Predicate Logic Research Articles

Propositions with Negative Predicates in Arabic Logic

ABSTRACT This paper explores a neglected category of propositions in Arabic logic, propositions with negative predicates (sālibat al-maḥmūl), by addressing two pivotal questions concerning this propositional form: first, whether it is possible to defend it as distinct from metathetic and simple negative propositions and second, whether affirmative instances of these propositions have existential import. The paper argues for the existence of two distinct and conflicting theories of existential import frequently implicit in the views of Arabic logicians: one centered on the copula and the other on the predicate.

A boolean-algebraic approach to completeness for normal modal predicate logics

This paper introduces an innovative methodology for demonstrating completeness for normal modal predicate logics. Traditional proofs typically involve constructing canonical models, wherein possible worlds are defined as maximal consistent sets possessing specific properties, with a heavy reliance on the Barcan Formula to affirm the existence of these worlds. Our approach deviates from the classical method by utilizing Boolean algebras and ultrafilters to construct models. Unlike conventional methods, our construction of possible worlds does not depend on the Barcan Formula; rather, these properties are ensured by Tarski’s Lemma. Furthermore, our proof distinguishes itself from other Boolean-algebraic completeness proofs in two key respects: it employs Kripke semantics instead of algebraic semantics and exclusively relies on ultrafilters, thereby offering a more concise approach. This methodology facilitates a natural extension from modal propositional logic to modal predicate logics and circumvents the added complexity of Q-filters. In our model, the equivalence class of each theorem of a normal modal predicate logic is a member of all worlds, while the equivalence class of each non-theorem is a member of some worlds. Consequently, the structure of these worlds ensures that non-theorems are false in the model.

Quantum networks theory

The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates in a robust manner. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the neighbourhood and partitioning between systems become both quantum, dynamical, and logical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.

Completely hereditarily atomic OMLS

Abstract An irreducible complete atomic oml of infinite height cannot be algebraic and have the covering property. However, modest departure from these conditions allows infinite-height examples. We use an extension of Kalmbach’s construction to the setting of infinite chains to provide an example of an infinite-height, irreducible, algebraic oml with the 2-covering property, and Keller’s construction provides an example of an infinite-height, irreducible, complete oml that has the covering property and is completely hereditarily atomic. Completely hereditarily atomic omls generalize algebraic omls suitably to quantum predicate logic.

ANSELM’S ONTOLOGICAL ARGUMENT AND GRADES OF BEING

AbstractAnselm described god as “something than which nothing greater can be thought” [1, p. 93], and Descartes viewed him as “a supreme being” [7, p. 122]. I first capture those characterizations formally in a simple language for monadic predicate logic. Next, I construct a model class inspired by Stoic and medieval doctrines of grades of being [8, 20]. Third, I prove the models sufficient for recovering, as internal mathematics, the famous ontological argument of Anselm, and show that argument to be, on this formalization, valid. Fourth, I extend the models to incorporate a modality fit for proving that any item than which necessarily no greater can be thought is also necessarily real. Lastly, with the present approach, I blunt the sharp edges of notable objections to ontological arguments by Gaunilo and by Grant. A trigger warning: every page of this writing flouts the old saw “Existence is not a predicate” and flagrantly.

Finite predicate-driven logic networks method for enhanced education data analysis

The subject matter of the study is intelligent data analysis in the field of academic information. The goal of the study is to create a mathematical model for analyzing students' academic information using the predicate-driven logic networks method, which allows taking into account both logical dependencies and probabilistic transitions between states. To achieve this goal, the following tasks were defined: analysis of the theoretical foundations of the logic networks method and predicate logic, integration of these approaches into a single mathematical model, development of approaches for its application in academic data analysis problems. The research used the methods of mathematical modeling, complex logical analysis, and method for constructing logic networks. The following results were obtained: a theoretical model was developed that integrates the principles of logic networks and predicate logic for analyzing student academic performance; the model accounts for both probabilistic transitions between states and logical dependencies among student parameters; the mathematical model also incorporates logical rules to enhance the accuracy of logical analysis within the academic context. The model was tested on a dataset of student performance, demonstrating its effectiveness in accurately predicting academic outcomes and confirming the validity of the integrated approach. Conclusions. The scientific novelty of the results obtained is as follows: 1) a theoretical model for analyzing student academic data was developed by integrating logic networks and predicate logic, allowing for the simultaneous consideration of probabilistic transitions and logical dependencies among student parameters; 2) the approach enhances the analysis process by incorporating logical rules into the probabilistic framework, providing a more nuanced and accurate tool for analyzing academic data; 3) this combined model offers a novel method for addressing complex logical analysis tasks in educational settings, paving the way for further research and practical applications in intelligent data analysis. The successful testing of the model on actual student data further underscores its potential as a powerful tool in educational data analysis.

Heterogeneous-branch integration framework: Introducing first-order predicate logic in Logical Reasoning Question Answering

The logical reasoning question-answering is a critical task in natural language processing, as it equips models with human-like logical reasoning intelligence. Existing approaches focus on extracting and leveraging the hidden logical structures within text. However, previous works explore partial logical relationships and neglect the holistic extraction within the text. Moreover, they struggle to fully model logical connections, including long-distance dependencies and local topology information. To address these issues, we propose a novel heterogeneous-branch integration framework. Our framework is based on first-order predicate logic theory and consists of three primary components. First, we construct two heterogeneous logical graphs to model logical relationships within and between propositions. Second, we propose a novel Graph-Masked Transformer with a novel graph-masked multi-head attention mechanism to enable distant node interactions and local sparse relationship modeling. Third, we propose a novel multi-branch fusion module to integrate information from multiple sources and generate answer predictions. The proposed heterogeneous-branch integration framework outperforms the VDGN method by 2.73% in accuracy on the ReClor dataset and 2.15% on the LogiQA dataset. Our code and models will be made available at https://github.com/starry-y/HBI.

Intertranslatability and Ground-Equivalence

AbstractWhen are logical theories equivalent? I discuss the notion of ground-equivalence between logical theories, which can be useful for various theoretical reasons, e.g., we expect ground-equivalent theories to have the same ontological bearing. I consider whether intertranslatability is an adequate criterion for ground-equivalence. Jason Turner recently offered an argument that first-order logic and predicate functor logic are ground-equivalent in virtue of their intertranslatability. I examine his argument and show that this can be generalized to other intertranslatable logical theories, which supports the following: intertranslatability implies ground-equivalence. I also argue, however, that this ground-equivalence argument can be challenged as it faces a dilemma. The dilemma arises because the argument allows two distinct readings, the ‘internal’ and the ‘external’ reading. I argue that the argument turns out to be unsuccessful in both readings. The upshot of this dilemma in both philosophy of logic and metaphysics is considered.

The Semantics of Stative Locative Events in Vietnamese: A Spatial Exploration

This study utilizes mathematical formalism and Figure-Ground argument mapping to categorize stative locative events in Vietnamese as surjective or bijective. Bijective events exhibit a one-to-one mapping of Figures and Grounds, while surjectives allow a single Figure to map onto multiple Grounds. Through analysis of Vietnamese sentences, the paper demonstrates how bijective and surjective locatives encode deterministic and ambiguous spatial relations between Figures and Grounds, respectively. This novel distinction aids in enhancing the understanding of locative event typologies and provides an explanatory model for cross-linguistic analysis. The proposed integrated formal framework combines predicate logic, set theory, and lexical semantics to account for complex locative meanings, with significant applications in computational and theoretical linguistics, as well as cognitive science.

A Version of Predicate Logic with Two Variables That has an Incompleteness Property

AbstractIn this paper, we consider predicate logic with two individual variables and general assignment models (where the set of assignments of the variables into a model is allowed to be an arbitrary subset of the usual one). We prove that there is a statement such that no general assignment model in which it is true can be finitely axiomatized. We do this by showing that the free relativized cylindric algebras of dimension two are not atomic.

Quantifiers and existence

There are some sentences that include expressions that refer to entities that do not exist. One example is this: Mary is in terror of werewolves. Some argue that this sentence cannot be translated into predicate logic. This may be seen as flaw in predicate logic. Against this, I argue in this paper that the problem is not predicate logic, but rather our commitments with the existence and nature of certain things. By revising some of these commitments, we can see that predicate logic is perfectly capable to deal with the problematic sentences.

Features of paper walls in the formation of the «Japanese house» microecosystem

The work relates to the topics of retroecology, namely, the features of microecosystems related to the housing of the peoples of the world. A unique structural element of a Japanese dwelling – a paper wall – is considered. It is shown that in terms of functional purpose, such a structure differs significantly from the well-known building walls. In this sense, the paper wall does not meet the generally accepted quality criteria. However, this element of housing, which is very popular in Japan, finds its admirers in other microecosystems. The paper highlights those well-known qualities of paper walls that appeal to the people of Japan, contrary to existing standards, but in full compliance with Japanese analog culture. Taking into account that the popularity of paper walls is influenced not by protection from climatic and other influences, but by the spirituality of the Japanese people, the mechanism of mathematical logic, namely, the model of predicate logic, was adopted as a tool for analyzing such structures. The formalized mathematical model made it possible to pay attention to some features in the preferences of the specified predicates and made it possible to clarify their priority for the user. It is not a contradiction that the Japanese paper wall is not designed to perform most of the known functions of an ordinary human dwelling wall. It has become preferable for the inhabitant of the country not as a protective or insulating structure, but mainly as a result of spiritual transformations in society. In particular, with the help of a specific building structure - a paper wall. Thus, the paper wall has become a component for the formation of mentality, psychology, and spirituality. That is, what is valued above all else in the microecosystem of the Japanese home

Exclusive Disjunctions With Three Disjuncts from First-Order Predicate Calculus

First-order predicate logic seems to be incompatible with the way people understand embedded exclusive disjunctions with three disjuncts. In classical logic, an exclusive disjunction with three disjuncts holds when the three disjuncts hold. However, it is hard to note that for people. The theory of mental models can explain this fact. According to that theory, individuals tend to process embedded exclusive disjunctions with three disjuncts intuitively. Thus, they only consider possible situations in which just one of the disjuncts is the case. The present paper tries to explain this problem within first-order predicate logic. The main point is that, in the latter logic, in inferences having, as its first premise, an embedded exclusive disjunction with three disjuncts, and, as its second premise, the first disjunct of that very exclusive disjunction, it is possible to infer none of the other two disjuncts.

TypeQL: A Type-Theoretic & Polymorphic Query Language

Relational data modeling can often be restrictive as it provides no direct facility for modeling polymorphic types, reified relations, multi-valued attributes, and other common high-level structures in data. This creates many challenges in data modeling and engineering tasks, and has led to the rise of more flexible NoSQL databases, such as graph and document databases. In the absence of structured schemas, however, we can neither express nor validate the intention of data models, making long-term maintenance of databases substantially more difficult. To resolve this dilemma, we argue that, parallel to the role of classical predicate logic for relational algebra, contemporary foundations of mathematics rooted in type theory can guide us in the development of powerful new high-level data models and query languages. To this end, we introduce a new polymorphic entity-relation-attribute (PERA) data model, grounded in type-theoretic principles and accessible through classical conceptual modeling, with a near-natural query language: TypeQL. We illustrate the syntax of TypeQL as well as its denotation in the PERA model, formalize our model as an algebraic theory with dependent types, and describe its stratified semantics.

Logical characterizations of algebraic circuit classes over integral domains

AbstractWe present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}^{}$ classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the $\mathrm{AC}_{R}$ and $\mathrm{NC}_R^{}$ hierarchy. Those generalizations apply to the Boolean $\mathrm{AC}$ and $\mathrm{NC}$ hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.

Wittgensteinian Predicate Logic and Compositionality

Wittgensteinian Predicate Logic and Compositionality

FGeo-SSS: A Search-Based Symbolic Solver for Human-like Automated Geometric Reasoning

Geometric problem solving (GPS) has always been a long-standing challenge in the fields of automated reasoning. Its problem representation and solution process embody rich symmetry. This paper is the second in a series of our works. Based on the Geometry Formalization Theory and the FormalGeo geometric formal system, we have developed the Formal Geometric Problem Solver (FGPS) in Python 3.10, which can serve as an interactive assistant or as an automated problem solver. FGPS is capable of executing geometric predicate logic and performing relational reasoning and algebraic computation, ultimately achieving readable, traceable, and verifiable automated solutions for geometric problems. We observed that symmetry phenomena exist at various levels within FGPS and utilized these symmetries to further refine the system’s design. FGPS employs symbols to represent geometric shapes and transforms various geometric patterns into a set of symbolic operation rules. This maintains symmetry in basic transformations, shape constructions, and the application of theorems. Moreover, we also have annotated the formalgeo7k dataset, which contains 6981 geometry problems with detailed formal language descriptions and solutions. Experiments on formalgeo7k validate the correctness and utility of the FGPS. The forward search method with random strategy achieved a 39.71% problem-solving success rate.

Requirement patterns in deductive verification of poST Programs

Process-oriented programming is one of the approaches to developing control software. A process-oriented program is defined as a sequence of processes. Each process is represented by a set of named states containing program code that define the logic of the process's behavior. Program execution is sequential execution of each of these processes in their current states at every iteration of the control cycle. Processes can interact through changing each other's states and shared variables. The paper expands a method for classifying temporal requirements for process-oriented programs in order to simplify and automate the deductive verification of such programs. The method consists of the following steps. At the first step, the requirements are formalized in a specialized language DV-TRL, a variant of typed first-order predicate logic with a set of interpreted types and predicate and functional symbols, that reflect specific concepts of control systems in a process-oriented paradigm. At the second step, the formalized requirements are divided into classes, each of which is defined by a pattern — a parametric formula of the DV-TRL language. The correctness conditions generated for process-oriented programs regarding requirements satisfying the same pattern have the same proof scheme. At the third step, appropriate proof schemes are developed. In our paper, we first give a brief introduction to the poST language, a process-oriented extension to the ST language of the IEC 61131-3 standard. Next, the DV-TRL language is defined. We also provide a collection of natural language requirements for several control systems. Then we define patterns that fully cover all the requirements of this collection. For each of these patterns we give an example of a formalized requirement from the collection and describe a scheme for proving the correctness conditions for this pattern. Statistics on the distribution of requirements from the collection across patterns reveals the most popular patterns. We also analyzed related works.

Variations on the Kripke Trick

AbstractIn the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate modal logic $$\textbf{QS5}$$ QS 5 that include the classical predicate logic $$\textbf{QCl}$$ QCl , Saul Kripke showed how a classical atomic formula with a binary predicate letter can be simulated by a monadic modal formula. We consider adaptations of Kripke’s simulation, which we call the Kripke trick, to various modal and superintuitionistic predicate logics not considered by Kripke. We also discuss settings where the Kripke trick does not work and where, as a result, decidability of monadic modal predicate logics can be obtained.

Concept logic trees: enabling user interaction for transparent image classification and human-in-the-loop learning

Interpretable deep learning models are increasingly important in domains where transparent decision-making is required. In this field, the interaction of the user with the model can contribute to the interpretability of the model. In this research work, we present an innovative approach that combines soft decision trees, neural symbolic learning, and concept learning to create an image classification model that enhances interpretability and user interaction, control, and intervention. The key novelty of our method relies on the fusion of an interpretable architecture with neural symbolic learning, allowing the incorporation of expert knowledge and user interaction. Furthermore, our solution facilitates the inspection of the model through queries in the form of first-order logic predicates. Our main contribution is a human-in-the-loop model as a result of the fusion of neural symbolic learning and an interpretable architecture. We validate the effectiveness of our approach through comprehensive experimental results, demonstrating competitive performance on challenging datasets when compared to state-of-the-art solutions.