source: §C.1.14 · p. 19

C.1.15 MT with ‘might’ (MTmi) · C.1.16 Complex Modus Ponens (CMP)

The sections test a conditional–modal inference about whether Mary was absent from a wedding and a complex conditional inference framed around NBA championship contenders. The supplied passages show zero-shot conditions and yes/no-frequency chart labels, but not readable model-by-model outcomes.

C.1.15 MT with ‘might’ (MTmi)

  • MTmi asks whether Mary’s absence follows from “If Mary was at the wedding, then Sue must have been” and “Sue might not have been at the wedding.”The supplied example passage cuts off before the question is complete.Source: §C.1.15 · p. 19

C.1.16 Complex Modus Ponens (CMP)

  • The cited passages report zero-shot conditions at temperatures 0 and 1, alongside chart labels for ‘Yes’ Frequency and ‘No’ Frequency; they do not identify these conditions or charts as CMP.Source: §C.1.16 · p. 20
  • CMP frames a nested conditional: if the Warriors do not win, then if the Lakers do not win, the Celtics will win; the Lakers are described as most likely to win, with the Warriors next and Celtics far behind.The question asks whether it follows that it is likely that, if the Lakers do not win, the…; the supplied passage truncates the final clause.Source: §C.1.16 · p. 20

C.1.17 ‘Must’ distribution over ‘or’ (MuDistOr) · C.1.18 ‘Might’ agglomeration (MiAg)

These subsections present model probes of two modal inference patterns: distributing “must” over “or” and agglomerating two “might” claims. The supplied passages identify example queries and experimental conditions but do not provide readable response frequencies.

C.1.17 ‘Must’ distribution over ‘or’ (MuDistOr)

  • MuDistOr asks whether “must” distributes across alternatives, using an envelope that must have been upstairs or under a bed as its example.Source: §C.1.17 · p. 21
  • The MuDistOr zero-shot display is labeled Temperature: 0 and distinguishes 'Yes' and 'No' frequencies.Source: §C.1.17 · p. 21
  • A second MuDistOr zero-shot display is labeled Temperature: 1 and uses the same inference example.Source: §C.1.17 · p. 21

C.1.18 ‘Might’ agglomeration (MiAg)

  • MiAg asks whether two claims that the envelope might be upstairs and might be under a bed license the combined claim that it might be upstairs under a bed.Source: §C.1.18 · p. 21
  • The MiAg zero-shot display is labeled Temperature: 0 and separates responses into 'Yes' and 'No' frequencies.Source: §C.1.18 · p. 21

C.1.19 Narrow-scope free choice (NSFC) · C.1.20 Wide-scope free choice (WSFC) · C.2 Performance summaries

The sections test narrow- and wide-scope free-choice inferences and present performance summaries for uncontroversial logical inference patterns. The supplied passages identify the test examples, model and response-frequency labels, and evaluation conditions, but do not provide readable model-level results.

C.1.19 Narrow-scope free choice (NSFC)

  • The NSFC example asks whether “John might clean his room or go to the lecture” licenses both modal claims conjoined.The source and proposed inference are presented as an example, not as a reported model result.Source: §C.1.19 · p. 21
  • The cited passages label 'Yes' and 'No' response frequencies on a percentage scale for zero-shot evaluation at temperatures 0 and 1.Source: §C.1.19 · p. 21

C.1.20 Wide-scope free choice (WSFC)

  • The passages specify a zero-shot evaluation at temperature 0 and show “Yes” and “No” response-frequency labels, but the excerpts do not report readable model-level outcomes.Source: §C.1.20 · p. 21
  • The WSFC example asks whether “John might clean his room or he might go to the lecture” licenses both possibilities under separate modal claims.Source: §C.1.20 · p. 21

C.2 Performance summaries

  • Figure 7 summarizes performance on uncontroversial logical inference patterns under different conditions at temperature 0.Source: Figure 7 · p. 22
  • Figure 8 summarizes performance on uncontroversial logical inference patterns under zero-shot and few-shot conditions at temperature 1.Source: Figure 8 · p. 22
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