Logical Connectives - Properties

Properties

Some logical connectives possess properties which may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are:

• Associativity: Within an expression containing two or more of the same associative connectives in a row, the order of the operations does not matter as long as the sequence of the operands is not changed.
• Commutativity: The operands of the connective may be swapped preserving logical equivalence to the original expression.
• Distributivity: A connective denoted by · distributes over another connective denoted by +, if a · (b + c) = (a · b) + (a · c) for all operands a, b, c.
• Idempotence: Whenever the operands of the operation are the same, the compound is logically equivalent to the operand.
• Absorption: A pair of connectives, satisfies the absorption law if for all operands a, b.
• Monotonicity: If f(a₁, ..., a_n) ≤ f(b₁, ..., b_n) for all a₁, ..., a_n, b₁, ..., b_n ∈ {0,1} such that a₁ ≤ b₁, a₂ ≤ b₂, ..., a_n ≤ b_n. E.g., .
• Affinity: Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., .
• Duality: To read the truth-value assignments for the operation from top to bottom on its truth table is the same as taking the complement of reading the table of the same or another connective from bottom to top. Without resorting to truth tables it may be formulated as g̃(¬a₁, ..., ¬a_n) = ¬g(a₁, ..., a_n). E.g., .
• Truth-preserving: The compound all those argument are tautologies is a tautology itself. E.g., ⊂. (see validity)
• Falsehood-preserving: The compound all those argument are contradictions is a contradiction itself. E.g., ⊄, ⊅. (see validity)
• Involutivity (for unary connectives): f(f(a)) = a. E.g. negation in classical logic.

For classical and intuitionistic logic, the "=" symbol means that corresponding implications "…→…" and "…←…" for logical compounds can be both proved as theorems, and the "≤" symbol means that "…→…" for logical compounds is a consequence of corresponding "…→…" connectives for propositional variables. Some of many-valued logics may have incompatible definitions of equivalence and order (entailment).

Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law.

In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic.