Introduction to Logic

The Conditional

Consider the following statement: "If you earn an A in logic, then I'll buy you a Yellow Mustang." It seems to be made up out of two simpler statements:

p: "You earn an A in logic," and

q: "I will buy you a Yellow Mustang."

What the original statement is then saying is this: if p is true, then q is true, or, more simply, if p, then q. We can also phrase this as p implies q, and we write pq.

Now let us suppose for the sake of argument that the original statement: "If you earn an A in logic, then I'll buy you a Yellow Mustang," is true. This does not mean that you will earn an A in logic; all it says is that if you do so, then I will buy you that Yellow Mustang. Thinking of this as a promise, the only way that it can be broken is if you do earn an A and I do not buy you a Yellow Mustang. In general, we ue this idea to define the statement pq.

Conditional The conditional pq, which we read "if p, then q" or "p implies q," is defined by the following truth table. p q pq T T T T F F F T T F F T The arrow "" is the conditional operator, and in pq the statement p is classed the antecedent, or hypothesis, and q is called the consequent, or conclusion. Notice that the conditional is a new example of a binary logical operator -- it assigns to each pair of statments p and q the new statement pq.

Notes

1. The only way that pq can be false is if p is true and q is false-this is the case of the "broken promise."

2. If you look at the truth table again, you see that we say that "pq" is true when p is false, no matter what the truth value of q. This again makes sense in the context of the promise — if you don't get that A, then whether or not I buy you a Corvette, I have not broken my promise. However, it goes against the grain if you think of "if p then q" as saying that p causes q. The problem is that there are really many ways in which the English phrase "if ... then ..." is used. Logicians have simply agreed that the meaning given by the truth table above is the most useful for mathematics, and so that is the meaning we shall always use. Shortly we'll talk about other English phrases that we interpret as meaning the same thing.

Here are some examples that will help to explain each line in the truth table.

Example 1 (True Implies True) is True

If p and q are both true, then pq is true. For instance:
If 1+1 = 2 then the sun rises in the east.

Here p: "1+1 = 2" and q: "the sun rises in the east."

Before we go on...

Notice that the statements need not have anything to do with one another. We are not saying that the sun rises in the east because 1+1 = 2, simply that the whole statement is logically true.

Example 2 True Can't Imply False

If p is true and q is false, then pq is false. For instance:
When it rains, I carry an umbrella.

Here p: "It is raining," and q: "I am carrying an umbrella." In other words, we can rephrase the sentence as: "If it is raining then I am carrying an umbrella." Now there are lots of days when it rains (p is true) and I forget to bring my umbrella (q is false). On any of those days the statement pq is clearly false.

Before we go on...

Notice that we interpreted "When p, q" as "If p then q."

The next example explains the last two lines of the truth table for the conditional.

Example 3 False Implies Anything

If p is false, then pq is true, no matter whether q is true or not. For instance:
If the moon is made of green cheese, then I am the King of England.

Here p: "the moon is made of green cheese," which is false, and q: "I am the King of England." The statement pq is true, whether or not the speaker happens to be the King of England (or whether, for that matter, there even is a King of England).

Before we go on...

"If I had a million dollars I'd be on Easy Street." "Yeah, and if my grandmother had wheels she'd be a bus." The point of the retort is that, if the hypothesis is false, the whole implication is true.

Example 3P Practice with the Conditional

Answer the following. If in doubt, use the truth table or consult the above three examples.

"If 1+1 = 3, then 1+2 = 3."   is	true false	.
"If the earth is round, then Mars is flat."   is	true false	.
"If the earth is flat, then Mars is flat."   is	true false	.
"If the earth is round, then Mars is round."   is	true false	.
"If the Mars is round, then I am the man in the moon."   is	true false	.

Looking at the truth table once more, notice that pq is true if either p is false or q is true (or both). Once more, the only way the implication can be false is for p to be true and q to be false. In other words, pq is logically equivalent to (~p)q. The following examples demonstrate this fact.

Example 4 Cogito; Ergo Sum

Let us take a look at Descartes' famous claim:
Cogito; ergo sum: "I think; therefore I am."

In order to conclude that "I am" from "I think," Descartes is making the following implicit assumption: "If I think, then I am." If Descartes does not think, then it doesn't matter whether he exists or not. If he does exist, then it doesn't matter whether he thinks or not. The only case that could contradict his assumption is the broken promise: if he thinks but does not exist.

Example 5 Switcheroo

Show that pq(~p)q.

Solution

We use a truth table to show this:

Same values

p q pq ~p (~p)q T T T F T T F F F F F T T T T F F T T T

Before we go on...

In other words, pq is true if either p is false or q is true. By DeMorgan's Law these statements are also equivalent to ~(p (~q)). The only way the conditional can be false is the case of the broken promise: when p is true and q is false.

The fact that we can convert implication to disjunction should surprise you. In fact, behind this is a very powerful technique. It is not too hard (using the truth table) to convert any logical statement into a disjunction of conjunctions of atoms or their negations. This is called disjunctive normal form, and is essential in the design of the logical circuitry making up digital computers.

For lack of a better name, we shall call the equivalence pq(~p)q the "Switcheroo" law.  

Example 5P Practice with Switcheroo

(a)	p: "If there is life on Mars, then we should fund NASA."
		Select one "Either there is life on Mars, or we should not fund NASA." "Either there is no life on Mars, or we should fund NASA." "Either there is no life on Mars, or we should not fund NASA." "Either there is life on Mars, or we should fund NASA." "If there is no life on Mars, then we should not fund NASA"
(b)	p: "If you are tall you will get into the lacrosse team."
		Select one "Unless you are tall you will not get into the lacrosse team." "Either you are tall or you will not get into the lacrosse team." "Either you are not tall or you won't get into the lacrosse team." "If you are not tall you can still get into the lacrosse team." "Either you will get into the lacrosse team, or you are not tall."
(c)	p: "We will fund NASA only if there is life on Mars."
		Select one "Either there is life on Mars or we will not fund NASA." "Either we will fund NASA or there is no life on Mars." "Either there is life on Mars or we will fund NASA." "If there is life on Mars, we will fund NASA." "If there is life on Mars, we may still not fund NASA."

Notice the difference between "if" and "only if." We say that "p only if q" means pq since, assuming that pq is true, p can be true only if q is also. In other words, the only line of the truth table that has pq true and p true also has q true. The phrasing "p is a sufficient condition for q" says that it suffices to know that p is true to be able to conclude that q is true. For example, it is sufficient that you get an A in logic for me to buy you a Corvette. Other things might induce me to buy you the car, but an A in logic would suffice. The phrasing "q is necessary for p" we'll come back to later (see Example 9).

Example 6 Rephrasing the Conditional

Rephrase the sentence "If it's Tuesday, this must be Belgium."

Solution

Here are various ways of rephrasing the sentence:

"Its being Tuesday implies that this is Belgium."

"This is Belgium if it's Tuesday."

"It's Tuesday only if this is Belgium."

"It can't be Tuesday unless this is Belgium."

"Its being Tuesday is sufficient for this to be Belgium."

"That this is Belgium is a necessary condition for its being Tuesday."

Q Does the commutative law hold for the conditional. In other words, is pq the same as qp?
A No. We can see this with the following truth table:

Different

p q pq qp T T T T T F F T F T T F F F T T

The columns corresponding to pq and qp are different, and hence the two statements are not equivalent. We call the statement qp the converse of pq.

The fact that a conditional can easily be confused with its converse is often used in advertising. For example, the slogan "Drink Boors, the designated beverage of the US Olympic Team" suggests that all US Olympic athletes drink Boors (i.e., if you are a US Olympic athlete, then you drink Boors). What it is trying to insinuate at the same time is the converse: that all drinkers of Boors become US Olympic athletes (if you drink Boors then you are a US Olympic athlete, or: it is sufficient to drink Boors to become a US Olympic athlete).

Although the conditional pq is not the same as its converse, it is the same as its so-called contrapositive, (~q)(~p). While this could easily be shown with a truth table (which you will be asked to do in an exercise) we can show this equivalence by using the equivalences we already know:

pq	(~p)q	Switcheroo
	q(~p)	Commutativity of
	~(~q)(~p)	Double Negative
	(~q)(~p)	Switcheroo

Example 7 Contrapositive

Rewrite the statement "If this grotesque animal is a Jersey cow, then it must be spotted" as its contrapositive

Solution

The given statement has the form pq, where p: "this grotesque animal is a Jersey cow," and q: "this grotesque animal is spotted." The contrapositive is the statement (~q)(~p), and can be worded as follows: "If this grotesque animal is not spotted, then it can't be a Jersey cow."

Example 8 Converse

Now give the converse of the statement in the previous example.

Solution

The converse of pq is qp, and can be stated as: "If this grotesque animal is spotted, then it must be a Jersey cow."

Example 9 Contrapositive

Give the contrapositive of the statement: "If you don't pay the ransom, you'll never see your Chia Pet again."

Solution

This is (~q)(~p) where p: "you will see your Chia Pet again" and q: "you do pay the ransom." The contrapositive is pq, or "If you will see your Chia Pet again, you must pay the ransom." A less awkward phrasing is "It is necessary for you to pay the ransom for you to see your Chia Pet again."

Before we go on...

When we say pq is true we do not mean that p must come before q or that p causes q. In fact, often we mean that q is the only possible cause of p, and so p is evidence that q has occurred. The phrasings "q is necessary for p" and "p only if q" are most natural in this case (try "p only if q" in this example).

Example 9P Practice with Converse and Contrapositive

For each of the given statements, enter the converse or contrapositive, as required. To enter them, use "AND" for , "OR" for ., and "IMPLIES" for . For instance, you could enter

(pq) (~r)     as     (p OR q) IMPLIES ~r

(a)	The converse of p(~q) is
(b)	The contrapositive of p(~q) is
(c)	The contrapositive of (~p)(~q) is

(d)	The converse of "If you drink Retrograde, you are cool." is
	Select one " If you don't drink Retrograde, you are uncool." " If you are cool, then you drink Retrograde." " If you are cool, then you don't drink Retrograde." " If you are uncool, then you don't drink Retrograde." " If you don't drink Retrograde, you are cool."
(e)	The contrapositive of "If you drink Retrograde, you are cool." is
	Select one " If you don't drink Retrograde, you are uncool." " If you are cool, then you drink Retrograde." " If you are cool, then you don't drink Retrograde." " If you are uncool, then you don't drink Retrograde." " If you don't drink Retrograde, you are cool."
(g)	The converse of the contrapositive of "If you drink Retrograde, you are cool." is
	Select one " If you don't drink Retrograde, you are uncool." " If you are cool, then you drink Retrograde." " If you are cool, then you don't drink Retrograde." " If you are uncool, then you don't drink Retrograde." " If you don't drink Retrograde, you are cool."

The Biconditional

We already saw that pq is not the same as qp. It may happen, however, that both pq and qp are true. For example, if p: "0 = 1" and q: "1 = 2," then pq and qp are both true because p and q are both false. The statement pq is defined to be the statement (pq)(qp). For this reason, the double headed arrow is called the biconditional. We get the truth table for pq by constructing the table for (pq)(qp), which gives us the following.

For the phrasing "p if and only if q,", remember that "p if q" means qp while "p only if q" means pq. For the phrasing "p is equivalent to q," the statements A and B are logically equivalent if and only if the statement AB is a tautology (why?). We'll return to that in the next section.

Example 10 Biconditional

(a) True or false? "1+1 = 3 if and only if Mars is a black hole."
(b) Rephrase the statement: "I teach math if and only if I am paid a large sum of money."

Solution

(a) True. The given statement has the form pq, where p: "1+1=3" and q: "Mars is a black hole." Since both statements are false, the biconditional pq is true.

(b) Here are some equivalent ways of phrasing this sentence:

"My teaching math is necessary and sufficient for me to be paid a large sum of money."

"For me to teach math it is necessary and sufficient that I be paid a large sum of money."

Sadly for our finances, none of these statements are true.