monotonicity: one construct, many systems

In order theory, a function between ordered sets is monotone, or order-preserving, when it respects the order: if x ≤ y then f(x) ≤ f(y), so increasing the argument never decreases the value. Birkhoff's Lattice Theory develops monotone maps as the structure-preserving morphisms of ordered sets. This order-preservation is the core construct that recurs, under the same name, in how adding premises affects conclusions and in how added support affects a collective choice.[1]

Unit

Whether x ≤ y guarantees f(x) ≤ f(y) throughout the domain

Recovery means

Not applicable

Measured by

Verified by checking that the order is preserved for all comparable pairs in the domain

Classical logical consequence is monotonic: adding premises to a valid argument never invalidates its conclusion, so the set of consequences only grows as the premise set grows. McCarthy's circumscription and Reiter's default logic are explicitly non-monotonic systems built to relax this, letting new information retract earlier conclusions. The property is the very same order-preservation (here the input is a premise set ordered by inclusion and the output is its consequence set), realised in a deductive system.[2][3]

Unit

Whether enlarging the premise set can only enlarge the set of conclusions

Recovery means

Not applicable

Measured by

Tested by whether every consequence of a premise set remains a consequence of every superset of it

In social choice theory, monotonicity requires that if an option wins and then gains support (with no voter ranking it lower), it must still win, so extra support can never harm it. May's 1952 theorem makes monotonicity one of the conditions that uniquely characterise simple majority rule. It is the same order-preservation construct, with voter support as the ordered input and the collective outcome as the order-respecting output.[4]

Unit

Whether increased support for an option can never worsen its collective outcome

Recovery means

Not applicable

Measured by

Checked as an axiom: raising an option in voters' preferences never turns its win into a loss