First-Order Logic

FOLLIE (for First-Order Logic for Language Inference and Entailment for French) connects French sentences, compositional first-order representations and automated natural language inference (NLI). The first French NLI-ready, NL-to-FOL resource, it contains 19,983 inference records and 36,987 non-empty sentence-slot--formula mappings from six heterogeneous NLI sources, retaining French lexical material and source problem structure. Its silver annotations are produced by a neural--symbolic pipeline, with 111 separately identified LLM-assisted additions. The construction combines morphosyntactic analysis, probability-conditioned categorial multitagging, grammatical proof search, discourse representation and first-order semantic construction. Stage-wise experiments show substantial gains in derivation coverage; analyses of conjunction distributions, logical-operator incidence and scope alternation characterise the resulting resource. We then develop an automatic inference engine whose input-conditioned background theories address lexical relations, quantification, quantities, event structure and temporal dependencies. Prover9 supplies proofs for entailment and contradiction, while Mace4 supports bounded consistency checks. Three-way accuracies are 65.37% on all 335 FraCaS-FR problems, 70.33% on 300 GQNLI-FR problems and 71.35% on 9,399 eligible SICK-FR records pooled across splits. Agreement with reference labels among committed decisions is 93.53%, 100.00% and 98.66%, at commitment coverage of 41.49%, 37.00% and 15.05%, respectively. Worked constructions, axiom schemas and proof analyses explain these outcomes and identify the contributions of scope, reference, lexical knowledge and numerical structure. FOLLIE provides a French experimental basis for studying the representations and assumptions that connect sentence interpretation to deductive inference.

Automated theorem generation (ATG) seeks to construct new theorems rather than merely prove given conjectures. Existing approaches typically generate candidate statements that require subsequent proof or formal verification. We propose a constructive framework for ATG in first-order logic based on triangular standard contradictions. The method constructs minimally unsatisfiable clause sets from selected first-order literals, then forms theorems by retaining some clauses as premises and taking the negation of one clause, or a conjunction of clauses, as the conclusion. Minimal unsatisfiability ensures that the remaining premises are satisfiable, avoiding trivial inference by explosion, while unsatisfiability of the full set guarantees the resulting entailment. Theorems are therefore correct by construction, without a separate proof procedure for each instance. Extending the previously developed $Δ_1$ generator, we introduce seven methods, $Δ_2$--$Δ_8$, forming a unified family of eight theorem-generation methods. Four generators, $Δ_1$--$Δ_4$, have been implemented and released. We also formulate a general $Δ$-class method encompassing the family. Combinatorial analysis demonstrates the capacity to generate large numbers of distinct theorems systematically. This framework offers a principled foundation for automated theorem discovery and the construction of logically guaranteed knowledge.

Solving machine learning problems is complex and typically reserved for experts. Over the past two decades, systems have emerged to support non-experts. Based on our review, we identify three categories: (1) fully automated AutoML systems, (2) expert cheat sheets for algorithm selection, and (3) decision-support systems using selection criteria (accuracy, transparency, data requirements). We propose a new platform combining categories 2 and 3 to deliver semi-automated, intelligent solution recommendations for non-experts. Unlike existing approaches that recommend a single algorithm, our platform suggests a complete pipeline tailored to the user's problem. It integrates expert-defined selection criteria with transfer learning and automatically extracts data characteristics (e.g., class imbalance, missing values) from user-provided datasets. The platform uses first-order logic to reason over its knowledge base and recommends suitable algorithms ranked by relevance. It features a user-friendly interface and connects to a crowdsourcing platform for ML experts, ensuring continuous updates. The platform is built incrementally, allowing seamless integration of new algorithms, criteria, and domain knowledge. To our knowledge, this is the first free, publicly accessible online framework that systematically captures and operationalizes expert knowledge to guide non-experts in solving ML problems in a structured, transparent manner.

The expressive limitations of message-passing Graph Neural Networks (GNNs) have motivated a wide range of more powerful graph learning architectures. We advocate Deep Homomorphism Networks (DHNs) as a model particularly well-suited for learning over relational databases, due to their close connection to important fragments of SQL such as conjunctive queries. We study the precise expressive power of DHNs by relating them to various natural fragments and extensions of first-order logic (FO). For DHNs with max, sum, and mean aggregations, we establish connections to the unary negation fragment (UNFO) and to the extensions of UNFO with counting quantifiers and with ratio quantifiers. We further relate sum-aggregation DHNs to the unary quantifier alternation fragment of FO and to an extension of FO with expressive counting. Through the classical correspondence between FO and SQL, these results also illuminate the relation between DHNs and SQL. They also enable us to study the decidability of two fundamental static analysis problems for DHNs, the emptiness problem and the subsumption problem. Finally, we confirm through experiments that the established differences in expressive power are reflected in the performance on suitable prediction tasks.

ULLER (Unified Language for LEarning and Reasoning) offers a unified first-order logic (FOL) syntax, enabling its knowledge bases to be used directly across a wide range of neurosymbolic systems. The original specification endows this syntax with three pairwise independent semantics: classical, fuzzy, and probabilistic, each accompanied by dedicated semantic rules. We show that these seemingly disparate semantics are all instances of one categorical framework based on monads, the very construct that models side effects in functional programming. This enables the modular addition of new semantics and systematic translations between them. As example, we outline the addition of generalised quantification in Logic Tensor Networks (LTN) to arbitrary (also infinite) domains by extending the Giry monad to probability spaces. In particular, our approach allows a modular implementation of ULLER in Python and Haskell, of which we have published initial versions on GitHub.

Defeasible statements are statements that are likely, or probable, or usually true, but may occasionally be false. Plausible reasoning makes conclusions from statements that are either facts or defeasible statements without using numbers. So there are no probabilities or suchlike involved. Seventeen principles of logics that do plausible reasoning are suggested and several important plausible reasoning examples are considered. There are 14 necessary principles and 3 desirable principles, one of which is not formally stated. A first-order logic, called Plausible Logic (PL), is defined that satisfies all but two of the desirable principles and reasons correctly with all the examples. As far as we are aware, this is the only such logic. PL has 8 reasoning algorithms because, from a given plausible reasoning situation, there are different sensible conclusions. This article is a condensation of my book `Plausible Reasoning and Plausible Logic' (PRPL), which is to be submitted. Each section of this article corresponds to a chapter in PRPL, and vice versa. The proofs of all the results are in PRPL, so they are omitted in this article.

Similarity in formal argumentation has recently gained attention due to its significance in problems such as argument aggregation in semantics and enthymeme decoding. While existing approaches focus on propositional logic, we address the richer setting of First-Order Logic (FOL), where similarity must account for structured content. We introduce a comprehensive framework for FOL argument similarity, built upon: (1) an extended axiomatic foundation; (2) a four-level parametric model covering predicates, literals, clauses, and formulae similarity; (3) two model families, one syntax-sensitive via language models, both integrating contextual weights for nuanced and explainable similarity; and (4) formal constraints enforcing desirable properties.

We introduce Qiana, a logic framework for reasoning on formulas that are true only in specific contexts. In Qiana, it is possible to quantify over both formulas and contexts to express, e.g., that ``everyone knows everything Alice says''. Qiana also permits paraconsistent logics within contexts, so that contexts can contain contradictions. Furthermore, Qiana is based on first-order logic, and is finitely axiomatizable, so that Qiana theories are compatible with pre-existing first-order logic theorem provers. We show how Qiana can be used to represent temporality, event calculus, and modal logic. We also discuss different design alternatives of Qiana.

Real world scenarios can be captured with lifted probability distributions. However, distributions are usually encoded in a table or list, requiring an exponential number of values. Hence, we propose a method for extracting first-order formulas from probability distributions that require significantly less values by reducing the number of values in a distribution and then extracting, for each value, a logical formula to be further minimized. This reduction and minimization allows for increasing the sparsity in the encoding while also generalizing a given distribution. Our evaluation shows that sparsity can increase immensely by extracting a small set of short formulas while preserving core information.