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



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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