Material covered in this chapter

3.1 The meaning of necessary and sufficient conditions

3.2 Conditionals and necessary and sufficient conditions

3.3 The relationship between conditionals and disguised conditionals

3.4 Contrapositive

3.1 The Meaning of Necessary and Sufficient Conditions
‘P is necessary for Q’ is equivalent to ‘P is required for Q’. In other words, for our purposes, necessary = required. ‘P is sufficient for Q’ is equivalent to ‘P is enough for Q’. In other words, for our purposes, sufficient = enough.
Example 3.1: An example of a necessary and a sufficient condition.

Being a bachelor is sufficient for being a male. Being male is necessary for being a bachelor.

Being a bachelor is sufficient for being a male, since being a bachelor is enough to be a male. Being a male is necessary for being a bachelor, since being a male is required for being a bachelor.
3.2 Conditionals and Necessary and Sufficient Conditions
Conditionals are sentences of this form: If p, then q. In the sentence, ‘If p then q’, ‘p’ is the antecedent and ‘q’ is the consequent. P is sufficient for Q. Q is necessary for P.
Example 3.2: A true conditional.

If it is rainy, then it is cloudy.

The antecedent, it is rainy, is sufficient for the consequent, it is cloudy, since it is enough for it to be cloudy that it is rainy. The consequent, it is cloudy, is necessary for the antecedent, it is rainy, because a requirement for rain is cloudiness. The conditional does not tell us the relationship between not P (it not being rainy) and Q (it being cloudy). Consider that it is not necessary for it to be not rainy for it to be cloudy (because there are cloudy rainy days), and it is not sufficient because sunny days are nonrainy days. Likewise, not being cloudy is neither necessary nor sufficient for it being rainy.
Example 3.3: A false conditional.

If it is cloudy, then it is rainy.

This conditional says that being cloudy is sufficient for it to be rainy. Clearly this is false, since it can be cloudy without it being rainy. Likewise, it says that being rainy is necessary for it being cloudy—again this is false. We can prove this by constructing a counterexample. A counterexample is a cloudy day with no rain. This counterexample shows that being cloudy is not enough for it to be rainy, which demonstrates that the conditional is false.
3.3 Disguised Conditionals
There are a number of sentences that do not appear to be conditional sentence, but which in fact have the same logical structure as conditional sentences. The table below lists a number of these disguised conditionals.
Disguised Conditional

Example

P =

Q =

Relation

Rewritten

Q unless P

It is not rainy unless it is cloudy.

cloudy

not rainy

Q is necessary for not P

Not P is sufficient for Q

If not P, then Q

Q, if P

It is cloudy, if it is rainy.

rainy 
cloudy 
Q is necessary for P

P is sufficient for Q

If P, then Q

Q provided that P

It is cloudy provided that it is rainy.

rainy

cloudy

Q is necessary for P

P is sufficient for Q

If P, then Q

P only if Q

It is rainy only if it is cloudy.

rainy

cloudy

Q is necessary for P

P is sufficient for Q

If P, then Q

When P then Q

When it is rainy, it is cloudy.

rainy

cloudy

Q is necessary for P

P is sufficient for Q

If P, then Q

All Ps are Qs

All rainy days are cloudy days.

rainy days

cloudy days

Q is necessary for P

P is sufficient for Q

If P, then Q

4. Contrapositive Being a bachelor is sufficient for being a male, since being a bachelor is enough to be a male. Being a male is necessary for being a bachelor, since it is required that to be a bachelor one is male.
3.4 Contrapositive
The contrapositive of ‘If P then Q = ‘If not Q then not P’. The two sentences are logically equivalent. This tells us that notQ is sufficient for notP, and notP is necessary for notQ.
Example 3.4: Contrapositive

The contrapositive of “If it is rainy, then it is cloudy” is “If it is not cloudy, then it is not rainy”.

Notice that in constructing a contrapositive you must perform two operations: (1) switch the consequent and the antecedent, and (2) put negations in front of the antecedent and the consequent. Students often forget to do one or the other.
