Three-valued logic¶
A three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value.
Core Idea¶
Three-valued logic is treated here as the recurring crossdomainmodelsstructuresrepresentations identity summarized by this source-grounded definition: A three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value. A three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value.
Scope of Application¶
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Logics. Just as in bivalent logic, where not all operators are given names and subsets of functionally complete operators are used, there may be functionally complete sets of ternary-valued operators.
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Motivation. Bruno de Finetti used a third value to represent when "a given individual does not know the [correct] response, at least at a given moment." Hilary Putnam used it to represent.
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Motivation. Similarly, Stephen Cole Kleene used a third value to represent predicates that are "undecidable by [any] algorithms whether true or false".
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Kleene and Priest logics. If the truth values 1, 0, and −1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic (where x + y uses addition, xy uses multiplication.
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Ternary Post logic. Along with minimum and maximum functions to define conjunction and disjunction connectives, respectively.
Clarity¶
A clear use of Three-valued logic names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is A three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value.
Manages Complexity¶
Three-valued logic compresses multiple crossdomainmodelsstructuresrepresentations details into a stable diagnostic relation. The source shows both the central mechanism—(This may be made clear by considering all possible truth tables for an arbitrary unary operator.—and the practical consequence—it may be defined either by appending one of the two equivalent axioms or equivalently to the axioms of intuitionistic logic, or by explicit truth tables for its operations.
Abstract Reasoning¶
- Type the carrier. Identify the crossdomainmodelsstructuresrepresentations entities to which the claim applies.
- State the relation. Use the source-grounded identity: A three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value.
- Check operation and conditions.
Knowledge Transfer¶
Within the home domain. Knowledge about Three-valued logic transfers literally when a new case preserves the same carrier type, relation, and recognition test. Just as in bivalent logic, where not all operators are given names and subsets of functionally complete operators are used, there may be functionally complete sets of ternary-valued operators. Bruno de Finetti used a third value to represent when "a given individual does not know the [correct] response, at least at a given moment." Hilary Putnam used it to represent values that cannot physically be decided. Beyond the home domain.
Relationships to Other Abstractions¶
Current abstraction Three-valued logic Domain-specific
Parents (1) — more general patterns this builds on
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Three-valued logic is a kind of Formal System Prime
A three-valued logic is a formal system specialized by a three-element truth-value semantics; it is not a kind of Omega-logic.
Hierarchy paths (2) — routes to 2 parentless roots
- Three-valued logic → Formal System → Formalization → Representation → Abstraction
- Three-valued logic → Formal System → Formalization → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Three-valued logic sits in a moderately populated region (49th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Formal Logic & Language Constructs (20 abstractions)
Nearest neighbors
- Two-Element Boolean Algebra — 0.87
- Gödel–Dummett Logic — 0.86
- Kripke–Platek set theory with urelements — 0.86
- Existential Instantiation — 0.86
- Valuation (logic) — 0.86
Computed from structural-signature embeddings · 2026-10-08