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Posted 30 August 2009

CONDITIONAL ASSERTIONS

The truth table for 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. 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 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.

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

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  4 is odd, then 3 = 3.

¨  If  4 is odd, then .

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 integerm 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  2 dividen then  4 divides  n. Then clearly  4, 8, 12 are all fourtunate. But so are  3 and  5. They are vacuously fourtunate!  On the other hand,  2 and 6 are not fourtunate.    The definition of “antisymmetric” (above) is another example of this.

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 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

For all

¨  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 3 only if x > 2.”  “Only if” suffers from a severe case of semantic contamination.

¨  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 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 0's after a certain point, then  r is rational.  For example, 3.42000… is the rational number 342/100.  The converse of this statement is that if a real number  r is rational, then its decimal expansion is all 0's after a certain point. This is false, as the decimal expansion of r = 1/3 shows.

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 0's after a certain place, then r is rational.”

The contrapositive of this statement is:

“If  r is not rational, then its decimal expansion does not have all 0's after any place.”  (See order of quantifiers).

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 change.

¨  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 1, 3 or 4 must hold.

The only possibility, then, is line 1, which says that  Q is true.

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

necessary and sufficient in Wikipedia.

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 3, a 4, a red dot and a blue dot are showing. You are told that, if any of the cards has an even integer on one side, it has a red dot on the other. What is the smallest number of cards you must turn over to verify this claim? Which ones should be turned over? Explain your answer.

Answer

You have to turn over the one marked 4 and the one marked with the blue dot.  You don’t have to turn over the other two.  I recommend you puzzle over this, taking into account the purple prose on this page, until you understand what is going on.

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 assertionx < 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