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CONDITIONAL
ASSERTIONS
The
truth table for conditional assertions
How
conditional assertions are worded
Assertions
related to a conditional assertion
How
to understand conditionals
Fallacies connected with conditional assertions
The truth table for conditional assertions
This section is concerned with logical constructions made with the connective called
the conditional operator. In mathematical English, applying the
conditional operator to P and Q produces a sentence that may be
written, “If P, then Q”, or “P implies
Q”.
(Fine
point.) Sentences of this
form are conditional
assertions.
Conditional assertions are at the very heart of mathematical reasoning. Mathematical proofs typically consist of chains of conditional assertions.
A conditional assertion “If P then Q” has the precise truth table shown here. The meaning of “If P then Q” is determined entirely by the truth values of P and Q and this truth table. The meaning is not determined by the usual English meanings of the words “if” and “then”.
The truth table can be summed up by saying:
A conditional is true unless
the hypothesis is true and the conclusion is false.
That
means that to prove “If P then Q” is
FALSE
you must show that P is TRUE(!) and Q is FALSE.
The purple statement just above is harder to believe in than
leprechauns. Some who are new to abstract math get into an
enormous amount of difficulty because they don’t take it seriously.
Example
“If n > 5, then n > 3” is true for all integers n.
¨ This means that “If 7 > 5 then 7 > 3” is true.
¨ It also means that “If 2 > 5 then 2 > 3” is true! If you really believe that “If n > 5, then n > 3” is true for all integers n, then you must in particular believe that “If 2 > 5 then 2 > 3” is true. That’s why the truth table for conditional assertions takes the form it does.
On the other hand, “If n > 5, then n > 8” is not true for all integers n. In particular, “If 7 > 5, then 7 > 8” is false.
This fits what
the truth table says, too.
For more about this,
see how to understand conditionals.
Usage
Conditionals such as “If P then Q” are also called implications, but be wary: that is a technical term and does not fit the meaning of “implication” in conversational English.
In symbolic logic, the
assertion can be written
¨
¨
¨
P
is the hypothesis
or antecedent of
the assertion and Q is the conclusion or consequent. (See fine
point 1.)
Worked Exercise
Which of these statements are true for all integers m?
a) If m + 5 = 7, then m = 2.
b) If ,
then m = 2.
Answer
(a) is true for all m.
(b) is false,
because the hypothesis is true and the conclusion is false for .
Worked Exercise
You have been given four cards each with an integer
on one side and a colored dot on the other. The cards are laid out on a table
in such a way that a
Answer
You have to turn over the one marked
Vacuous truth
The last two lines of the truth table for conditional assertions mean that if the hypothesis of the assertion is false, then the assertion is automatically true.
In the case that “If P then Q” is true because P is false, the assertion is said to be vacuously true.
The word “vacuous” refers to the fact that in that case the conditional assertion says nothing interesting about either the hypothesis or the conclusion. In particular, the conditional assertion may be true even if the conclusion is false (because of the last line of the truth table).
Example
Both these statements are vacuously true!
¨
If
¨
If .
Examples
¨
If A
is any set then . Proof.
¨ Let x and y be real numbers. Then if x < y and y < x, then x = y. This says that the relation “<” is antisymmetric.
Remarks
Although vacuous truth may be disturbing when you first see it, making either statement in the example false would result in even more peculiar situations. For example, if decided that “If P then Q” must be false when P and Q are both false, you would then have to say that this statement
“For any integers m and n, if m > 5 and 5 > n, then m > n,”
is not always true (substitute 3 for m and 4 for n and you get both P and Q false). This would surely be an unsatisfactory state of affairs.
Most of the time in mathematical writing the conditional assertions which are actually stated involve assertions containing variables, and the claim is typically that the assertion is true for all instances of the variables. Assertions involving statements without variables occur only implicitly in the process of checking instances of the assertions. That is why a statement such as, “If 3 > 5 and 5 > 4, then 3 > 4” seems awkward and unfamiliar.
Definitions involving vacuous truth
Vacuous truth can cause surprises in connection with certain
concepts which are defined using a conditional assertion. Let's
look at a made-up example here: to say that a natural number n is fourtunate (the spelling is intentional) means that
if
If you see a conditional statement that seems wrong
check whether it is vacuously true.
How conditional assertions are worded
A conditional assertion may be worded in various ways. It takes some practice to get used to understanding all of them as conditional. The five most common ways of wording a conditional assertion with hypothesis P and conclusion Q are:
¨ If P, then Q.
¨ P only if Q.
¨ P implies Q.
¨ P is a sufficient condition for Q.
¨ Q is a necessary condition for P.
Example
¨ If x > 3, then x > 2.
¨ x > 3 only if x > 2.
¨ x > 3 implies x > 2.
¨ That x > 3 is sufficient for x > 2.
¨ That x > 2 is necessary for x > 3.
All these statements mean the same thing.
Remarks
¨
Watch out particularly for only
if: it is easy to read the statement “P only if Q” backward when
it occurs in the middle of a mathematical argument. “x > 3 only if x > 2” means exactly the same thing as
“If x > 3, then x > 2.” It may help to think of the wording as
reading: “ x can be greater than
¨ See also let and fine point 2 .
Universally true conditionals
An assertion that a mathematical object of one kind
A is necessarily also of kind B is a disguised universally true
conditional. So is an assertion that an
object with property P must also have property Q. Such assertions use words
such as every, all and each.
Expressing universally true conditionals in math English
The sentences listed in the example above provide ways of expressing universally true conditionals in English. You may also use most of the forms listed in the section on general universally true assertions:
¨ For every function f, if f is differentiable then it is continuous.
¨ For any function f, if f is differentiable then it is continuous.
¨ For all functions f, if f is differentiable then it is continuous.
¨ For each function f, if f is differentiable then it is continuous.
In any of these sentences, the “for all”
phrase may come after the main clause.
Definite and indefinite descriptions can also be used:
¨ If the function f is differentiable, then it is continuous.
¨ If a function f is differentiable, then it is continuous.
Disguised conditionals
There are other ways of expressing universal conditionals
that are disguised, because they
are not conditional assertions in English. Let C(f) mean
that f is continuous and and D(f) mean that f is differentiable. The (true) assertion “ ” can be said in the following ways:
¨ Every differentiable function is continuous.
¨ Any differentiable function is continuous.
¨ All differentiable functions are continuous.
¨ Each differentiable function is continuous.
¨ Differentiable functions are continuous. Or: differentiable functions are always continuous.
¨
A differentiable
function is continuous.
¨
The differentiable functions are continuous. I believe this usage is obsolescent. I don’t
think younger native-English-speaking Americans would use it. (Warning: This claim is not based on
lexicographical research.)
Watch out for the purple forms. Beginning abstract math students sometimes
don’t recognize them as universal.
Assertions related to a conditional assertion
Converse
The converse of a conditional assertion “If P then Q” is “If Q then P”.
A conditional
assertion may be true and its converse false, or vice versa.
Example
If it’s a cow, it eats grass, but if it eats grass it might not be a cow.
Example
The converse of
If x > 3, then x > 2
is
If x > 2, then x > 3
The first is true for all real numbers x, whereas there are real numbers for which the second one is false.
Example
If the decimal
expansion of a real number r is
all
Contrapositive
The contrapositive of a conditional assertion “If P then Q” is “If not Q then not P.”
A conditional assertion and its contrapositive are both true or both false.
Example
The contrapositive of
If x > 3, then x > 2
is (after a little translation)
If then
These two statements are equivalent.
Example
Let's look again at the (true) assertion:
“If the decimal
expansion of a real number r is
all
The contrapositive of this statement is:
“If r is not
rational, then its decimal expansion does not have all
In other words, no matter how far out you go in the decimal expansion of a real number
that is not rational, you can
find a nonzero entry further out. This statement is true because it is the contrapositive of a
true statement.
How to understand conditionals
As you can see from the preceding discussions, statements of the form “If P then Q” don’t quite mean the same thing in math as they do in ordinary English.
¨ In ordinary English, “If P then Q” can suggest order of occurrence. For example, “If we go outside, the neighbors will see us” implies that the neighbors will see us after we go outside.
¨ “If
P then Q” can also suggest causation. The preceding example has the connotation that the
neighbors will see us because we went outside.
Because of the semantic contamination caused by these connotations, it may be hard to believe that in math English a conditional says exactly the same thing as its contrapositive, or that “If P then Q” means exactly the same as “P only if Q”.
Example
All three of these statements mean identically the same thing in math texts:
¨ If n > 7, then n > 4.
¨ n > 7 only if n > 4.
¨ If n is not greater than 4, then n is not greater
than 7 (or: If then
.) (Contrapositive)
You need to understand
this so well that it is part of your
unconscious reaction to conditionals. How can you
gain that intuitive understanding? One
way is by doing abstract math regularly for several years! (This is how you gain expertise in anything,
of course.) But it may help to remember that when doing proofs, we must take the rigorous view of mathematical
objects:
¨ Math objects don’t cause
anything to happen.
The integers
(like all math objects) just sit there, not doing anything and not affecting anything. 10 is
not greater than 4 because
it is greater than 7. Both facts, and
are eternally true. (Eternal is
how we think of them