Is an interpretation always true?

An interpretation is a way to assign meanings. Particular statements may be true or false under it. It is a model of a knowledge base only when all required statements hold.

Can two symbols refer to one object?

Yes, in ordinary first-order semantics this is possible unless additional assumptions or constraints prohibit it.

Is this the same as interpretability?

No. Interpretability concerns human understanding of a model's behavior; interpretation here concerns the semantics of formal symbols and statements.

AI Glossary / Governance, ethics and safety / Intermediate

Interpretation

Definition

An interpretation, in formal logic, assigns meanings to the symbols of a language so that statements can be evaluated as true or false. In a basic first-order setting, it specifies a domain of objects and how constants and predicates refer to those objects and their relationships. This is different from explaining a machine-learning prediction.

What it means

A formal expression such as Connected(a, b) has a structure, but its symbols need meanings before it describes a particular situation. The letters might refer to two devices, and Connected might mean that a direct communication link exists between them.

Another interpretation could assign the same symbols to two rooms and use Connected for a doorway relationship. The written expression is unchanged, but what it describes has changed.

Interpretations therefore separate syntax, the arrangement of symbols, from semantics, their meaning. The intended interpretation is the one a designer wants to represent. Formal reasoning may consider other interpretations as well, especially when determining what follows from a set of statements.

How it works

Begin with a nonempty domain: the objects under discussion. Assign each constant symbol to an object in that domain. Assign each predicate a property or relation over the appropriate number of objects. If the language contains function symbols, those also need suitable mappings.

A statement can then be evaluated under that interpretation. A predicate applied to particular objects is true when those objects stand in the assigned relation. More complex statements combine these evaluations using logical operators and, where present, quantifiers.

An interpretation satisfying every statement in a knowledge base is a model of that knowledge base. A logical consequence must hold in every model, not merely in a convenient example. For formulas containing free variables, their values also depend on a variable assignment; an interpretation alone does not choose those values in the usual first-order treatment.

Three practical examples

1. A facilities knowledge base assigns a room identifier to a particular room and defines Adjacent as sharing a doorway, rather than merely being on the same floor.

2. An equipment ontology distinguishes the relation Operates from Owns. A contractor can operate an asset without owning it, so the two predicates have different interpretations.

3. A planning system represents Before as an ordering between tasks. Its intended meaning must distinguish a strict precedence requirement from a suggested scheduling preference.

Think of it this way

Think of labels placed on a collection of objects, together with a key explaining the relationships drawn between them. The key turns marks into claims about the collection.

The analogy is limited. A formal interpretation provides a precise mathematical basis for evaluating statements, while an ordinary key can leave much implicit. Different labels may refer to the same object unless the representation adds constraints that rule this out. Familiar-looking names alone do not settle identity or meaning.

Worked example

Consider a toy domain containing three meeting rooms: East, West, and North. Assign constant a to East and constant b to West. Interpret Adjacent as sharing a direct doorway. Suppose East and West share a doorway, while North shares one with neither.

Under this interpretation, Adjacent(a, b) is true. Now keep the doorway relationship unchanged but assign b to North. The same written statement, Adjacent(a, b), is false under the second interpretation.

This does not mean that the building changed. The assignment of a symbol changed. To describe the building consistently, a knowledge-base designer must maintain the intended mapping between symbols and rooms.

If a set of formal statements does not distinguish these possibilities, a reasoning system may have to consider both. Adding a clear assertion or constraint can exclude an unwanted possibility, provided that the added statement accurately reflects the intended situation.

Illustrative case study

Fictional case: Juniper Engineering combines two equipment catalogs for a rule-based maintenance assistant. Both use a relation called PartOf. One means direct physical containment; the other includes components nested several levels below an assembly.

The merged representation initially treats the relations as identical. A rule intended for direct components therefore applies more broadly than its designers expected. The team separates direct membership from the broader containment relation and documents how each is derived.

This invented case illustrates the importance of semantic agreement. It claims no measured business outcome. Fixing the interpretation of one relation does not validate all facts in the catalogs; identifiers, missing records, and other relationship meanings still need their own checks.

Common misconceptions and mistakes

An interpretation is not necessarily an explanation in everyday language, and it is not automatically the designer's intended interpretation. A model in logic is an interpretation satisfying statements; it is not necessarily a trained machine-learning model.

The names of symbols do not force particular meanings. Nor should a system assume that two differently named constants denote different objects unless its formalism or stated assumptions provide that rule. Such assumptions must be made explicit when they affect what the system can conclude.

Business use

Businesses encounter interpretation issues when integrating knowledge graphs, rules, and structured data definitions. Similar labels across departments may conceal different domains, relationships, or criteria.

Review the meaning of important symbols with domain specialists. Use concrete examples that make competing interpretations visible, and record identity and relationship assumptions. This helps teams distinguish a reasoning error from a mismatch in what the representation was meant to say. It also makes later changes to business terminology easier to assess before updating dependent rules.

Check your understanding

Connected(a, b) changes truth value when b is assigned to a different object. Must the underlying real-world relationship have changed?

Answer: No. A different symbol assignment can change the statement's truth under the interpretation even when the domain's relationships stay the same.

Related terms

Learn next

Next, study intended interpretation and logical consequence. Write one simple relationship statement, assign its symbols to concrete objects, and construct a second interpretation in which the statement has a different truth value.

Sources

David Poole and Alan Mackworth, Artificial Intelligence lecture 12.2, formal semantics: https://artint.info/slides/ch12/lect2.pdf The author slides were read. This entry covers the formal-logic sense while retaining the supplied governance category. The source AIMA glossary did not expose the specific entry during retrieval; editorial scope review remains appropriate. Examples are original.

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