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Consider the following statement: "If you earn an A in logic, then I'll buy you a Corvette." 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 Corvette."
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 Corvette," 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 Corvette. 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 Corvette. In general, we ue this idea to define the statement pq.
Conditional
The conditional p
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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.
If p and q are both true, then pq is true. For instance:
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.
If p is true and q is false, then pq is false. For instance:
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."
If p is false, then pq is true, no matter whether q is true or not. For instance:
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.
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, p
q is logically equivalent to (~p)
q. The following examples demonstrate this fact.
Let us take a look at Descartes' famous claim:
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.
Show that pq
(~p)
q.
Solution
We use a truth table to show this:
![]() | ![]() | |||
Before We Go On ...
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 in Example 5 the "Switcheroo" law.
Switcheroo Law
The Switcheroo law is the logical equivalence
![]() ![]() ![]() In words, it expresses the equivalence between saying "if p is true, then q must be true" and saying "either p is not true, or else q must be true." |
Use the Switcheroo law to transform the following statement into a disjunction: "If 0 = 1, then I am the Queen of Sheba."
Solution
The given statement has the form pq, where p: "0 = 1" and q: "I am the Queen of Sheba." The Switcheroo law says that this is equivalent to (~p)
q. In words, this becomes: "either 0
1, or I am the Queen of Sheba."
Some Phrasings of the Conditional
Each of the following is equivalent to the conditional p
|
Notice the difference between "if" and "only if." We say that "p only if q" means pq since, assuming that p
q is true, p can be true only if q is also. In other words, the only line of the truth table that has p
q 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 11).
Rephrase the sentence "If it's Tuesday, this must be Belgium."
Solution
Here are various ways of rephrasing the sentence:
Answer No. We can see this with the following truth table:
![]() | ![]() | ||
The columns corresponding to pq and q
p are different, and hence the two statements are not equivalent.
We call the statement qp the converse of p
q. Again, a conditional and its converse are not equivalent. This fact hardly deters the sales pitches used in the promotions of certain products. 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:
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."
Now give the converse of the statement in the previous example.
Solution
The converse of pq is q
p, and can be stated as: "If this grotesque animal is spotted, then it must be a Jersey cow."
Show, using the Switcheroo Law, that ~(pq)
p
(~q), and apply it to rewrite the negation of the statement "If this grotesque animal is a Jersey cow, then it must be spotted."
Solution
Applying the Switcheroo Law to ~(pq) gives:
~(p![]() | ![]() ![]() | (Switcheroo) |
![]() ![]() | (De Morgan) | |
![]() ![]() | (Double Negative) |
Using this to negate the given statement yields: "You're lying. This grotesque animal is a Jersey cow, and you can plainly see for yourself that it is not spotted."
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 p
q, 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).
The Biconditional We already saw that pq is not the same as q
p. It may happen, however, that both p
q and q
p are true. For example, if p: "0 = 1" and q: "1 = 2," then p
q and q
p are both true because p and q are both true. The statement p
q is defined to be the statement (p
q)
(q
p). For this reason, the double headed arrow
is called the biconditional. We get the truth table for p
q by constructing the table for (p
q)
(q
p), which gives us the following.
Biconditional
The biconditional p
The arrow " |
Note
From the truth table, we see that, for p |
Some Phrasings of the Bionditional
Each of the following is equivalent to the biconditional p
p is necessary and sufficient for q. p is equivalent to q. Notice that p |
For the phrasing "p if and only if q,", remember that "p if q" means qp while "p only if q" means p
q. For the phrasing "p is equivalent to q," the statements A and B are logically equivalent if and only if the statement A
B is a tautology (why?). We'll return to that in the next section.
True or false? "1+1 = 3 if and only if Waner is Alexander the Great."
Solution
True. The given statement has the form pq, where p: "1+1=3" and q: "Waner is Alexander the Great." Since both statements are false (one being evidence of delusions of grandeur on the part of one of the authors), the biconditional p
q is true.
Rephrase the statement: "I teach math if and only if I am paid a large sum of money."
Solution
Here are some equivalent ways of phrasing this sentence:
"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.
![]() | 2. Logical Equivalence, Tautologies and Contradictions | ![]() | Section 3 Exercises | ![]() | 4. Tautological Implications and Tautological Equivalences | ![]() | Main Logic Page | ![]() | "Real World" Page |
![]() | Stefan Waner (matszw@hofstra.edu) | ![]() | Steven R. Costenoble (matsrc@hofstra.edu) |