Necessity and Sufficiency

• ola:: describes the relationship between clauses of a conditional statement
• A condition can be necessary, sufficient, or both necessary and sufficient.
  • but never sufficient + unnecessary
• Any conditional relationship consists of
  • at least one sufficient condition an
  • at least one necessary condition
  • one of them is the antecedent, one of them is the consequent
• eg: male/brother
  • being a male
    • is a necessary condition for being a brother,
    • but it is not sufficient for being a brother (must also be sibling)
  • while being a male+sibling is a necessary AND sufficient condition for being a brother
• eg: omelette egg
  • having an egg
    • is necessary for omelette
    • but insufficient — without milk, is just fried egg
  • being an omelette
    • is sufficient and necessary to contain egg
• eg: persecution
  • “Christian persecution is sufficient for being real Christian”
  • but it is not necessary — paralysed grandma
  • but it is not sufficient — crazy people get persecuted

Necessity

• implicational = necessary
  • Necessary conditions must be true for another statement to be true.
• if consequent is true, the antecedent must be true
  • backwards validation
• In order for human beings to live
  • it is necessary that they have air
  • (but that’s insufficient, there are other needs)

Sufficiency

• conditional = sufficient
  • Sufficient conditions guarantee the truth of another statement.
  • “if and only if”…
• if the antecedent is true, the consequent must be true
  • forwards validation
• If humans are living, then it is sufficient to say there is air

Duality

• For any statements S and N
  • the assertion that “N is necessary for S”, is equivalent to
  • the assertion that “S is sufficient for N”
• junctions?
  • disjunctions (using “or”) of sufficient conditions may achieve necessity
  • conjunctions (using “and”) of necessary conditions may achieve sufficiency
• every mathematical predicate N with the set T(N) of objects, events, or statements for which N holds true;
  • then asserting the necessity of N for S
    • is equivalent to claiming that T(N) is a superset of T(S),
  • e.g. egg necessary for omelettes = egg dishes may be omelette dishes
  • while asserting the sufficiency of S for N
    • is equivalent to claiming that T(S) is a subset of T(N).
  • e.g. omelette is sufficient for egg presence = omelette dishes are always egg dishes

Venn Diagram

• $A \cap B$ is sufficient but unnecessary for being in $A$
• $A$ is necessary but insufficient for being in $A \cap B$
• $A$ and $B$ are necessary AND sufficient for being in $A \cap B$

Examples

• N ∩ S: iff and only if
• ¬N ∩ S: Not required
• N ∩ ¬S: Not enough

Consequent: Number is even.

Consequent: Person is alive

Consequent: Test score meets a passing grade of 40.