Here's a breakdown of how these concepts are understood in formal semantics:
1. Intersection in Formal Semantics:
* Intersection is a logical operation that combines two sets (or predicates) and produces a new set containing only elements that are present in *both* original sets. It's represented by the symbol "∩".
* Example:
* Set A: {Dogs, Cats, Birds}
* Set B: {Dogs, Horses, Fish}
* A ∩ B = {Dogs}
2. Relative Intersection:
* Relative Intersection is a variation of intersection that takes into account the context of the sentence or phrase. It's not a universally accepted term, but it is used to describe cases where the intersection is defined based on the surrounding linguistic elements.
* Example:
* Consider the sentence: "The cat is sitting on the mat."
* The phrase "on the mat" restricts the set of cats we're considering to those that are physically on the mat. This restriction is a form of relative intersection.
3. Pure Intersection:
* Pure Intersection is often used in contrast to "Relative Intersection" and refers to the standard, unconstrained intersection operation, where the intersection is determined solely by the sets themselves, without regard to context.
* Example:
* "Dogs" and "Animals" are sets with a pure intersection, which would include all dogs, as all dogs are also animals.
In Summary:
The terms "relative intersection" and "pure intersection" are not standard linguistic terms but are used in formal semantics to differentiate between intersection operations that are influenced by context and those that are not.
It's important to remember that these concepts are tools for analyzing the meaning of words and sentences in a formal and precise way. They are not directly related to everyday language usage.