Necessary and Sufficient Conditions
ONLY-IF Proposition
The proposition "p only if q" means that the proposition p can take place only if proposition q takes place also. That is, if the proposition q does not take place, then the proposition p cannot take place.
Formally, if p and q are propositions,
p only if q means "if not q then not p"
or equivalently,
if p then q
by logical equivalence between a proposition and its contrapositive.
Biconditional
Given sentential variables p and q, the biconditional of p and q is "p if, and only if, q." We symbolize the biconditional of p and q by p ↔ q. It is true if both p and q have the same truth values and is false if p and q have opposite truth values.
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
According to the definitions of if and only if, saying "p if, and only if, q" should mean the same as saying both "p if q" and "p only if q." That is,
p ↔ q ≡ (p → q) ∧ (q → p)
The following truth table shows that this is, in fact, the case.
| p | q | p → q | q → p | p ↔ q | (p → q) ∧ (q → p) |
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | T | F | F | F |
| F | F | T | T | T | T |
Since column 5 and column 6 have the same truth values and so p ↔ q ≡ (p → q) ∧ (q → p).
In other words, the right side can be read "if p then q, and if q then p," and, by definition, so can the left side.
We can take this one step further, since
p ↔ q ≡ (p → q) ∧ (q → p)
Earlier it was noted that p → q ≡ ~p ∨ q. Therefore, we have
p ↔ q ≡ (~p ∨ q) ∧ (~q ∨ p)
which tells us that any proposition form containing → or ↔ is logically equivalent to one containing only ~, ∧, and ∨.
Necessary and Sufficient Conditions
In logic and in mathematics, it is often important to determine whether the conditions in the hypothesis of a proposition are necessary or sufficient to justify its conclusion. This is done by ascertaining the truth or falsity of the statement and its converse, and then applying the following principles.
If r and s are propositions:
r is a sufficient condition for s means "if r
then s." i.e., r → s.
r is a necessary condition for
means "if not r then not s." i.e., ~r → ~s ≡ s→ r.
In other words, r is a necessary condition for s also means "if s then r." Consequently is a necessary and sufficient condition for s means "r if, and only if, s." That is, r ↔ s. Putting it differently, to say "r is a sufficient condition for s" means that the occurrence of r is sufficient to guarantee the occurrence of s. On the other hand, to say "r is a necessary condition for s" means that if r does not occur, then s cannot occur either.