Copyright © Norman Swartz, 1997
Department of Philosophy
Simon Fraser University

The Concepts of
Necessary Conditions and Sufficient Conditions

Contents
 Introduction
 Definition of "necessary condition"
 Definition of "sufficient condition"
 Necessary conditions that are not jointly sufficient
 Sufficient conditions that are not necessary
 The concept of converse relations
 "Is a necessary condition for" and "is a sufficient condition for" are converse relations
 Four possible combinations
 Different kinds (or modes) of necessary condition

1. Introduction
Everyone is familiar with the concept of a necessary condition. For
example, we all know that air is necessary for (human) life. Without
air, there is no (human) life. Similarly, a microscope (or some other
instrument) is necessary for human beings to see viruses. (Viruses
are too small to be seen by the naked eye.)
Similarly, everyone is familiar with the concept of a sufficient
condition. For example, it suffices (i.e. it is sufficient for) an
object's having four sides that it is a square. Or, again, it is
sufficient for your having something to drink that you have a glass of
CocaCola®.

2. Definition of "necessary condition"
Definition: A condition A is said to be
necessary for a condition B, if (and only if)
the falsity (/nonexistence /nonoccurrence) [as the case may be]
of A guarantees (or brings about) the falsity (/nonexistence
/nonoccurrence) of B.


So common is this notion of necessary
condition that there are, not surprisingly, a great many ways to express
that something is a necessary condition. Here are a number of examples, all
 more or less  saying the same thing:
 "Air is necessary for human life."
 "Human beings must have air to live."
 "Without air, human beings die (i.e. do not live)."
 "If a human being is alive, then that human being has air (to breathe)."

In an `ifthen' statement (such as the last example immediately above), the
clause that follows the "then" (i.e. the socalled `consequent') states the
necessary condition for the antecedent (i.e. the clause immediately following
the "if"). Thus that some human being has air (to breathe) is a
necessary condition for that human being's being alive.
Let's look at some further examples (again, all saying pretty much the same
thing). Notice the use of "ifthen" (in the sixth [highlighted] example):

"Having a microscope (or some other instrument) is a necessary condition
for (our) seeing viruses."

"A microscope (or a similar instrument) is needed to see viruses."

"Human beings must use (have) a microscope to see viruses."

"Human beings cannot see viruses without a microscope."

"Anyone who sees viruses has (uses) a microscope."

"If someone sees viruses, then that person uses a
microscope." NOTE

"Without a microscope, a person cannot see viruses."

"If a person does not have (the use of) a microscope, then that person does
not see viruses."

"Whoever lacks a microscope does not see viruses."

"One must have a microscope to see viruses."
But of course, as we well know, in general a necessary condition is not a
sufficient condition. All sorts of conditions may be necessary
for others, but do not  by themselves  suffice for, or
guarantee, those others.

3. Definition of "sufficient condition"
Definition: A condition A is said to be sufficient
for a condition B, if (and only if) the
truth (/existence /occurrence) [as the case may be] of A
guarantees (or brings about) the truth (/existence /occurrence) of B.

For example, while air is a necessary condition for human life, it is by
no means a sufficient condition, i.e. it does not, by itself, i.e. alone,
suffice for human life. While someone may have air to breathe, that person
will still die if s/he lacks water (for a number of days), has taken poison,
is exposed to extremes of cold or heat, etc. There are, in fact, a very great
many conditions that are necessary for human life, and no one  or even just
a few of them  will suffice for [or guarantee] human life.
Or, further, consider the property of having four sides. While having
four sides is a necessary condition for something's being a square,
that single condition is not, by itself, sufficient (to guarantee)
something's being a square, i.e. some foursided things (e.g. trapezoids)
are not squares. There are several necessary conditions for something's
being a square, and all of these must be satisfied for something's being
a square:
 x has (exactly) four sides
 each of x's sides is straight
 x is a closed figure
 x lies in a plane
 each of x's sides is equal in length to each of the others
 each of x's interior angles is equal to the others (they are
each right [i.e. 90^{o}] angles)
 the sides of x are joined at their ends

The foregoing is a complete set of necessary conditions, i.e.
the set comprises a set of sufficient condition for x's being square.
Frequently the terminology of "individually necessary" and "jointly sufficient"
is used. One might say, for example, "each of the members of the foregoing
set is individually necessary and, taken all together, they are
jointly sufficient for x's being a square."
Caution: In this example, we have been able, with ease, to
list a set of individually necessary conditions that is also
sufficient for something's being a square. However, we must not
generalize from this simple example and believe that it is
usually, or often, a straightforward task to specify sets of
conditions that are individually necessary and jointly
sufficient. Sometimes it is much easier to specify (some, or
many, of the) necessary conditions even though we are unable to
specify a set that is jointly sufficient. Other times, the
converse is true: for some cases it will be easier to specify
sufficient conditions without our being able to specify
individually necessary ones.

4. Necessary conditions that are not sufficient
Example 4.1  A set of conditions that are individually necessary
without being jointly sufficient.
Thomas White, the author of a recent textbook in philosophy,
attempted to use as
his example the specifying of the necessary and sufficient
conditions for hearing music from a Walkman®. Here is the
list of necessary conditions that White offered (irrelevant
conditions have been here removed, and the list has been
renumbered):
 The Walkman is in good working order.
 The batteries are good. [ Note 1 ]
 The earphones are plugged in.
 The tape has music on it and is in good working condition.
 You operate the controls correctly.
 Discovering Philosophy, by Thomas I. White, p. 25
(Englewood Cliffs, NJ: Prentice Hall), 1991.
White then goes on to make a toohasty claim: "... taken all
together they are sufficient " (ibid., p. 25).
Unfortunately, for his illustrative purposes, the list is by no
means sufficient. Here are just a few of the many additional
necessary conditions that my own students, in previous years,
have offered:
 The listener must not be deaf.
 The ambient (surrounding) sound must not drown out the
earphones.
 The listener must be wearing the earphones, or must be close
enough to them, to hear the music.
 There must be nothing blocking the sound in the listener's ears.
 The tape must be inserted correctly; the door of the Walkman
must be closed; and the tape must not be at the end of the reel
(more specifically, it must be positioned so that some of the
parts of the tape which contain recorded music will pass over the
playback heads).
 The earphones are in good working order.
 The listener does not die in the time between operating the controls
correctly and the music's emerging from the earphones.
Even this expanded list is not complete. I imagine that will
little effort, you yourself can come up with still further
necessary conditions. Indeed, there does not seem to be any
practical limit to the number of necessary conditions.

Conclusion: Sometimes (as in the case of hearing music from
a Walkman), it is (far) easier to specify necessary
conditions than sufficient ones.


5. Sufficient conditions that are not necessary
Example 5.1  A set of conditions that are (jointly) sufficient
without being individually necessary.
A sufficient condition for travelling from Calgary to
Vancouver would be your taking an uneventful trip as a passenger
on a regularly scheduled air flight. But while that method of
getting from the one city to the other would suffice, it is by no
means necessary. There are all sorts of other conditions
that would also suffice for your getting from Calgary to
Vancouver.
 You could take VIA rail; or
 You could travel by car; or
 You could hike; or
 You could ride a bicycle; or
 You could travel on horseback; or
 ..., ...

Example 5.2  A(nother) set of conditions that are (jointly) sufficient
without being individually necessary.
If you'll forgive the morbid example, think of all the ways
a person might die: having his/her head chopped off; being at
`GroundZero' when a nuclear bomb is detonated; tearing a gaping
hole in one's space suit while on a `spacewalk' on the Moon;
etc., etc. But none of these conditions is a necessary
condition for a person's dying. Indeed almost everyone dies
without having satisfied one of these unusual sufficient
conditions.

Conclusion: Sometimes, it is easier to specify sufficient
conditions than necessary ones.


6. The concept of converse relations
Here are some examples of twoplace relations:
 ... is larger than ..
 ... is taller than ...
 ... is above ...
 ... is a parent of ...
 ... is a child of ...
 ... loves ...
 ... employs ...
 ... detests ...
Examples of threeplace relations:
 ...is between ... and ...
 ...travels to ... by ...
 ...sends ... to ...
Certain twoplace relations are converses of one another. For
example, each member of the following pairs is a converse of the other
member:
 ... is a parent of ...
 ... is a child of ...
 ... is taller than ...
 ... is shorter than ...
 ... is above ...
 ... is below ...
 ... loves ...
 ... is loved by ...
Definition: Two (twoplace) relations, R_{1}
and R_{2}, are converses of one another,
if and only if, (1) xR_{1}y (e.g. Sandra is taller
than Louise) guarantees yR_{2}x (e.g. Louise
is shorter than Sandra), and (2) yR_{2}x
guarantees xR_{1}y.


These twoplace relations are not converses
of one another:
 ... is a daughter of ...
 ... is a parent of ...
The `trouble' with this case is that although "x is a daughter of y"
does indeed guarantee "y is a parent of x", the second condition
stipulated in the definition for "converse relation" does not hold.
For it is false that "y is a parent of x" guarantees "x is a
daughter of y".
Or, consider this case:
 ... is not taller than ...
 ... is taller than ...
Again, these two relations are not converses of one another.
In this instance the first condition of the definition of "converse
relation" does not hold. From "x is not taller than y" it does not
follow that "y is shorter than x". Perhaps x and y are exactly the
same height. If so, "x is not taller than y" will be true, but "y
is taller than x" will be false.

7. "Is a necessary condition for" and "is a sufficient condition for"
are converse relations

If x is a necessary condition for y, then y is a
sufficient condition for x.
And, equivalently,
If y is a sufficient condition for x, then x is a
necessary condition for y.


Let's look at some examples that illustrate the claim in the box above:

"Since having a microscope is necessary for seeing viruses, then seeing viruses
guarantees that one has a microscope (i.e. suffices for having a microscope)."

Similarly, "Since having air to breathe is necessary for human life, if follows
that the existence of human life (demonstrates, assures, guarantees, i.e.)
suffices for the existence of air."

"Since every square must have four sides (i.e. since having four
sides is a necessary condition for being a square), being a square
is a sufficient condition (but not a necessary one) for something's
having four sides." Put another way: "All squares (must) have four
sides; but not all foursided things [e.g. trapezoids] are squares."

"Being a father is a sufficient condition for being male, and being male
is a necessary condition for being a father." (But being a father is not
a necessary condition for being a male [e.g. James Dean was not a father,
but was male]; and being a male is not a sufficient condition for being a
father [again the case of James Dean].)

8. Four possible combinations

Pick any two conditions whatsoever. The relationship
between the two conditions must be exactly one of the
following four possibilities:
 The first is a necessary, but not a sufficient,
condition for the second; or
 The first is a sufficient, but not a necessary,
condition for the second; or
 The first is both a necessary and a sufficient
condition for the second; or
 The first is neither a necessary nor a
sufficient condition for the second.


Examples 8.1  The first is a necessary, but not a sufficient, condition for the
second.

"Sam's being a male is a necessary, but not a sufficient condition, for being
a father."

"A table's having four sides is a necessary, but not a sufficient, condition
for its being square."

"Being more than 6 feet (183 centimeters) tall is a necessary, but not a
sufficient condition for being 6 feet 3 inches (190.5 centimeters)
tall."

"Owning a 1996 Chevrolet Cavalier is a necessary, but not a
sufficient condition, for owning a red, 6cylinder, 1996
Chevrolet Cavalier."

"Having a ticket in a lottery is a necessary, but not a sufficient condition,
for winning that lottery."
Examples 8.2  The first is a sufficient, but not a necessary, condition for the
second.

"Sam's being a father is a sufficient, but not a necessary condition, for
being a male."

"A table's being square is a sufficient, but not a necessary condition, for
its having four sides."

"Being 6 feet 3 inches tall (190.5 centimeters) tall is a sufficient,
but not a necessary condition for being more than 6 feet (180
centimeters) tall."

"Owning a red, 6cylinder, 1996 Chevrolet Cavalier is a sufficient,
but not a necessary condition, for owning a 1996
Chevrolet Cavalier."

"Winning a lottery is a sufficient, but not a necessary condition, for having
a ticket."
Examples 8.3  The first is both a necessary and a sufficient
condition for the second.

"Sam's being a father is both a necessary and a
sufficient condition for his being a male parent."

"Frankie's being older than Johnny is both a necessary
and a sufficient condition
for Johnny's being younger than Frankie."

"Today's being neither Saturday nor Sunday is both a
necessary and a sufficient
condition for today's being a weekday."

[TRICKY, but true.]
"X'sbeinganecessaryconditionforY is both a necessary
and a sufficient condition for
Y'sbeingasufficientconditionforX."
Examples 8.4  The first is neither a necessary nor a sufficient
condition for the second.

"John's loving Pamela is neither a necessary nor a
sufficient condition for Pamela's loving John."

"Having a married brother is neither a necessary nor a
sufficient condition for being an aunt."

"Being the smartest student in a class is neither a
necessary nor a sufficient
condition for achieving the highest grade in that class."

"Aziz's playing baseball well is neither a necessary
nor a sufficient condition
for his having a driver's license."

"Wanting to succeed is neither a necessary nor a
sufficient condition for success."

Practice Exercise #1
First Set of Practice Exercises
on Necessary Conditions and Sufficient Conditions.

Practice Exercise #2
Second Set of Practice Exercises
on Necessary Conditions and Sufficient Conditions.

9. Different kinds (or modes) of necessary condition
(To be written. ...)


NOTES
 Strictly, although the batteries' being good is usually a
necessary condition for a Walkman's working, it is not absolutely
necessary. For it is possible to operate a Walkman without
batteries, e.g. by using a plugin ACtoDC adapter. What
is necessary, along these lines, is that there be a proper
power source for the Walkman. [ Resume ]

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