Regular modal logic¶
A classical modal logic closed under a rule lifting conjunction-preserving implication through necessity and containing the duality of necessity and possibility.
Core Idea¶
Every normal modal logic is regular but regularity need not include necessitation or the full distribution axiom K, neighborhood-semantic conventions vary and regular must not be conflated with regularity of Kripke frames. The inference rule ensures that whenever a conjunction entails a conclusion, jointly necessary premises entail its necessity; duality defines possibility through negated necessity while preserving a weaker-than-normal modal algebra. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Regular modal logic belongs to modal logic and is useful where the analyst can specify the typed modal logic carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the classical propositional base, box and diamond modal operators, duality axiom, regularity inference rule from A and B implying C to boxed A and boxed B implying boxed C, closure under substitution and modus ponens, inclusion relations with normal and classical modal logics, neighborhood-frame semantics and omitted necessitation or K principles are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the classical propositional base, box and diamond modal operators, duality axiom, regularity inference rule from A and B implying C to boxed A and boxed B implying boxed C, closure under substitution and modus ponens, inclusion relations with normal and classical modal logics, neighborhood-frame semantics and omitted necessitation or K principles are explicit the center of the account.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Regular modal logic. Regular modal logic compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed modal logic carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the classical propositional base, box and diamond modal operators, duality axiom, regularity inference rule from A and B implying C to boxed A and boxed B implying boxed C, closure under substitution and modus ponens, inclusion relations with normal and classical modal logics, neighborhood-frame semantics and omitted necessitation or K principles are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of modal logic because they reuse the typed modal logic carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The inference rule ensures that whenever a conjunction entails a conclusion, jointly necessary premises entail its necessity; duality defines possibility through negated necessity while preserving a weaker-than-normal modal algebra., and type the carrier, state every parameter and convention in the definition, test that the classical propositional base, box and diamond modal operators, duality axiom, regularity inference rule from A and B implying C to boxed A and boxed B implying boxed C, closure under substitution and modus ponens, inclusion relations with normal and classical modal logics, neighborhood-frame semantics and omitted necessitation or K principles are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Regular modal logic Domain-specific
Parents (1) — more general patterns this builds on
-
Regular modal logic is a kind of Inference Prime
The proposed strict upward parent is
prime:inference.
Hierarchy path (1) — routes to 1 parentless root
- Regular modal logic → Inference → Rationality → Normativity → Constraint
Neighborhood in Abstraction Space¶
Regular modal logic sits in a crowded region of the domain-specific corpus (26th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Boolean & Modal Logic (15 abstractions)
Nearest neighbors
- Modal algebra — 0.92
- Doxastic logic — 0.92
- Logical equality — 0.91
- Paraconsistent logic — 0.91
- BCK algebra — 0.91
Computed from structural-signature embeddings · 2026-09-08