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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
Note: Some
of the narrative formats used for proving conditional assertions are discussed
in Forms of Proof.
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.
Conditionals and Truth Sets
“If P(x) then Q(x)” means that . We say P(x) is stronger than Q(x), meaning that it
puts more
requirements on Q(x).
This means that the objects that make P true make Q true, so
there might be more
objects making Q true than
making P true.
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.
See also fine
point 1.
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 you 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
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A conditional assertion may be worded in various ways. It takes some practice to get used to understanding all of them as conditional. The most common ways of wording a conditional assertion with hypothesis P and conclusion Q are:
¨ If P, then Q.
¨ Q, if P.
¨ P only if Q.
¨ P implies Q.
¨ P is a sufficient condition for Q.
¨ Q is a necessary condition for P.
In mathematical
logic, the assertion may be written in any of these ways:
¨
¨
(see fine point 1).
Usage
In the assertion “If P, then Q”:
¨ P is the hypothesis or antecedent of the assertion. It is a constraint or condition (!) that holds in the very narrow context of the assertion. In other words, when “If P, then Q” is asserted, you don’t know that P is true. Exception: The idea of the direct method of proof is to assume that P is true during the proof.
¨ Q is the conclusion or consequent. . It is also incorrect to assume that Q is true anywhere else except in the assertion.
¨
The logical symbols are frequently used when writing on the
blackboard, but are not common in texts, except for texts in mathematical
logic.
¨
means that
(more here), so there is a
notational clash between implication written “
” and inclusion written “
” . This is a kind of reversal (see Galois correspondence) that confused many of my
students.
Example
¨ If x > 3, then x > 2.
¨ x > 2 if x > 3 (the hypothesis is a postcondition).
¨ 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.
¨
¨
¨
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 , fine point 1 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 as a postcondition.
¨ If f is differentiable then it is
continuous, for any function f. [or for
every function f, for all functions f, for each function f.]
¨ If f is differentiable then it is continuous, where f is any function. (See also here.)
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.) (Note)
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.
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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
I am not making
a claim about “reality”).
¨ When you look at the integers, every time you find one
greater than 7 it turns out to be greater than 4. That is how to think about “If n > 7, then n > 4”.
¨ You can’t find one greater than 7 that is not greater
than 4 (it happens that n > 7 only if n > 4).
¨ Every time you look at one less than or equal to 4 it
turns out to be less than or equal to 7 (contrapositive).
These three
observations are about the same set of facts about a bunch of things (integers) that just sit there
in their various relationships without changing, moving or doing anything. If you keep these remarks in mind, you will
eventually have a natural, unforced understanding of conditionals in math.
Remark
None of this
means you have to think of mathematical objects as dead and fossilized all the time. Feel free to think of them using all the metaphors
and imagery you know, except
when you are reading or formulating a proof written in mathematical English. Then you have to be rigorous!
Method of deduction: Modus ponens
The truth
table for conditional assertions may be summed up by saying: The conditional
assertion “If P, then Q” is true unless P is true and Q is false.
This fits with the major use of conditional assertions in reasoning:
Method of deduction
¨ If you know that a conditional assertion is true and
¨ you know that its hypothesis is true,
¨ then you know its conclusion is true.
In symbols:
(1) When “If P then Q” and P are both true,
(2) then Q must be true as well.
This notation means that if the statements in
(1)
are true, the statement in (2) has to be
true too.
This fact is called modus
ponens and is the
most used method of deduction of all.
Remark
That modus ponens is valid is a consequence of the truth table:
¨ If P is true that means that one of the first two lines of the truth table holds.
¨
If the assertion “If P then Q” is true, then
one of lines
The only possibility, then, is line
Remark
Modus ponens is not a method of proving conditional assertions. It is a method of using a conditional assertion in the proof of another assertion. Methods for proving conditional assertions are found in the chapter on forms of proof.
Uses of modus ponens
A theorem (call it Theorem T) in a mathematical text generally takes the form of an conditional assertion: "If [hypothesis] is true, then [conclusion]." The theorem will then typically be applied in the proof of some subsequent theorem using modus ponens. In the application, the author will verify that the hypothesis of Theorem T are true, and then will be able to assert that the conclusion is true.
Equivalence
See necessary and sufficient in
Wikipedia.
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Exercises
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
Appendix
Fine Point 1
If you know some logic, you may know that there is a subtle difference between the statements
¨ “If P then Q”
¨ “P implies Q”.
Here is a concrete example:
a) “If x < 2, then x is positive.”
b) “x < 2 implies x is positive”.
Note that the subject of sentence (a) is the (variable) number x, but the subject of sentence (b) is the assertion “x < 2 “. Behind this is a distinction made in formal logic between the material conditional “if…then” (roughly: true in all examples) and logical consequence (Q can be proved given P). I will ignore the distinction here, as most mathematicians do except when they are proving things about logic.
In some texts, denotes the material conditional and
denotes logical consequence. Return.
Fine Point 2
The variables in a hypothesis are constrained by the hypothesis. For example, in the statement “If x > 3, then x > 2”, the x is constrained to be bigger than 2. The scope is the entire conditional statement, so that the statement “x > 2” is under the constraint that x > 3. This fits with the truth table. Note that in “x > 3 only if x > 2”, the constraint applies to “x > 3” even though the constraint is applied afterward. In any case, if the whole conditional statement is true, the truth table means the constraint doesn’t add any knowledge about the conclusion. That’s why this is a fine point instead of a purple prose! Return
Note
A statement such as “The decimal expansion of an irrational number does not terminate” is not an example of this form. The expression getting the universal quantification is “irrational number” and so it is an example of the same type as “A differentiable function is continuous.” Return