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Predicate (logic)
Definition
In logic, a predicate represents a property of an individual or a relation among individuals. A predicate symbol combined with the appropriate arguments forms an atomic formula, such as Approved(document1) or Reviews(person1, document1). Under a classical interpretation, that formula is true or false. Predicates provide structure for expressing facts and rules about particular objects rather than treating every statement as an unrelated proposition.
What it means
The word 'predicate' is used at several closely related levels: the symbol written in a formula, the relation it denotes, or a condition containing variables. Keeping these levels distinct prevents confusion. Reviews is a symbol; Reviews(Ada, report1) is an atomic formula; the interpretation specifies whether the named person reviews the named report. A meaningful-looking name does not automatically give a computer the intended meaning. The model must specify what the arguments represent and when the relation holds.
How it works
Choose a domain of individuals and symbols for referring to them. Give each predicate a number of arguments, called its arity, and an interpretation. A unary predicate represents a property; a binary predicate represents a relation involving two arguments. Argument order matters unless the relation has an appropriate symmetry property.
Variables let a formula discuss unspecified individuals. Quantifiers then express statements about all individuals or at least one individual in the relevant domain. Connectives combine atomic formulas into larger claims. For example, a rule can connect being a document and having approval to another property. A function symbol has a different role: it denotes an object from its arguments, while a predicate expresses whether a relation holds. In the classical setting, evaluating a formula and establishing that a knowledge base entails it are also distinct tasks.
Three practical examples
- A document knowledge base uses ReviewedBy(document, person) to record who reviewed which item. Consistent argument order allows rules and queries to use the same relation reliably.
- A robot's symbolic model uses At(robot, location) and Clear(location). Planning rules can combine these facts without reducing every robot-location pair to an unrelated, unexplained label.
- A catalog represents Compatible(component, device). A rule engine can answer compatibility questions from declared facts and rules, while the interpretation must distinguish tested compatibility from a manufacturer's claim if those are different business concepts.
Think of it this way
A predicate resembles a carefully labeled form with a fixed set of blanks. '___ reviews ___' has two blanks with different roles. Filling them produces a specific assertion. The form's label and blank order make the assertion understandable, but evidence or a stipulated model is still needed to decide whether it is true.
Worked example
Consider a finite domain containing two people, Ada and Ben, and two reports, R1 and R2. Use Person and Report as unary predicates. Person is true exactly for Ada and Ben; Report is true exactly for R1 and R2.
Define Reviews(x, y) as true exactly for the ordered pairs (Ada, R1) and (Ben, R2). All other pairs are false in this fully specified toy interpretation. Reviews(Ada, R1) is therefore true, while Reviews(R1, Ada) is false. Reversing the arguments changes the claim.
Now evaluate: 'Every report has a person who reviews it.' Formally, for every y, Report(y) implies that there exists an x with Person(x) and Reviews(x, y). This is true. Ada supplies a witness for R1, and Ben supplies a witness for R2. For individuals that are not reports, the implication places no review requirement on them.
Compare: 'There is a person who reviews every report.' Formally, there exists an x such that Person(x) and, for every y, Report(y) implies Reviews(x, y). This is false. Ada does not review R2, and Ben does not review R1. The first statement allows different reviewers for different reports; the second requires one shared reviewer.
Add the pair (Ada, R2). Both statements now become true, with Ada witnessing the second. There are still four domain individuals and sixteen possible ordered pairs for the binary relation, but only three pairs are true. The truth of a relation depends on its specified interpretation, not on assuming that every possible pair holds.
Illustrative case study
In a fictional document portal, a dashboard labels the original two-report example 'single reviewer coverage complete.' Its query actually checks only whether every report has some reviewer. The records are accurate; the query answers the wrong question.
The team writes the two quantified statements separately and tests them against the four-individual interpretation. The first succeeds and the second fails, exposing the mismatch. It then adds Ada's review of R2 and confirms that the second statement changes to true. This small counterexample gives a clearer specification than relying on the dashboard label. The team also documents that Reviews means a completed review, rather than an assignment to review later.
Common misconceptions and mistakes
A predicate name is not an explanation of how truth is established. Argument order, arity and domain conventions require explicit agreement. A predicate is also not necessarily a learned classifier or a piece of executable code, although software can implement checks corresponding to predicates.
Do not confuse an unrecorded fact with a false fact. The worked example stipulates a complete interpretation. An incomplete knowledge base may leave a claim undecided unless a closed-world convention applies. Finally, exchanging 'for every' and 'there exists' can change the meaning, as the reviewer example demonstrates.
Business use
Predicates support explicit knowledge models, rule systems and structured queries. Define each relation in ordinary language, including argument roles, time scope and the evidence required for an assertion. Distinguish 'assigned,' 'completed' and 'approved' when they describe different events. Use small counterexamples to test whether rules express the intended business question. When information is incomplete, document whether missing assertions remain unknown or are treated as false, because that choice changes query results and subsequent decisions.
People also ask
Can a predicate have more than two arguments?
Yes; arity can be larger.
Is a predicate with variables already a complete claim?
Its truth generally depends on a variable assignment unless the variables are bound by quantifiers.
Does a predicate have to describe a physical object?
No. Domains can contain abstract individuals such as numbers or other modeled entities.
Check your understanding
Why is Reviews(R1, Ada) false in the example?
Answer: That ordered pair is absent from the fully specified relation.
Does every report having a reviewer imply one person reviews them all?
Answer: No.
What changes after adding Reviews(Ada, R2)?
Answer: Ada becomes a reviewer of both reports, satisfying the shared-reviewer statement.
Related terms
- Relation
- Arity
- Interpretation
- Domain
- Entailment
- Closed-world assumption
- Predicate logic
- atomic formula
- quantifier
- free variable
Learn next
Remove every review of R2 and reevaluate both quantified statements. Then examine how your conclusions would change if missing review records represented unknown information instead of a fully specified false relation.
More in Graphs and symbolic AI
- Precomputation - Precomputation means calculating information before the moment it is needed, then storing it for later reuse.
- Precondition - A precondition is a condition that must hold for an action to be applicable in a specified model.
- Precondition axioms - Precondition axioms are logical statements requiring an action's preconditions to hold whenever that action occurs in a planning model.
- Predicate indexing
- Predictive learning - Predictive learning, in the world-model sense associated with Yann LeCun, learns from observations to predict aspects of an environment's future state.
- Preference constraints - Preference constraints express which assignments are more desirable in a constraint problem, rather than simply declaring which assignments are permitted.
Sources
Poole and Mackworth, Predicate Calculus: https://artint.info/3e/html/ArtInt3e.Ch15.S3.html Cornell University, Predicate Logic: https://www.cs.cornell.edu/courses/cs3110/2014fa/lectures/21/lec21.html
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