Does satisfying one clause satisfy a CNF formula?

No. Every clause in the conjunction must be satisfied.

Is a clause always propositional?

No. First-order clauses use predicates and may contain variables, although this article's worked example is propositional.

AI Glossary / Graphs and symbolic AI / Intermediate

Clause

Definition

A clause, in the logic used by many symbolic AI systems, is a disjunction of literals: statements joined by logical OR, where each statement is an atom or its negation. A clause is satisfied when at least one of its literals is true. Clauses are basic building blocks in satisfiability solving and automated theorem proving.

What it means

An atom is a basic proposition such as Approved. A literal is either that proposition or its negation, such as NOT Approved. The expression NOT Approved OR Reviewed is therefore a clause containing two literals. It expresses that approval cannot hold without review.
Several clauses can be joined by AND to form conjunctive normal form, often abbreviated CNF. The distinction matters: the solver must satisfy every clause, but only one literal within each clause needs to be true. This entry uses the logical meaning of clause, rather than a grammatical clause or a contractual provision.

How it works

Translate the relevant conditions into logical expressions, then represent them as clauses when the reasoning method requires that form. Each assignment of truth values makes a clause true or false. If every literal in one clause is false, that assignment violates the whole CNF formula containing it.
Solvers can exploit the structure. A unit clause contains one literal, so satisfying it requires that literal to be true. If all but one literal in a clause become false, the remaining literal is forced. Resolution combines clauses containing complementary literals to derive further consequences. First-order clauses can also contain variables and predicates, bringing unification and implicit universal quantification into the picture.

Three practical examples

A configuration checker represents incompatible options with NOT OptionA OR NOT OptionB, forbidding their simultaneous selection.
A workflow model represents a prerequisite with NOT Release OR Reviewed, allowing release only when review is true.
A theorem prover converts a knowledge base and the negation of a proposed conclusion into clauses, then searches for a contradiction.

Think of it this way

Think of a clause as a condition with several alternative ways to satisfy it. A collection of clauses is a checklist where every line must pass, even though each line may offer alternatives. The analogy is limited because a negative literal is a logical condition, not an instruction to perform an action, and the alternatives can interact across many lines.

Worked example

Let A mean Approved, R mean Reviewed, and D mean DocumentsPresent. Use three clauses: NOT A OR R; NOT R OR D; and A. The third is a unit clause, so A must be true. In the first clause, NOT A is then false, forcing R to be true. In the second, NOT R becomes false, forcing D to be true.
The assignment A = true, R = true, D = true satisfies all three clauses. Now add the unit clause NOT D. It requires D to be false, contradicting the consequence above. No assignment can satisfy all four clauses together.
Notice that the first clause alone does not require approval. If A is false, NOT A is true and that clause is satisfied regardless of R. Also, it does not say that review causes approval. These distinctions prevent a common translation error: a prerequisite is not the same as a sufficient condition or a causal relationship.

Illustrative case study

At fictional Birch Configurations, a team translates product rules into a satisfiability model. One rule says that selecting a premium feature requires an additional component. During review, the team notices that someone has reversed the implication, requiring the feature whenever the component is present.
Writing the rule as a clause makes the difference visible through a small truth table. The team corrects the translation and retains the distinguishing assignment as a review example. This fictional case illustrates checking the meaning of a rule before trusting the solver; it does not claim any measured business savings.

Common misconceptions and mistakes

OR within a clause is normally inclusive: more than one literal may be true. A clause saying A OR B therefore does not, by itself, mean exactly one of them. That requires an additional condition preventing both.
Do not confuse a clause with a conjunction of literals. Nor should an empty clause be treated as harmless missing data: in standard clausal logic it is false, because it has no literal that could satisfy it. Conversely, an empty collection of clauses is satisfied. A Horn clause has at most one positive literal, not necessarily just one literal in total. Finally, encodings with fresh variables can preserve satisfiability without having exactly the same assignments over the expanded vocabulary.

Business use

Clauses can give rule-heavy applications a precise representation that a solver can check for consistency. The business value depends on faithful translation. Keep a plain-language statement beside each rule and review a permitted and a forbidden example. When the overall set is inconsistent, investigate the conflicting rules instead of interpreting the solver's failure as a mysterious technical problem. A logically consistent rule set can still encode an unwanted policy, so domain review remains necessary.

Check your understanding

Does NOT A OR R force R to be true when A is false?

Answer: No. NOT A is already true, so the clause is satisfied whether R is true or false. It forces R only when A is true.

Related terms

Learn next

Learn unit propagation next. Take the three-clause example, change the unit clause A to NOT D, and trace what the remaining clauses imply. Then study resolution to understand how clause-based proofs derive contradictions.

Sources

Primary sources: Russell and Norvig, original propositional reasoning chapter, https://aima.cs.berkeley.edu/newchap07.pdf ; Peter Flach, Simply Logical, predicate-to-clausal conversion, https://book.simply-logical.space/src/text/appendices/b_1.html ; original AIMA logic implementation documentation, https://aima.cs.berkeley.edu/lisp/doc/overview-LOGIC.html . The approval example and fictional case are original illustrations.

Previous: Classifier head Next: Clefts