We consider the problem of answering queries about formulas of first-order logic based on background knowledge partially represented explicitly as other formulas, and partially represented as examples independently drawn from a fixed probability distribution. PAC semantics, introduced by Valiant, is one rigorous, general proposal for learning to reason in formal languages: although weaker than classical entailment, it allows for a powerful model theoretic framework for answering queries while requiring minimal assumptions about the form of the distribution in question. To date, however, the most significant limitation of that approach, and more generally most machine learning approaches with robustness guarantees, is that the logical language is ultimately essentially propositional, with finitely many atoms. Indeed, the theoretical findings on the learning of relational theories in such generality have been resoundingly negative. This is despite the fact that first-order logic is widely argued to be most appropriate for representing human knowledge. In this work, we present a new theoretical approach to robustly learning to reason in first-order logic, and consider universally quantified clauses over a countably infinite domain. Our results exploit symmetries exhibited by constants in the language, and generalize the notion of implicit learnability to show how queries can be computed against (implicitly) learned first-order background knowledge.

Automating the translation of natural language to first-order logic (FOL) is crucial for knowledge representation and formal methods, yet remains challenging. We present a systematic evaluation of fine-tuned LLMs for this task, comparing architectures (encoder-decoder vs. decoder-only) and training strategies. Using the MALLS and Willow datasets, we explore techniques like vocabulary extension, predicate conditioning, and multilingual training, introducing metrics for exact match, logical equivalence, and predicate alignment. Our fine-tuned Flan-T5-XXL achieves 70% accuracy with predicate lists, outperforming GPT-4o and even the DeepSeek-R1-0528 model with CoT reasoning ability as well as symbolic systems like ccg2lambda. Key findings show: (1) predicate availability boosts performance by 15-20%, (2) T5 models surpass larger decoder-only LLMs, and (3) models generalize to unseen logical arguments (FOLIO dataset) without specific training. While structural logic translation proves robust, predicate extraction emerges as the main bottleneck.

Due to its expressiveness and unambiguous nature, First-Order Logic (FOL) is a powerful formalism for representing concepts expressed in natural language (NL). This is useful, e.g., for specifying and verifying desired system properties. While translating FOL into human-readable English is relatively straightforward, the inverse problem, converting NL to FOL (NL-FOL translation), has remained a longstanding challenge, for both humans and machines. Although the emergence of Large Language Models (LLMs) promised a breakthrough, recent literature provides contrasting results on their ability to perform NL-FOL translation. In this work, we provide a threefold contribution. First, we critically examine existing datasets and protocols for evaluating NL-FOL translation performance, revealing key limitations that may cause a misrepresentation of LLMs' actual capabilities. Second, to overcome these shortcomings, we propose a novel evaluation protocol explicitly designed to distinguish genuine semantic-level logical understanding from superficial pattern recognition, memorization, and dataset contamination. Third, using this new approach, we show that state-of-the-art, dialogue-oriented LLMs demonstrate strong NL-FOL translation skills and a genuine grasp of sentence-level logic, whereas embedding-centric models perform markedly worse.

In particular, he proposed a semantics to capture the quality possessed by the output of (probably approximately correct) P AC-learning algorithms when formulated in a logic.

Currently, there is a lack of rigorous theoretical system for systematically generating non-trivial and logically valid theorems. Addressing this critical gap, this paper conducts research to propose a novel automated theorem generation theory and tool. Based on the concept of standard contradiction which possesses unique deductive advantages, this paper defines and proves, for the first time, a new logical structure known as rectangular standard contradiction. Centered on this structure, a complete Automated Theorem Generation (ATG) theory is put forward. Theoretical proofs clarify two core properties of rectangular standard contradiction: first, it is a standard contradiction (necessarily unsatisfiable); second, it exhibits non-redundancy (the remaining clause set becomes satisfiable after removing any clause). Leveraging these properties, this paper proves that partitioning a rectangular standard contradiction into a premise subset $A$ and negation of its complement $H$, a valid theorem $A \vdash \neg H$ can be formed, and all such theorems are logically equivalent. To implement this theory, an efficient template-based ATG algorithm is designed, and a Rectangular Automated Theorem Generator is developed. This research enables machines to transition from "verifiers" to "discoverers", opening up new avenues for fundamental research in the fields of logic and artificial intelligence.

Artificial Intelligence (AI) is a powerful new language of science as evidenced by recent Nobel Prizes in chemistry and physics that recognized contributions to AI applied to those areas. Yet, this new language lacks semantics, which makes AI's scientific discoveries unsatisfactory at best. With the purpose of uncovering new facts but also improving our understanding of the world, AI-based science requires formalization through a framework capable of translating insight into comprehensible scientific knowledge. In this paper, we argue that logic offers an adequate framework. In particular, we use logic in a neurosymbolic framework to offer a much needed semantics for deep learning, the neural network-based technology of current AI. Deep learning and neurosymbolic AI lack a general set of conditions to ensure that desirable properties are satisfied. Instead, there is a plethora of encoding and knowledge extraction approaches designed for particular cases. To rectify this, we introduced a framework for semantic encoding, making explicit the mapping between neural networks and logic, and characterizing the common ingredients of the various existing approaches. In this paper, we describe succinctly and exemplify how logical semantics and neural networks are linked through this framework, we review some of the most prominent approaches and techniques developed for neural encoding and knowledge extraction, provide a formal definition of our framework, and discuss some of the difficulties of identifying a semantic encoding in practice in light of analogous problems in the philosophy of mind.

Reinforcement Learning with Verifiable Rewards (RLVR)-based post-training of Large Language Models (LLMs) has been shown to improve accuracy on reasoning tasks and continues to attract significant attention. Existing RLVR methods, however, typically treat all tokens uniformly without accounting for token-level advantages. These methods primarily evaluate performance based on final answer correctness or Pass@K accuracy, and yet make claims about RL post-training leading to improved reasoning traces. This motivates our investigation into the effect of RL post-training on intermediate tokens which are not directly incentivized. To study this, we design an experimental setup using the GRPO algorithm with Qwen-2.5-0.5B model on the GSM8K dataset. We introduce trace coherence, a First-Order Logic (FOL)-based measure to capture the consistency of reasoning steps by identifying errors in the traces. We distinguish between trace validity and trace coherence, noting that the former implies logical soundness while the latter measures local coherence via lack of errors. Our results show that RL post-training overall improves trace coherence with the most significant gains on problems where the base model fails but the RL model succeeds. Surprisingly, RL enhances local coherence without necessarily producing valid or correct solutions. This highlights a crucial distinction: improved local coherence in reasoning steps does not guarantee final answer correctness. We argue that claims of improved reasoning via RL must be examined with care, as these may be based on improved trace coherence, which may not translate into fully valid mathematical proofs.

Neural Information Processing Systems http://nips.cc/

Efforts to apply transformer-based language models (TLMs) to the problem of reasoning in natural language have enjoyed ever-increasing success in recent years. The most fundamental task in this area to which nearly all others can be reduced is that of determining satisfiability. However, from a logical point of view, satisfiability problems vary along various dimensions, which may affect TLMs' ability to learn how to solve them. The problem instances of satisfiability in natural language can belong to different computational complexity classes depending on the language fragment in which they are expressed. Although prior research has explored the problem of natural language satisfiability, the above-mentioned point has not been discussed adequately. Hence, we investigate how problem instances from varying computational complexity classes and having different grammatical constructs impact TLMs' ability to learn rules of inference. Furthermore, to faithfully evaluate TLMs, we conduct an empirical study to explore the distribution of satisfiability problems.

Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus "grassroots mathematics". We begin with a brief review of first-order logic with counting operators and cardinality comparisons. This system is known to be of high complexity, and drowns out finer aspects of the combination of logic and counting. We move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting. We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced. Next, we investigate a series of strengthenings, again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.