Hyponymy is a sense relation
that serves to relate word-concepts in a hierarchical
fashion. Examples are:
The more specific concept is known as the hyponym, and the more general concept is known as the hypernym or superordinate. Apple is the hyponym of fruit and fruit is the superordinate of Apple.
Typically in semantics we can view things from two points of view, either from the Extensional viewpoint (relating the meanings of words to the outside world) or from the Intensional viewpoint, (relating word meaning to mental concepts). From an extensional view, the superordinate is the class of items that includes the class of hyponyms. If one is the subset of the other, then they are in a hyponymical relation. By this thinking, if the class of flowers contains the class of daisies then they are in the superordinate-hyponym relation. (true)
From the intensional view, the hyponym is more a more detailed mental concept than that of the superordinate. The mental concept flower is rather vague, perhaps only containing the fact that it is a plant, containing usually small, colorful petals. The concept for daisy is much more detailed, however, containing the specific colors, height, season of appearance, etc.
There are two problems that arise when we try to formalize the definition. Typically hyponymy is expressed as asymmetic entailment: It's an apple entails that it's a fruit, but not vice versa. It's a square entails that it's a rectangle, but not the reverse. The problem is that just because a pair is hyponmyic, it doesn't mean that the superordinate is always appropriate:
"The fact that it was a cow surprised the driver."
"The fact that it was an animal surprised the driver."
In this case, the fact that it was a cow doesn't entail the fact that the driver was surprised because it was an animal. Additionally, we have problems with negation: "It's not a cow" doesn't entail "it's not an animal."
The second problem stems from the fact that the definition of entailment used in linguistics states that only logically necessary, context-independent relations may count as entailment. The problem comes from the fact that speakers typically judge knife:cutlery and dog:pet to be valid relationships, just like dog:animal, but these relationship aren't always true. There are many hunting knives which aren't cutlery, just as there are many wild dogs that aren't pets. What seems to be happening is since no context or domain is being specified, subjects are responding assuming a typical, everyday context. The relations stated above are true, for an everyday context. The problem comes from the fact that linguistic entailment demands context-independence.
Despite the problems that are encountered in constructing a formal definition of meronymy, it still remains an exceedingly useful tool in understanding word sense.
See also: Meronymy, Synonymy.
Cruse, Alan, Meaning in Language: An Introduction to Semantics and Pragmatics, Oxford University Press, Oxford, 2000.
Thanks to Gritchka for pointing out that superordinates are also called "hypernyms".