Valuation (logic)¶

In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12751
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Mathematical Logic, Model Theory → Mathematics

Core Idea¶

Valuation (logic) is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.

In logic and model theory, a valuation can be. In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables. In first-order logic and higher-order logics, a structure, (the interpretation) and the corresponding assignment of a truth value to each sentence in the language for that structure (the valuation proper).

The interpretation must be a homomorphism, while valuation is simply a function. In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives.

For Valuation (logic), the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols.
Constitutive relation — A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language.
Operating condition — In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.
Recognition evidence — In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives.
Admissible variation — In this context, a valuation begins with an assignment of a truth value to each propositional variable.
Characteristic consequence — This assignment can be uniquely extended to an assignment of truth values to all propositional formulas.
Failure boundary — Formulas are built out of atomic formulas using logical connectives and quantifiers.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.
Not an over-broad reading. In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.
Not an over-broad reading. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives.
Not an over-broad reading. In this context, a valuation begins with an assignment of a truth value to each propositional variable.
Not automatically Truth value. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Valuation (logic) applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Mathematical logic. In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols.
Mathematical logic. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language.
Notation. If v is a valuation, that is, a mapping from the atoms to the set { t, f } , then the double-bracket notation is commonly used to denote a valuation; that is, [![\phi]!]_v = v(\phi) for a propositional formula \phi .
Documented setting. The interpretation must be a homomorphism, while valuation is simply a function.
Mathematical logic. In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.
Mathematical logic. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Valuation (logic) names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. The strongest recognition evidence in the frozen account is: In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Valuation (logic) compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—a structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language.—and the practical consequence—this assignment can be uniquely extended to an assignment of truth values to all propositional formulas. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.
Check operation and conditions. In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.
Demand recognition evidence. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives.
Test variation. Change an implementation or setting while preserving in this context, a valuation begins with an assignment of a truth value to each propositional variable.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Valuation (logic) transfers literally when a new case preserves the same carrier type, relation, and recognition test. In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language.

Beyond the home domain. No canonical parent is asserted for Valuation (logic). An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema; recognition evidence → In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives

Applied / In Practice¶

In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Mathematical logic; invariant → In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema; boundary → the case exits the class when in mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema

Structural Tensions¶

T1 — Stable identity versus admissible variation. In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. In this context, a valuation begins with an assignment of a truth value to each propositional variable. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. This assignment can be uniquely extended to an assignment of truth values to all propositional formulas. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Valuation (logic) literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Valuation (logic) distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Valuation (logic) is structural-leaning. Its structural side is the repeatable organization summarized by In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. It further constrains recognition and variation through: In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Valuation (logic) literal. Its documented scope includes the condition that In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. Another bounded application condition is that A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—In this context, a valuation begins with an assignment of a truth value to each propositional variable.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Function (Mapping).

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Valuation (logic). The reviewed identity is: In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Valuation (logic) Domain-specific

Parents (1) — more general patterns this builds on

Valuation (logic) is a kind of Function (Mapping) Prime

A logical valuation is an assignment mapping formulas or sentences to truth values under a schema; it is not itself a logic system and is unrelated to Omega-logic.

Hierarchy path (1) — routes to 1 parentless root

Valuation (logic) → Function (Mapping)

Neighborhood in Abstraction Space¶

Valuation (logic) sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Formal Logic & Language Constructs (20 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema?
Truth value. A semantic value assigned to a proposition or formula to represent its status with respect to truth under a specified logic and interpretation. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Finite-Valued Logic. A logic is characterized by a finite logical matrix: finitely many semantic values, designated values defining consequence, and truth functions interpreting its connectives. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Logical equality. A truth-functional connective that is true exactly when its two propositions have the same truth value. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Valuation (logic) remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Valuation_(logic) (revision 1300832123).
Preserved source candidate: https://books.google.com/books?id=LTOfZn728-EC&pg=PA155

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.