📘 Predicate logic
📌 What is Predicate Logic?
Predicate Logic (also called First-Order Logic or FOL) is an extension of Propositional Logic that allows reasoning with objects, variables, and quantifiers (like “for all”, “there exists”).
It is not fuzzy by default, but understanding Predicate Logic is essential because Fuzzy Logic can be extended to work on predicates—leading to Fuzzy Predicate Logic or Fuzzy First-Order Logic.
📌 Why Predicate Logic Matters in Fuzzy Systems?
• Helps in expressing more complex, real-world reasoning using variables and relations (not just true/false propositions).
• It becomes the foundation for fuzzy knowledge representation, especially in expert systems and natural language processing.
✅ Components of Predicate Logic:
| Component | Description | Example |
|---|---|---|
| Predicate | A function that describes a property or relation | Tall(x) — “x is tall” |
| Variables | Entities over which predicates operate | x, y, etc. |
| Quantifiers | Used to describe logical statements with multiple objects | ∀ (for all), ∃ (there exists) |
| Logical Connectives | AND (∧), OR (∨), NOT (¬), Implication (→), Equivalence (↔) | Tall(x) ∧ Smart(x) |
🧠 Example: Classical Predicate Logic Statement
∀x (Human(x) → Mortal(x))
Meaning: “All humans are mortal.”
📈 Limitation of Classical Predicate Logic:
• It assumes binary truth: Either something is true (1) or false (0).
• In real-world applications, such crisp logic may not suffice.
• e.g., Is a person 170 cm tall really Tall or Not Tall? It’s subjective.
🌫️ Fuzzy Predicate Logic (Fuzzy First-Order Logic)
➤ What is Fuzzy Predicate Logic?
It extends classical predicate logic by allowing truth values to be fuzzy (i.e., between 0 and 1).
So instead of:
Tall(John) = True (1)
You can have:
Tall(John) = 0.7 → Meaning “John is 70% tall” based on the context.
🔢 Example: Fuzzy Predicate
| Statement | Membership Value |
|---|---|
| Tall(Om) | 0.8 |
| Hot(Weather) | 0.6 |
These fuzzy predicates use membership functions just like in fuzzy sets.
📚 Applications of Fuzzy Predicate Logic
| Domain | Application |
|---|---|
| Expert Systems | Representing fuzzy knowledge base |
| NLP | Handling vague sentences (e.g., “Some students are very intelligent”) |
| Healthcare | Diagnosis with symptom-to-disease fuzzy relations |
| Robotics | Rule-based behavior using fuzzy predicates |
✍️ Exam-Ready Answer Summary:
Predicate Logic is a form of logical reasoning that uses predicates, variables, and quantifiers to express statements about objects and their properties. It forms the basis of knowledge representation in AI. Fuzzy Predicate Logic extends classical predicate logic by allowing truth values between 0 and 1, making it suitable for real-world reasoning involving uncertainty, vagueness, and partial truth.
🎯 Mnemonic: “PQVLC”
• P – Predicates
• Q – Quantifiers
• V – Variables
• L – Logical Connectives
• C – Crisp → Fuzzy Transition (Fuzzy Predicate Logic)