Most (all?) languages use particular morphosyntactic forms with a certain meaning to express possession, as in English examples (1).
(1) (a) Most teachers' cars are not luxury models.
(b) Four cats of Tom's wandered off last week.
Although the term possessive is sometimes extended to include other expressions that use the forms without the meaning or express the meaning in another form, possessives in the strict sense are usefully distinguished from cases such as those illustrated in (2).
(2) (a) John's not remembering that he had met her annoyed Mary.
(b) The cars that most teachers have are not luxury models.
It is particularly helpful to distinguish consistently between possessives as in
(3) (a) Two people's photographs were on the mantel.
(b) Photographs of two people's were on the mantel.
and relational noun complements like that in (4).
(4) Photographs of two people were on the mantel.
The photographs mentioned in (3) could depict any number of people, unlike those mentioned in (4).
The talk will present a largely compositional analysis of what possessives like those in (1) and (3) mean, which also explains the more complex varieties in (5). (5) (a) Four of two crooked politicians' luxury cars were impounded. (b) Four luxury cars of two crooked politicians' were impounded. The analysis provides diagnostics for distinguishing all these possessives from relational noun complements, as in (4), and also from other things that are more naturally called possessives, viz. the modifying possessives (or classifying genitives) in (6). (6) (a) He prefers child's play to hard work. (b) There are many pockets in most gardeners' aprons.