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Necessary and
Sufficient Conditions
Sandra LaFave
A is necessary
for B means that B can’t occur without A. Whenever you have B, you have A.1
A is sufficient
for B means A guarantees B. Whenever you have A, you have B.2
Examples:
| A
|
B |
Condition |
| Being female
|
Being pregnant
|
A is necessary
for B, since being pregnant (B) can't occur without being female (A). A is not sufficient for B, however, because being female doesn't guarantee being pregnant. |
| Getting an A
|
Passing the class
|
A is sufficient for
B, since getting an A in a class guarantees that you pass. But getting an A is not necessary in order to pass; you could still pass with a different passing grade (B or C). |
| Buying a lottery ticket |
Winning the lottery
|
A is necessary
for B, since you can't win without buying a ticket. But, alas, buying a ticket doesn't guarantee that you win, so A is not sufficient for B. |
| Decapitation |
Death
|
A is sufficient for
B, since decapitation guarantees death. But decapitation is not necessary for death, since you can die in many ways other than decapitation. |
Think how this distinction expands your ability to express complicated stuff in a snappy and elegant way. Buying a lottery ticket is necessary but not sufficient for winning the lottery, right? Suppose I say "Plato believes virtue is sufficient for happiness, whereas Aristotle believes virtue is necessary but not sufficient for happiness." I've just said something very important about Plato and Aristotle's ethical beliefs.
Now some of you might be thinking this is confusing because if you reverse the terms A and B, you get what might appear to be an opposite result. For example, if A is necessary for B (being female is necessary for being pregnant, for example), then B looks to be sufficient for A (i.e., being pregnant guarantees that you are female). Good job! You are correct, but the whole thing is completely consistent (i.e., not contradictory at all). The more you think about it, the clearer (and cooler) it gets. If you want more on this reversal, see here.
To build a “formal” or
“essential” definition of X (the kind Socrates and Plato are looking for), you
need to specify what properties all and
only X’s have. Then you’ll know the
Form of X, or X-ness itself. Each property in your list of properties must be
necessary, and the properties together must be sufficient. For example, here is how we get a formal
definition of “square”:
| A |
B |
Condition |
| Being equilateral
(E) |
Being a square
|
A is necessary
for B |
| Being a rectangle
(R) |
Being a square
|
A is necessary
for B |
| Being both E and
R |
Being a square
|
A is necessary
and sufficient for B |
All squares are
equilateral, all squares are rectangular, and only squares are equilateral and
rectangular.
Plato’s basic idea is
that everything in the world fits into some class, and the classes can be
precisely defined. In other words, Plato presupposes that all concepts are what
Wittgenstein calls closed: i.e., Plato believes it is possible to
specify the necessary and sufficient conditions for membership in the class of
things denoted by any concept. This basic idea of Plato’s had a profound effect
on Western thought; it led people to think we could, just by thinking, say
exactly what goodness or beauty or philosophy or persons are. Until quite
recently, the idea defined philosophy’s task: figure out those conditions.
Ludwig Wittgenstein revolutionized Western philosophy by arguing that this
underlying idea is wrong. He points out that most concepts are open: i.e., that
we define concepts in terms of paradigms, and argue that a particular thing
either is or is not a member of the class in question on the basis of
resemblance or lack of resemblance of the thing in question to the paradigm.
1. In logic terms, A is necessary for B if and only if (iff or "↔") B can't occur without A. A is necessary for B iff whenever you have B, you
have A, or, in logic symbols, (x)(Bx → Ax).
2. Whenever you have A, you have B. In logic symbols, (x)(Ax → Bx). Anything A is B.
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