A polarity item (PolI) is a word or phrase that appears only in the context of another word or grammatical construction, called its licensor. For example the licensors of the determiner ANY in English include negation (SUZIE DIDN'T LICK ANY CAT, NO DOG LICKED ANY CAT, but *SUZIE LICKED ANY CAT), possibility operators (ANY CAT CAN SLEEP, but *ANY CAT SLEPT, *ANY CAT MUST SLEEP), and simple interrogation (DID ANY CAT SLEEP?), etc.
The distribution of PolIs can be explained in OT by assuming that they have counterparts, call them superior items (SupIs), which outrank them for the expression of particular meanings. So if both ANY and EVERY mean A, but EVERY >> ANY, then EVERY CAT SLEPT is preferred to ANY CAT SLEPT as the expression of the meaning AcSc (a not-so-innocent shorthand for Ax(Cx->Sx)).
When a PolI is licensed, it generally occurs in an environment that does not exclude its SupI. For example SUZIE DIDN'T LICK ANY CAT and SUZIE DIDN'T LICK EVERY CAT; ANY CAT CAN SLEEP and EVERY CAT CAN SLEEP; and DID ANY CAT SLEEP? and DID EVERY CAT SLEEP? are all grammatical. However they all have different meanings. When ANY occurs, A has scope over its licensing operator, whereas when EVERY occurs with a licensor for ANY, the licensing operator has scope over A. So SUZIE DIDN'T LICK ANY CAT means Ac~Lsc whereas SUZIE DIDN'T LICK EVERY CAT means ~AcLsc; ANY CAT CAN SLEEP means AcDSc whereas EVERY CAT CAN SLEEP means DAcSc; and DID ANY CAT SLEEP? means Ac?Sc whereas DID EVERY CAT SLEEP? means ?AcSc.
These results can be accounted for by constraints that associate the grammatical constructions consisting of ANY together with its various licensors with semantic constructions of the form AO, where O is the operator associated with a particular licensor L. For example, theconstraint NEG-ANY associates NOT+ANY and NO+ANY constructions with A~; D-ANY associates CAN+ANY constructions with AD; and Q-ANY associates inversion+ANY constructions with A? Each of these constraints outranks EVERY. The effect of D-ANY is shown in the following tableaux.
ANY CAT CAN SLEEP D-ANY EVERY ANY
=> AcDSc *
DAcSc *! *
EVERY CAT CAN SLEEP D-ANY EVERY ANY
=> DAcSc *
AcDSc *! *
The various ANY-licensing constraints presented here are also strictly ranked: ?ANY >> NEG-ANY >> D-ANY. Both ?-ANY and D-ANY license the occurrence of ANY in subject position; NEG-ANY does not (*ANY CAT DIDN'T SLEEP). However, ANY is licensed subject position in negative questions (DIDN'T ANY CAT SLEEP? DID ANY CAT NOT SLEEP?), showing that ?-ANY >> NEG-ANY. Moreover, ANY is not licensed in subject position in negative possibility statements (*ANY CAT CAN'T SLEEP), showing that NEG-ANY >> D-ANY.