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 p→q.
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. If we think 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 p→q.
The conditional p→q, which we read "if p, then q" or "p implies q," is defined by the following truth table.
The arrow "→" is the conditional operator, and in p→q the statement p is called 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 p→q.
1. The only way that p→q 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 "p→q" 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 Mustang, 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.
Here p: "1+1 = 2" and q: "The sun rises in the east."
Notice that the statements p and q 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.
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 p→q is clearly false.
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.
Here p: "the moon is made of green cheese," which is false, and q: "I am the King of England." The statement p→q 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).
"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 p→q 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.
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.
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.
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."
|Some Phrasings of the Conditional
Each of the following is equivalent to the conditional p→q.
Notice the difference between "if" and "only if." We say that "p only if q" means p→q 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 9).
Q Does the commutative law hold for the conditional. In other words, is p→q the same as q→p?
A No. We can see this with the following truth table:
The columns corresponding to p→q and q→p are different, and hence the two statements are not equivalent. We call the statement q→p the converse of p→q.
The statement q→p is called the converse of the statement p→q. A conditional and its converse are not equivalent.
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 p→q 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:
The statement (~q) → (~p) is called the contrapositive of the statement p→q. A conditional and its contrapositive are equivalent.
We already saw that p→q 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 false. 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.
The biconditional p↔q, which we read "p if and only if q" or "p is equivalent to q," is defined by the following truth table.
The arrow "↔" is the biconditional operator.
Note that, from the truth table, we see that, for p↔q to be true, both p and q must have the same truth values; otherwise it is false.
| Some Phrasings of the Bionditional
Each of the following is equivalent to the biconditional p↔q.
p is necessary and sufficient for q.
p is equivalent to q.
Notice that p↔q is logically equivalent to q↔p (you are asked to show this as an exercise), so we can reverse p and q in the phrasings above.
For the phrasing "p if and only if q,", remember that "p if q" means q→p 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.
(b) 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.