More on necessary/sufficient
conditions
1
From Lecture 2: Other equivalents
to “if p then q.”
✤
1. If p then q
✤
2. q, if p.
✤
3. p only if q
✤
4. q is a necessary condition for p.
✤
5. p is a sufficient condition for q.
2
✤
Note that the order in which p, q occur
in the formulations is important.
✤
These locutions are deeply important
for understanding much academic and
technical writing that you will
encounter in future years.
✤
E.g. when an economist says that suchand-such condition (p) is a sufficient
condition for inflation (q), it’s crucially
important that you understand that has
the same truth-conditions as “if p then
q” and not the same truth conditions as
“if q then p.”
Examples/Non-Examples of
Necessary and Sufficient Conditions
UCI students
✤
Examples:
✤
Being a history major at UCI is a sufficient
condition for being a student at UCI.
✤
Being a history major at UCI is not a
necessary condition for being a student at
UCI (because, for instance, you might be a
math major).
✤
A necessary condition for being in math
honors society is being a math major.
✤
Being a member of chess club is not a
sufficient condition for being a math
major (because some history majors are
members of chess club).
Math
Honors
Society
History
Majors
Math Majors
Chess Club
Members
3
How to visualize
✤
1. If p then q
✤
To visualize this:
✤
2. q, if p.
✤
You can draw pictures of the p’s and the
q’s like circles or squares on a page.
✤
3. p only if q
✤
Then “if p then q” being true corresponds
to the p shape fitting inside the q shape.
✤
Hence, “p is a sufficient condition for q” is
true iff the p shape fits inside the q shape.
✤
And “p is a necessary condition for q” is
true iff the p shape contains the q shape.
✤
4. q is a necessary condition for p.
✤
5. p is a sufficient condition for q.
4
Examples from the Slides re.
Tautologies
5
First Example
of a Tautology
✤
6
1
2
3
4
5
p
q
¬p
(p∧¬p)
¬(p ∧ ¬p)
T
T
T
F
F
T
F
F
2nd Example of
a Tautology
✤
7
1
2
3
4
5
p
q
(p∧q)
p
((p∧q)➝p)
T
T
T
F
F
T
F
F
3rd Example of
a Tautology
✤
8
1
2
3
4
p
q
¬p
p ∨ ¬p
T
T
T
F
F
T
F
F