Artificial Intelligence with PHP
  • Getting Started
    • Introduction
    • Audience
    • How to Read This Book
    • Glossary
    • Contributors
    • Resources
    • Changelog
  • Artificial Intelligence
    • Introduction
    • Overview of AI
      • History of AI
      • How Does AI Work?
      • Structure of AI
      • Will AI Take Over the World?
      • Types of AI
        • Limited Memory AI
        • Reactive AI
        • Theory of Mind AI
        • Self-Aware AI
    • AI Capabilities in PHP
      • Introduction to LLM Agents PHP SDK
      • Overview of AI Libraries in PHP
    • AI Agents
      • Introduction to AI Agents
      • Structure of AI Agent
      • Components of AI Agents
      • Types of AI Agents
      • AI Agent Architecture
      • AI Agent Environment
      • Application of Agents in AI
      • Challenges in AI Agent Development
      • Future of AI Agents
      • Turing Test in AI
      • LLM AI Agents
        • Introduction to LLM AI Agents
        • Implementation in PHP
          • Sales Analyst Agent
          • Site Status Checker Agent
    • Theoretical Foundations of AI
      • Introduction to Theoretical Foundations of AI
      • Problem Solving in AI
        • Introduction
        • Types of Search Algorithms
          • Comparison of Search Algorithms
          • Informed (Heuristic) Search
            • Global Search
              • Beam Search
              • Greedy Search
              • Iterative Deepening A* Search
              • A* Search
                • A* Graph Search
                • A* Graph vs A* Tree Search
                • A* Tree Search
            • Local Search
              • Hill Climbing Algorithm
                • Introduction
                • Best Practices and Optimization
                • Practical Applications
                • Implementation in PHP
              • Simulated Annealing Search
              • Local Beam Search
              • Genetic Algorithms
              • Tabu Search
          • Uninformed (Blind) Search
            • Global Search
              • Bidirectional Search (BDS)
              • Breadth-First Search (BFS)
              • Depth-First Search (DFS)
              • Iterative Deepening Depth-First Search (IDDFS)
              • Uniform Cost Search (UCS)
            • Local Search
              • Depth-Limited Search (DLS)
              • Random Walk Search (RWS)
          • Adversarial Search
          • Means-Ends Analysis
      • Knowledge & Uncertainty in AI
        • Knowledge-Based Agents
        • Knowledge Representation
          • Introduction
          • Approaches to KR in AI
          • The KR Cycle in AI
          • Types of Knowledge in AI
          • KR Techniques
            • Logical Representation
            • Semantic Network Representation
            • Frame Representation
            • Production Rules
        • Reasoning in AI
        • Uncertain Knowledge Representation
        • The Wumpus World
        • Applications and Challenges
      • Cybernetics and AI
      • Philosophical and Ethical Foundations of AI
    • Mathematics for AI
      • Computational Theory in AI
      • Logic and Reasoning
        • Classification of Logics
        • Formal Logic
          • Propositional Logic
            • Basics of Propositional Logic
            • Implementation in PHP
          • Predicate Logic
            • Basics of Predicate Logic
            • Implementation in PHP
          • Modal Logic
          • Temporal Logic
        • Informal Logic
        • Semi-formal Logic
      • Set Theory and Discrete Mathematics
      • Decision Making in AI
    • Key Application of AI
      • AI in Astronomy
      • AI in Agriculture
      • AI in Automotive Industry
      • AI in Data Security
      • AI in Dating
      • AI in E-commerce
      • AI in Education
      • AI in Entertainment
      • AI in Finance
      • AI in Gaming
      • AI in Healthcare
      • AI in Robotics
      • AI in Social Media
      • AI in Software Development
      • AI in Adult Entertainment
      • AI in Criminal Justice
      • AI in Criminal World
      • AI in Military Domain
      • AI in Terrorist Activities
      • AI in Transforming Our World
      • AI in Travel and Transport
    • Practice
  • Machine Learning
    • Introduction
    • Overview of ML
      • History of ML
        • Origins and Early Concepts
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • Key Terms and Principles
      • Machine Learning Life Cycle
      • Problems and Challenges
    • ML Capabilities in PHP
      • Overview of ML Libraries in PHP
      • Configuring an Environment for PHP
        • Direct Installation
        • Using Docker
        • Additional Notes
      • Introduction to PHP-ML
      • Introduction to Rubix ML
    • Mathematics for ML
      • Linear Algebra
        • Scalars
          • Definition and Operations
          • Scalars with PHP
        • Vectors
          • Definition and Operations
          • Vectors in Machine Learning
          • Vectors with PHP
        • Matrices
          • Definition and Types
          • Matrix Operations
          • Determinant of a Matrix
          • Inverse Matrices
          • Cofactor Matrices
          • Adjugate Matrices
          • Matrices in Machine Learning
          • Matrices with PHP
        • Tensors
          • Definition of Tensors
          • Tensor Properties
            • Tensor Types
            • Tensor Dimension
            • Tensor Rank
            • Tensor Shape
          • Tensor Operations
          • Practical Applications
          • Tensors in Machine Learning
          • Tensors with PHP
        • Linear Transformations
          • Introduction
          • LT with PHP
          • LT Role in Neural Networks
        • Eigenvalues and Eigenvectors
        • Norms and Distances
        • Linear Algebra in Optimization
      • Calculus
      • Probability and Statistics
      • Information Theory
      • Optimization Techniques
      • Graph Theory and Networks
      • Discrete Mathematics and Combinatorics
      • Advanced Topics
    • Data Fundamentals
      • Data Types and Formats
        • Data Types
        • Structured Data Formats
        • Unstructured Data Formats
        • Implementation with PHP
      • General Data Processing
        • Introduction
        • Storage and Management
          • Data Security and Privacy
          • Data Serialization and Deserialization in PHP
          • Data Versioning and Management
          • Database Systems for AI
          • Efficient Data Storage Techniques
          • Optimizing Data Retrieval for AI Algorithms
          • Big Data Considerations
            • Introduction
            • Big Data Techniques in PHP
      • ML Data Processing
        • Introduction
        • Types of Data in ML
        • Stages of Data Processing
          • Data Acquisition
            • Data Collection
            • Ethical Considerations in Data Preparation
          • Data Cleaning
            • Data Cleaning Examples
            • Data Cleaning Types
            • Implementation with PHP
          • Data Transformation
            • Data Transformation Examples
            • Data Transformation Types
            • Implementation with PHP ?..
          • Data Integration
          • Data Reduction
          • Data Validation and Testing
            • Data Splitting and Sampling
          • Data Representation
            • Data Structures in PHP
            • Data Visualization Techniques
          • Typical Problems with Data
    • ML Algorithms
      • Classification of ML Algorithms
        • By Methods Used
        • By Learning Types
        • By Tasks Resolved
        • By Feature Types
        • By Model Depth
      • Supervised Learning
        • Regression
          • Linear Regression
            • Types of Linear Regression
            • Finding Best Fit Line
            • Gradient Descent
            • Assumptions of Linear Regression
            • Evaluation Metrics for Linear Regression
            • How It Works by Math
            • Implementation in PHP
              • Multiple Linear Regression
              • Simple Linear Regression
          • Polynomial Regression
            • Introduction
            • Implementation in PHP
          • Support Vector Regression
        • Classification
        • Recommendation Systems
          • Matrix Factorization
          • User-Based Collaborative Filtering
      • Unsupervised Learning
        • Clustering
        • Dimension Reduction
        • Search and Optimization
        • Recommendation Systems
          • Item-Based Collaborative Filtering
          • Popularity-Based Recommendations
      • Semi-Supervised Learning
        • Regression
        • Classification
        • Clustering
      • Reinforcement Learning
      • Distributed Learning
    • Integrating ML into Web
      • Open-Source Projects
      • Introduction to EasyAI-PHP
    • Key Applications of ML
    • Practice
  • Neural Networks
    • Introduction
    • Overview of NN
      • History of NN
      • Basic Components of NN
        • Activation Functions
        • Connections and Weights
        • Inputs
        • Layers
        • Neurons
      • Problems and Challenges
      • How NN Works
    • NN Capabilities in PHP
    • Mathematics for NN
    • Types of NN
      • Classification of NN Types
      • Basic NN
        • Simple Perceptron
        • Implementation in PHP
          • Simple Perceptron with Libraries
          • Simple Perceptron with Pure PHP
      • NN with Hidden Layers
      • Deep Learning
      • Bayesian Neural Networks
      • Convolutional Neural Networks (CNN)
      • Recurrent Neural Networks (RNN)
    • Integrating NN into Web
    • Key Applications of NN
    • Practice
  • Natural Language Processing
    • Introduction
    • Overview of NLP
      • History of NLP
        • Ancient Times
        • Medieval Period
        • 15th-16th Century
        • 17th-18th Century
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • Key Concepts in NLP
      • Common Challenges in NLP
      • Machine Learning Role in NLP
    • NLP Capabilities in PHP
      • Overview of NLP Libraries in PHP
      • Challenges in NLP with PHP
    • Mathematics for NLP
    • NLP Techniques
      • Basic Text Processing with PHP
      • NLP Workflow
      • Popular Tools and Frameworks for NLP
      • Techniques and Algorithms in NLP
        • Basic NLP Techniques
        • Advanced NLP Techniques
      • Advanced NLP Topics
    • Integrating NLP into Web
    • Key Applications of NLP
    • Practice
  • Computer Vision
    • Introduction
  • Overview of CV
    • History of CV
    • Common Use Cases
  • CV Capabilities in PHP
  • Mathematics for CV
  • CV Techniques
  • Integrating CV into Web
  • Key Applications of CV
  • Practice
  • Robotics
    • Introduction
  • Overview of Robotics
    • History and Evolution of Robotics
    • Core Components
      • Sensors (Perception)
      • Actuators (Action)
      • Controllers (Processing and Logic)
    • The Role of AI in Robotics
      • Object Detection and Recognition
      • Path Planning and Navigation
      • Decision Making and Learning
  • Robotics Capabilities in PHP
  • Mathematics for Robotics
  • Building Robotics
  • Integration Robotics into Web
  • Key Applications of Robotics
  • Practice
  • Expert Systems
    • Introduction
    • Overview of ES
      • History of ES
        • Origins and Early ES
        • Milestones in the Evolution of ES
        • Expert Systems in Modern AI
      • Core Components and Architecture
      • Challenges and Limitations
      • Future Trends
    • ES Capabilities in PHP
    • Mathematics for ES
    • Building ES
      • Knowledge Representation Approaches
      • Inference Mechanisms
      • Best Practices for Knowledge Base Design and Inference
    • Integration ES into Web
    • Key Applications of ES
    • Practice
  • Cognitive Computing
    • Introduction
    • Overview of CC
      • History of CC
      • Differences Between CC and AI
    • CC Compatibilities in PHP
    • Mathematics for CC
    • Building CC
      • Practical Implementation
    • Integration CC into Web
    • Key Applications of CC
    • Practice
  • AI Ethics and Safety
    • Introduction
    • Overview of AI Ethics
      • Core Principles of AI Ethics
      • Responsible AI Development
      • Looking Ahead: Ethical AI Governance
    • Building Ethics & Safety AI
      • Fairness, Bias, and Transparency
        • Bias in AI Models
        • Model Transparency and Explainability
        • Auditing, Testing, and Continuous Monitoring
      • Privacy and Security in AI
        • Data Privacy and Consent
        • Safety Mechanisms in AI Integration
        • Preventing and Handling AI Misuse
      • Ensuring AI Accountability
        • Ethical AI in Decision Making
        • Regulations & Compliance
        • AI Risk Assessment
    • Key Applications of AI Ethics
    • Practice
  • Epilog
    • Summing-up
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On this page
  • Classification (REDO !)
  • Key Characteristics
  • Advantages
  • Disadvantages
  • Applications of Predicate Logic in AI
  1. Artificial Intelligence
  2. Mathematics for AI
  3. Logic and Reasoning
  4. Formal Logic

Predicate Logic

Predicate logic, also known as First-Order Logic (FOL), is a fundamental tool in AI used to represent complex relationships between objects and their properties. Unlike propositional logic, which deals with simple true/false statements, predicate logic allows reasoning about objects, their attributes, and their interconnections. This capability makes it an essential component in knowledge representation, expert systems, and automated reasoning.

In this chapter, we will explore the basics of predicate logic, its syntax and semantics, the role of quantifiers, and its practical applications in AI.

Classification (REDO !)

Category
Value

By Structure

Formal Logic - uses symbols to represent statements and logical relationships.

By Reasoning Direction

Deductive Logic - from general to specific, the truth of the conclusion is logically certain if the premises are true.

By Update Behavior

Usually Monotonic Logic - once something is proven true in propositional logic, adding more facts can’t make it false.

By Modality

Non-Modal – it does not include modal operators like “must” or “might”.

By Degree of Truth

Non-fuzzy (Crisp) Logic – truth values are binary (true/false).

By Time Awareness

Non-temporal – It does not reason over time.

Key Characteristics

  • Use of Quantifiers: Predicate logic uses two main quantifiers:

    • ∀ (for all) – Universal quantifier

    • ∃ (there exists) – Existential quantifier

  • Variables and Predicates: It allows the use of variables (like x, y) and predicates to describe statements such as "x is a cat" or "x loves y."

  • Functions and Constants: Predicate logic can include functions (e.g., father_of(x)) and constants (e.g., Socrates).

  • Syntax and Semantics: Syntax defines how statements are formed. Semantics explains what these statements mean in a logical system.

  • More Expressive Than Propositional Logic: Predicate logic can represent statements involving objects and their relationships, which propositional logic cannot.

Advantages

  • High Expressiveness: Can represent complex statements about the world, including relationships between multiple objects.

  • Supports Reasoning: Helps AI systems make logical inferences, answer queries, and validate rules.

  • Used in Knowledge Representation: Suitable for building expert systems, rule-based engines, and semantic web technologies.

  • Supports Automation: Logical statements can be verified and solved using automated reasoning tools like Prolog or theorem provers.

Disadvantages

  • Computational Complexity: Predicate logic problems can be hard to solve and may require a lot of processing time and memory.

  • Undecidability: In general, first-order logic is undecidable, meaning there is no algorithm that can solve all problems or decide the truth of every statement.

  • Difficult to Scale: Applying predicate logic to large datasets or dynamic environments can be challenging.

  • Learning Curve: The use of symbols, rules, and formal grammar can be difficult for beginners to understand and use correctly.

Applications of Predicate Logic in AI

Predicate logic is used in AI to represent complex relationships between objects and to perform advanced reasoning. It allows systems to make logical inferences using variables, functions, and quantifiers like "for all" or "there exists." This makes it ideal for expert systems, knowledge representation, semantic analysis in natural language processing, and automated theorem proving. Its expressiveness supports more intelligent behavior in AI compared to simpler propositional logic.

Example:

  • Expert Systems: Predicate logic is used to represent rules and facts in systems that simulate human reasoning.

  • Natural Language Processing (NLP): Logical forms can represent meanings of sentences for tasks like question answering and semantic analysis.

  • Theorem Proving: Automatic provers use predicate logic to verify mathematical theorems or logical properties of software.

  • Ontology and Semantic Web: Used in defining structured knowledge (like RDF and OWL) for understanding and sharing information across systems.

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