Produced by Charles Wells Revised 2017-01-19 Introduction to this website website TOC website index blog Back to top of Mathematical Reasoning chapter
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CONTENTS The truth table for conditionals |
This section is concerned with logical constructions made with the connective called the conditional operator. In mathematical English, applying the conditional operator to assertions $P$ and $Q$ produces an assertion that may be written, “If $P$, then $Q$”, or “$P$ implies $Q$”. 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.
Some of the narrative formats used for proving conditional assertions are discussed in Forms of Proof.
A conditional assertion “If $P$ then $Q$” has the precise truth table shown here.
| $P$ | $Q$ | If $P$ then $Q$ | |
| T | T | T | |
| T | F | F | |
| F | T | T | |
| F | F | T |
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 is summed up by this purple pronouncement:
The Prime Directive of conditional assertions:
A conditional assertion 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 Prime Directive 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.
The statement “if $n\gt 5$, then $n\gt 3$” is true for all integers $n$.
For more about this, see Understanding conditionals.
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 $2\gt 5$ then $2\gt 3$” seems awkward and unfamiliar.
It is unfamiliar and occurs rarely. I mention it here because of the occurrence of vacuous truths, which do occur in mathematical writing.
The set $\{x|P(x)\}$ is the set of exactly all $x$ for which $P(x)$ is true. It is called the truth set of $P(x)$.
“If $P(x)$ then $Q(x)$” means that $\{x|P(x)\}\subseteq \{x|Q(x)\}$. We say $P(x)$ is stronger than $Q(x)$, meaning that $P$ puts more requirements on $x$ than $Q$ does. The objects $x$ that make $P$ true necessarily make $Q$ true, so there might be objects making $Q$ true that don't make $P$ true.
The statement "$x\gt4$" is stronger than the statement "$x\gt\pi$". That means that $\{x|x\gt4\}$ is a proper subset of $\{x|x\gt\pi\}$. In other words, $\{x|x\gt4\}$ is "smaller" than $\{x|x\gt\pi\}$ in the sense of subsets. For example, $3.5\in\{x|x\gt\pi\}$ but $3.5\notin\{x|x\gt4\}$. This is a kind of reversal (a Galois correspondence) that confused many of my students.
"Smaller" means the truth set of the stronger statement omits elements that are in the truth set of the weaker statement. In the case of finite truth sets, "smaller" also means it has fewer elements, but that does not necessarily work for infinite sets, such as in the example above, because the two truth sets $\{x|x\gt4\}$ and $\{x|x\gt\pi\}$ have the same cardinality.
Making a statement stronger
makes its truth set "smaller".
In the assertion “If $P$, then $Q$”:
Conditionals such as “If $P$ then $Q$” are also called implications , but be wary: "implication" is a technical term and does not fit the meaning of the word in conversational English.
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 the vacuous case the conditional assertion says nothing interesting about either $P$ or $Q$. In particular, the conditional assertion may be true even if the conclusion is false (because of the last line of the truth table).
Both these statements are vacuously true!
If $A$ is any set then $\emptyset\subseteq A.$ Proof (rewrite by definition): You have to prove that if $x\in\emptyset$, then $x\in A$. But the statement "$x\in\emptyset$" is false no matter what $x$ is, so the statement "$\emptyset\subseteq A$" is vacuously true.
Vacuous truth can cause surprises in connection with certain concepts which are defined using a conditional assertion.
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 $m\gt 5$ and $5\gt n$, then $m\gt n$”
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.
A conditional assertion may be worded in various ways. It takes some practice to get used to understanding all of them as conditional.
Our habit of swiping English words and phrases and changing their meaning in an unintuitive way causes many problems for new students, but I am sure that the worst problem of that kind is caused by the way conditional assertions are worded.
The most common ways of wording a conditional assertion with hypothesis $P$ and conclusion $Q$ are:
Since "$P(x)\supset Q(x)$" means that $\{x|P(x)\}\subseteq \{x|Q(x)\}$, there is a notational clash between implication written “$\supset $” and inclusion written “$\subseteq $”. This is exacerbated by the two meanings of the inclusion symbol "$\subset$".
Math logic is notorious for the many different symbols used by different authors with the same meaning. This is in part because it developed separately in three different academic areas: Math, Philosophy and Computing Science.
All the statements below mean the same thing. In these statements $n$ is an integer variable.
These ways of wording conditionals cause problems for students, some of them severe. They are discussed in the section Understanding conditionals.
The logical symbols "$\to$", "$\Rightarrow$", "$\supset$" are frequently used when writing on the blackboard, but are not common in texts, except for texts in mathematical logic.
If you know some logic, you may know that there is a subtle difference between the statements
Here is a concrete example:
Note that the subject of sentence (1) is the (variable) number $x$, but the subject of sentence (2) is the assertion "$x\lt2$". Behind this is a distinction made in formal logic between the material conditional “if $P$ then $Q$” (which means that $P$ and $Q$ obey the truth table for "If..then") 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, $P\Rightarrow Q$ denotes the material conditional and $P\to Q$ denotes logical consequence.
A conditional assertion containing a variable that is true for any value of the correct type of that variable is a universally true conditional assertion. It is a special case of the general notion of universally true assertion.
These are all assertions of the form "If $P(x)$, then $Q(x)$". In (1), the hypothesis is the assertion "$x\lt5$"; in (2), it is the assertion "$n^2$ is even", using an adjective to describe property that $n^2$ is even; in (3), it is the assertion "$x$ is an integer", using a noun to assert that $x$ has the property of being an integer. (See integral.)
The sentences listed in the example above provide ways of expressing universally true conditionals in English. They use "for all" or "for any", You may also use these forms (compare in this discussion of universal assertions in general.)
Sometimes mathematicians write, "If the function $f$ is differentiable, then it is continuous." At least sometimes, they mean that every function that is differentiable is continuous. I suspect that this usage occurs in texts written by non-native-English speakers.
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
“For all $f$, if $D(f)$, then $C(f)$”
can be said in the following ways:
The converse of a conditional assertion “If $P$ then $Q$” is “If $Q$ then $P$”.
Whether a conditional assertion is true
has no bearing on whether its converse it true.
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.
The contrapositive of "If $x > 3$, then $x > 2$" is (after a little translation) "If $x\leq2$ then $x\leq3$." For any number $x$, these two statements are both true or both false.
This means that if you prove "If not $Q$ then not $P$", then you have also proved "If $P$ then $Q$."
You can prove an assertion by proving its contrapositive.
This is called the contrapositive method and is discussed in detail in this section.
So a conditional assertion and its contrapositive have the same truth value. Two assertions that have the same truth value are said to be equivalent. Equivalence is discussed with examples in the Wikipedia article on necessary and sufficient.
As you can see from the preceding discussions, statements of the form “If $P$ then Q” don’t mean the same thing in math that they do in ordinary English. This causes semantic contamination.
In ordinary English, “If $P$ then $Q$” can suggest order of occurrence. For example, “If we go outside, then the neighbors will see us” implies that the neighbors will see us after we go outside.
Consider "If $n\gt7$, then $n\gt5$." If $n\gt7$, that doesn't mean $n$ suddenly gets greater than $7$ earlier than $n$ gets greater than $5$. On the other hand, "$n\gt5$ is necessary for $n\gt7$" (which remember means the same thing as "If $n\gt7$, then $n\gt5$) doesn't mean that $n\gt5$ happens earlier than $n\gt7$. Since we are used to "if...then" having a timing implication, I suspect we get subconscious dissonance between “If $P$ then $Q$” and "$Q$ is necessary for $P$" in mathematical statements, and this dissonance makes it difficult to believe that that can mean the same thing.
“If $P$ then $Q$” can also suggest causation. The the sentence, "If we go outside, the neighbors will see us" has the connotation that the neighbors will see us because we went outside.
The contrapositive is "If the neighbors won't see us, then we don't go outside." This English sentence seems to me to mean that if the neighbors are not around to see us, then that causes us to stay inside. In contrast to contrapositive in math, this means something quite different from the original sentence.
For some instances of the use of "if...then" in English, the truth table is different.
Consider: "If you eat your vegetables, you can have dessert." Every child knows that this means they will get dessert if they eat their vegetables and not otherwise. So the truth table is:
| $P$ | $Q$ | If $P$ then $Q$ | |
| T | T | T | |
| T | F | F | |
| F | T | F | |
| F | F | T |
In other words, $P$ is equivalent to $Q$. It appears to me that this truth table corresponds to English "if...then" when a rule is being asserted.
These examples show:
The different ways of expressing a conditional assertion
may mean different things in English
but they always mean the same thing in math --
the meaning given by the truth table
How can I get to the stage where I automatically understand conditional assertions?
You need to understand the equivalence of these formulations 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! (Of course, this is how you gain expertise in anything.) In other words, Practice, Practice!
But it may help to remember that when doing proofs, we must take the rigorous view of mathematical objects:
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 $7$. There is no "because" in rigorous math. Both facts, $10\gt4$ and $10\gt7$, are eternally true.
Eternal is how we think of them – I am not making a claim about “reality”.
These three observations describe 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:
In symbols:
(1) When “If $P$ then $Q$” and $P$ are both true,
(2) then $Q$ must be true as well.
Modus Ponens is the most used method of deduction of all. It is the proof machine that makes Rewrite according to the definition work.
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 Forms of proof.
Assertions $P$ and $Q$ are equivalent if the conditional assertions "If $P$ then $Q$" and "If $Q$ then $P$" are both true or both false. The truth table is:
| $P$ | $Q$ | $P$ equiv $Q$ | |
| T | T | T | |
| T | F | F | |
| F | T | F | |
| F | F | T |
Observe that this table differs from the truth table for conditional assertions only in the third line.
The statement that $P$ and $Q$ are equivalent can be worded in these ways:
Thanks to Eugene Mondkar for catching an error.
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