Modern AI systems often excel at pattern recognition, but many real-world tasks require something more precise: the ability to represent facts, rules, and relationships in a form that machines can reason over reliably. Knowledge representation (KR) tackles this challenge by providing formal languages that make meaning explicit and checkable. Among these, first-order logic (FOL) remains one of the most influential foundations because it balances expressive power with a well-defined notion of truth. If you have explored symbolic reasoning in an AI course in Delhi, you have likely encountered FOL as a core tool for building systems that can prove statements, verify constraints, and detect contradictions.
This article explains the formal syntax and semantics of first-order logic and shows how these connect to automated theorem proving.
Why First-Order Logic Matters in Knowledge Representation
Knowledge representation aims to capture “what is true” about a domain in a structured way. FOL is especially useful because it can express:
- Objects (people, devices, transactions)
- Properties (isStudent(x), isPrime(n))
- Relations (Parent(x, y), Connected(a, b))
- General rules using quantifiers, such as “all” or “there exists”
In practical terms, this means you can encode rules like: “Every employee must have a manager,” or “If a user resets a password, they must verify identity.” Unlike informal text, logical statements have precise meaning and can be checked systematically.
Formal Syntax: The Building Blocks of FOL
Syntax defines what counts as a well-formed formula. Think of syntax as the grammar of logic.
Terms
A term denotes an object in the domain.
- Constants: specific named objects (e.g., alice, 0, delhi)
- Variables: placeholders (e.g., x, y)
- Function symbols: build new terms from terms (e.g., fatherOf(x), add(x, y))
Examples of terms: x, alice, fatherOf(alice), add(2, 3).
Predicates (Atomic Formulas)
A predicate represents a property or relation. When applied to terms, it forms an atomic formula:
- Student(alice)
- Parent(x, y)
- Greater(add(x, 1), y)
Logical Connectives and Quantifiers
Complex statements are built using:
- Negation: ¬P
- Conjunction: P ∧ Q
- Disjunction: P ∨ Q
- Implication: P → Q
- Biconditional: P ↔ Q
- Universal quantifier: ∀x P(x) (“for all x”)
- Existential quantifier: ∃x P(x) (“there exists an x”)
A well-formed formula (WFF) example:
- ∀x (Human(x) → Mortal(x))
- ∃y (City(y) ∧ LocatedIn(y, India))
A key syntactic idea is scope: quantifiers bind variables only within their scope. This matters because changing quantifier order can change meaning dramatically.
Formal Semantics: What the Formulas Mean
Semantics defines truth: when is a formula considered true or false?
Interpretations (Structures)
An interpretation provides:
- A domain of objects (the “universe” you talk about)
- A mapping for constants to domain objects
- A mapping for function symbols to actual functions over the domain
- A mapping for predicate symbols to relations over the domain
For instance, if the domain is all people in an organisation:
- alice maps to a specific person
- Manager(x, y) is true when x manages y
- employeeId(x) returns a unique identifier
Variable Assignments and Satisfaction
Because formulas contain variables, semantics uses assignments (which map variables like x to specific domain objects). A formula is satisfied under an interpretation and assignment if it evaluates to true.
Quantifiers are defined through satisfaction:
- ∀x P(x) is true if P(x) is true for every object in the domain.
- ∃x P(x) is true if there exists at least one object making P(x) true.
This precise definition is the reason FOL is so valuable: it prevents “hand-wavy” meaning and enables systematic reasoning.
From Logic to Automated Theorem Proving
Automated theorem proving (ATP) tries to determine whether a conclusion logically follows from a set of premises.
Entailment and Validity
- A knowledge base entails a statement if the statement is true in every interpretation where the knowledge base is true.
- A statement is valid if it is true in all interpretations (independent of any knowledge base).
ATP systems often convert formulas into forms that are easier to work with, such as clause normal form. A common method is resolution, which proves a goal by contradiction:
- Add the negation of the goal to the knowledge base
- Apply inference rules to derive a contradiction
- If a contradiction is derived, the goal is proven
Many learners first connect these ideas in an AI course in Delhi, where KR and inference are taught together: representation defines meaning, and inference exploits that meaning.
Practical Considerations and Common Pitfalls
FOL is powerful, but practical KR requires care:
- Expressiveness vs. efficiency: richer expressions can be harder to prove automatically.
- Modelling errors: small mistakes in quantifiers or variable scope can flip meaning.
- Closed-world vs. open-world assumptions: logic itself does not assume unknown facts are false; many applications do, so the distinction must be managed explicitly.
A good modelling habit is to test the knowledge base with examples and counterexamples, ensuring the rules capture what you intended.
Conclusion
First-order logic remains a central language for knowledge representation because its syntax precisely defines what can be stated, and its semantics precisely defines what those statements mean. This combination makes automated theorem proving possible: machines can transform logical formulas, apply inference rules, and verify whether conclusions follow from premises. If your goal is to design reliable reasoning systems—whether for verification, compliance rules, or structured decision support—mastering FOL is a strong step, and an AI course in Delhi that covers KR and theorem proving can provide a practical pathway to applying these foundations in real systems.
