Normal modal logic¶

A normal modal logic contains all propositional tautologies and the distribution axiom K, and is closed under uniform substitution, modus ponens, and necessitation.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11025
Domain group: Humanities
Origin domain: Philosophy
Subdomains: Modal Logic, Philosophical Logic → Philosophy

Core Idea¶

A normal modal logic is a set of modal formulas containing all propositional tautologies and every instance of the distribution axiom □(A → B) → (□A → □B), while remaining closed under uniform substitution, modus ponens, and necessitation. The smallest such system is K; stronger normal systems add axiom schemata such as T, 4, 5, B, or D. Normality is a structural package, not a claim that one particular modal theory is uniquely standard. It ensures that necessity distributes over implication and that theorems may be necessitated.

Scope of Application¶

Documented setting. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g.
Common normal modal logics. The notation refers to the table at Kripke semantics § Common modal axiom schemata.
Common normal modal logics. Frame conditions for some of the systems were simplified: the logics are sound and complete with respect to the frame classes given in the table, but they may correspond to a.
Common normal modal logics. The following table lists several common normal modal systems.
Documented setting. A normal modal logic contains all propositional tautologies and the distribution axiom K, and is closed under uniform substitution, modus ponens, and necessitation.

Clarity¶

A clear use of Normal modal logic names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is A normal modal logic contains all propositional tautologies and the distribution axiom K, and is closed under uniform substitution, modus ponens, and necessitation.

Manages Complexity¶

Normal modal logic compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—frame conditions for some of the systems were simplified: the logics are sound and complete with respect to the frame classes given in the table, but they may correspond to a larger class of frames.—and the practical consequence—detachment rule (modus ponens): A\to B, A \in.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: A normal modal logic contains all propositional tautologies and the distribution axiom K, and is closed under uniform substitution, modus ponens, and necessitation.
Check operation and conditions. The following table lists several common normal modal systems.
Demand recognition evidence. A normal modal logic contains all propositional tautologies and the distribution axiom K, and is closed under uniform substitution, modus ponens, and necessitation.

Knowledge Transfer¶

Within the home domain. Knowledge about Normal modal logic transfers literally when a new case preserves the same carrier type, relation, and recognition test. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g. The notation refers to the table at Kripke semantics § Common modal axiom schemata. Beyond the home domain. No canonical parent is asserted for Normal modal 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.

Relationships to Other Abstractions¶

Current abstraction Normal modal logic Domain-specific

Parents (1) — more general patterns this builds on

Normal modal logic is a kind of Formal System Prime

Normal modal logic is a domain-specific kind of formal system under its frozen identity and differentia. Complete-catalog comparison found the corresponding live broader identity.

Hierarchy paths (2) — routes to 2 parentless roots

Normal modal logic → Formal System → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space¶

Normal modal logic sits in a sparse region of the domain-specific corpus (69^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Formal Models & Logical Foundations (33 abstractions)

Nearest neighbors

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