2.7: Conditionals and the Word If

The word if is not in the list of premise indicator words. You cannot rely on if to indicate a premise. Here is why. In argument A below, the word if is followed by a premise, but in argument B it is part of the conclusion.

A. If, as we know, all men are mortal and Jeremiah is a man, not a god, then he is mortal, too.

B. If a mercury thermometer is given prolonged heating, it will break. This is because prolonged heating will cause the mercury to expand a great deal. But the thermometer will break apart whenever the mercury expands this much.

Figure \(\PageIndex{1}\)

Let's examine argument B more carefully. Does it assume that a mercury thermometer is actually given prolonged heating? No. Notice also that the conclusion is not that the mercury thermometer will actually break, but only that it will break if heated. The conclusion is an if-then statement: if the thermometer is heated, then it will break. So, the if is not indicating a premise, nor is it indicating a conclusion; it is performing another function. These if-then statements are called conditional statements or conditionals. When we say, “If we cancel the picnic, I’ll be happy,” we are offering a conditional, but not offering an argument.

Worse yet, the occurrence of the word "if" in a sentence is not a reliable indicator that the sentence contains a conditional. For example, the sentence, "If you don't mind, you're standing on my foot" is not a conditional. It is a special idiom in English and is not a conditional because it cannot be rewritten equivalently as "P implies Q."

A statement can be a conditional even if the companion word then is not present. For example:

• If the Campbell's Soup Company puts less salt in its soup, sales of Campbell's soup will increase.

Does it follow from this conditional claim that Campbell's Soup Company does put less salt in its soup? No. Is the speaker committed to the claim that sales of Campbell's soup will increase? No, the commitment is only to an increase on the condition that the company does something about the salt. That is why conditionals are called "conditionals."

Should you conclude from the original conditional statement that, if Campbell's sales do not increase, then the company failed to put less salt in its soup? Yes, this last conditional statement, follows with certainty from the original conditional statement. It is the contrapositive of the original statement. We will explore contrapositives in more detail in Chapter 11.

Conditionals have a standard form which is “If A, then B.”

Often conditionals are expressed in other ways. For example, here is a conditional that contains neither an “if” nor a “then:”

• The larger a star the quicker it burns up and dies.

Rewriting it in standard form produces:

• If a star is larger, then it burns up and dies quicker.

Exercise \(\PageIndex{1}\)

Figure \(\PageIndex{1}\)

The Governor of Alaska (on left)

Suppose you were to learn for certain that if a person is the governor of Alaska, then he or she is a U.S. citizen. If so, can you be absolutely sure that if somebody is not a U.S. citizen, then he or she is not the governor of Alaska?

Answer

Yes, you can be sure. This is the contrapositive of the original conditional.

Is the following conditional making a true statement about the real world?

• If President John F. Kennedy was born in Bangladesh, then he was born in Asia.

Yes, it is true, and it is true even though both the if-part and the then-part are false. There’s a big lesson with that example:

Figure \(\PageIndex{1}\)

We will explore the logic of conditionals in more depth in Chapter 11.

Exercise \(\PageIndex{1}\)

Answer "yes" or "no, not always" to these conditional claims:

a. If it's an apple, then it's a fruit.
b. If it's a fruit, then it's an apple.
c. It's an apple if it's a fruit.
d. It's a fruit if it's an apple.
e. It's not a fruit if it's not an apple.
f. It's not an apple if it's not a fruit.
g. If the current president of the United States were also the leader of Pakistan, then the president would be the leader of an Asian country.
h. If the tallest building in the U.S. is only 15 feet tall, then there is no building in the U.S. taller than 30 feet.
i. If Joshua Dicker or his dad, Stuart, are invited, then Joshua Dicker's dad is invited.

Answer

(a) yes (b) no (c) no (d) yes (e) no (f) yes (g) yes (h) yes (i) no. In (i), if the or were and, then the answer would be yes.