- #1
Logical Dog
- 362
- 97
If an object type ◌ exists and within the set of these objects exist some which have both well defined properties A and B, and some which only have the property A. There is at least one ◌ which does not have both properties.
Thus say we create the set of such objects such that A or B (inclusive or) is true. And we create the set of ◌ such that property B is true. It then becomes that:
[tex]\left \{ \cdot | B(\cdot ) = true \right \} \subset \left \{ \cdot | B(\cdot ) \vee A(\cdot ) = true \right \} [/tex]
We can then say that A is a necessary but not sufficient condition for B. If there do not exist any where only one of a or b apply then:
[tex]\left \{ \cdot | B(\cdot ) = true \right \} = \left \{ \cdot | B(\cdot ) \vee A(\cdot ) = true \right \} [/tex]
In which case it becomes that A is a necessary and sufficient condition for B. Or, b is a neccesarry or sufficient condition for A. (due to symmetry of equivalence).
Thus say we create the set of such objects such that A or B (inclusive or) is true. And we create the set of ◌ such that property B is true. It then becomes that:
[tex]\left \{ \cdot | B(\cdot ) = true \right \} \subset \left \{ \cdot | B(\cdot ) \vee A(\cdot ) = true \right \} [/tex]
We can then say that A is a necessary but not sufficient condition for B. If there do not exist any where only one of a or b apply then:
[tex]\left \{ \cdot | B(\cdot ) = true \right \} = \left \{ \cdot | B(\cdot ) \vee A(\cdot ) = true \right \} [/tex]
In which case it becomes that A is a necessary and sufficient condition for B. Or, b is a neccesarry or sufficient condition for A. (due to symmetry of equivalence).