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. In such sentences, P is called the hypothesis and Q is the conclusion.
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 value 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:
“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.
For more about this, see how to understand conditionals.
“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.
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).
¨ 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.
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.
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
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).
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.
¨ 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.
¨ 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.
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
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.
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 the function f is differentiable, then it is continuous.
¨ If a function f is differentiable, then it is continuous.
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 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.
The converse of a conditional assertion “If P then Q” is “If Q then P”.
If it’s a cow, it eats grass, but if it eats grass it might not be a cow.
The converse of
If x > 3, then x > 2
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.
If the decimal
expansion of a real number r is
The contrapositive of
If x > 3, then x > 2
is (after a little translation)
Let's look again at the (true) assertion:
“If the decimal
expansion of a real number r is
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.
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 , or that “If P then Q” means exactly the same as “P only if Q”.
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.
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!
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.
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.
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
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.
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.
Which of these statements are true for all integers m?
a) If m + 5 = 7, then m = 2.
b) If , then m = 2.
(a) is true for all m.
(b) is false, because the hypothesis is true and the conclusion is false for .
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
You have to turn over the one marked
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.
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
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