Boolean & Modal Logic

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Abstractions about Boolean and modal algebras, normal forms, predicate calculi, deductive closure, distributivity, and expressive completeness.

15 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• BCK algebra — An algebra with a binary implication-like operation and a distinguished zero satisfying the BCI identities plus an axiom that enforces the BCK weakening condition.
• Boolean algebra — An algebraic structure with conjunction, disjunction and complementation satisfying laws that model two-valued logic and set operations.
• Deductive closure — The smallest superset of a set of formulas that contains every formula derivable from it under a specified consequence relation or proof system.
• Disjunctive normal form — A Boolean formula represented as a disjunction of conjunctions of literals—an OR of AND terms.
• Functional completeness — The property of a set of Boolean connectives from which every Boolean function can be expressed by composition.
• Inclusion (Boolean algebra) — The canonical partial order on a Boolean algebra, where a≤b exactly when a∧¬b=0, equivalently a∧b=a or a∨b=b.
• Modal algebra — A Boolean algebra equipped with a unary normal meet-preserving modal operator, providing algebraic semantics for normal propositional modal logics.
• Monadic predicate calculus — The function-free fragment of first-order logic whose predicate symbols all have arity one.
• Monadic second-order logic — The fragment of second-order logic that permits quantification over individual elements and unary predicates or sets, but not arbitrary higher-arity relations.
• Negation normal form — A logical formula form using only conjunction, disjunction and literals, with every negation applied directly to an atomic proposition.
• Prenex normal form — A first-order formula form in which all quantifiers occur in one leading prefix followed by a quantifier-free matrix.
• Principle of distributivity — The propositional-logic laws stating that conjunction distributes over disjunction and disjunction distributes over conjunction.
• Product term — A Boolean expression formed as the conjunction of one or more literals, with a minterm as the special case containing every variable exactly once.
• 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.
• Unate function — A Boolean function that is monotone in each variable after independently choosing whether that variable is interpreted positively or negatively.