Logical NOR¶
The Boolean connective that is true exactly when all its operands are false, equivalently the negation of disjunction; repeated NOR alone is functionally complete for propositional logic.
Core Idea¶
NOR denies that any input is true. Its binary truth table has exactly one true case, and De Morgan's law identifies it with the conjunction of the negated operands.
Its importance exceeds one connective: self-NOR gives negation, and further compositions construct OR, AND, and every Boolean function. Hardware and software realizations preserve the truth function only under declared width, polarity, and state conventions.
Scope of Application¶
- Propositional logic. Defines joint denial and functional bases.
- Boolean algebra. Supports identities and normal-form transformations.
- Digital logic. Implements NOR gates and NOR-only circuits.
- Programming and bit operations. Applies elementwise to fixed-width bit patterns when defined.
Clarity¶
State operands and arity, truth-value system, inclusive OR, notation and precedence, proposition or bit-vector domain, word width and signedness, unknown-state handling, desired truth table, hardware polarity and voltage mapping, fan-in, propagation timing, gate-count/depth metric, and whether a named symbol denotes NOR or XNOR in the source. Inclusion test: Require the truth-functional operation negating inclusive OR, with output true exactly when no operand is true, or a correctly mapped gate/bitwise realization of that operation. Exclusion test: Exclude NAND, exclusive NOR/equivalence, negation of each input followed by OR, natural-language 'neither...nor' when scope differs, bitwise complement without OR, and a low-output wired-OR circuit lacking the specified logic convention. Nearest boundary: XNOR is true when inputs are equal, including the both-true row; NOR is true only on the all-false row. Exit condition: Meaning changes with arity, inclusive versus exclusive OR, proposition versus bit-vector domain, operator precedence and scope, two-valued versus many-valued logic, active-high/active-low hardware convention, fan-in, and treatment of unknown or high-impedance states. Common misclassifications: It is not NAND. It is not XNOR or logical equivalence. It negates the whole disjunction, not merely one operand. Logical NOR and bitwise NOR require different operand domains. Nearest named distinctions: NAND: Negates conjunction and is false only when all inputs are true. XNOR: Returns true when inputs are equal. OR: Returns true when at least one input is true. Bitwise NOR: Applies NOR independently at each bit position of fixed-width operands.
Manages Complexity¶
A tiny truth table can produce large compositional expressions. Many-valued hardware states, active-low naming, bit widths, and ambiguous symbols create errors outside ideal Boolean semantics.
Abstract Reasoning¶
- Fix the operand domain, arity, and truth-value convention.
- Compute inclusive disjunction of all operands.
- Negate that result and verify the all-false true row.
- Use algebraic identities or truth tables to transform larger expressions.
- For implementation, map truth to physical or bit-level states and evaluate cost and timing.
Knowledge Transfer¶
Functional-basis reasoning transfers to NAND and other complete operator sets. Binary NOR identities do not transfer unchanged to fuzzy, probabilistic, quantum, or multivalued logics without defining their disjunction and negation.
Relationships to Other Abstractions¶
Current abstraction Logical NOR Domain-specific
Parents (1) — more general patterns this builds on
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Logical NOR presupposes Boolean algebra Domain-specific
Logical NOR presupposes Boolean algebra: the parent's defining role is necessary to the child's frozen mechanism or criterion.
Hierarchy path (1) — routes to 1 parentless root
- Logical NOR → Boolean algebra → Representation → Abstraction
Neighborhood in Abstraction Space¶
Logical NOR sits in a crowded region of the domain-specific corpus (25th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Logical Connectives & Formal Systems (13 abstractions)
Nearest neighbors
- NOR logic — 0.92
- Logical or — 0.91
- Switching circuit theory — 0.91
- Logical Operation — 0.89
- Ternary Operation — 0.88
Computed from structural-signature embeddings · 2026-10-08