Learning Objectives
- Basic Truth Tables for
- Conditional
- Biconditional
- Working with the Conditional Statement
- Converse
- Inverse
- Contrapositive
Conditional
This is sometimes called an implication. A conditional is a logical compound statement in which a statement p, called the antecedent, implies a statement q, called the consequent.
A conditional is written as p → q and is translated as “if p, then q”.
The English statement “If it is raining, then there are clouds is the sky” is a conditional statement. It makes sense because if the antecedent “it is raining” is true, then the consequent “there are clouds in the sky” must also be true.
Notice that the statement tells us nothing of what to expect if it is not raining; there might be
clouds in the sky, or there might not. If the antecedent is false, then the consequent becomes
irrelevant.
Example
Suppose you order a team jersey online on Tuesday and want to receive it by Friday so you can
wear it to Saturday’s game. The website says that if you pay for expedited shipping, you will
receive the jersey by Friday. In what situation is the website telling a lie?
There are four possible outcomes:
1) You pay for expedited shipping and receive the jersey by Friday
2) You pay for expedited shipping and don’t receive the jersey by Friday
3) You don’t pay for expedited shipping and receive the jersey by Friday
4) You don’t pay for expedited shipping and don’t receive the jersey by Friday
Only one of these outcomes proves that the website was lying: the second outcome in which you pay for expedited shipping but don’t receive the jersey by Friday. The first outcome is exactly what was promised, so there’s no problem with that. The third outcome is not a lie because the website never said what would happen if you didn’t pay for expedited shipping;
maybe the jersey would arrive by Friday whether you paid for expedited shipping or not. The fourth outcome is not a lie because, again, the website didn’t make any promises about when the jersey would arrive if you didn’t pay for expedited shipping.
It may seem strange that the third outcome in the previous example, in which the first part is false but the second part is true, is not a lie. Remember, though, that if the antecedent is false, we cannot make any judgment about the consequent. The website never said that paying for expedited shipping was the only way to receive the jersey by Friday.
Example
A friend tells you “If you upload that picture to Facebook, you’ll lose your job.” Under what conditions can you say that your friend was wrong?
There are four possible outcomes:
1) You upload the picture and lose your job
2) You upload the picture and don’t lose your job
3) You don’t upload the picture and lose your job
4) You don’t upload the picture and don’t lose your job
There is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still keep your job. In the last two cases, your friend didn’t say anything about what would happen if you didn’t upload the picture, so you can’t say that their statement was wrong. Even if you didn’t upload the picture and lost your job anyway, your
friend never said that you were guaranteed to keep your job if you didn’t upload the picture; you might lose your job for missing a shift or punching your boss instead.
Aconditional statementtells us that if the antecedent is true, the consequent cannot be false. Thus, a conditional statement is only false when a true antecedent implies a false consequent.
Examples
Another example is living in an apartment and paying rent.p → q where p is I live in an apartment and q is then I pay rent. What are the outcomes?
- I do live in an apartment and I pay rent, then the situation is true (no eviction!)
- I live in an apartment and I don’t pay rent, then the situation is false (eviction, broken promise)
- I don’t live in an apartment but I do pay rent, then the situation is true (though why would you do it?)
- I don’t live in an apartment and I don’t pay rent, then the situation is true (no promise broken)
With conditional situations, we also have the following:
Related Statements
The original conditional is “if p, then q” p → q
The converse is “if q, then p” q → p
The inverse is “if not p, then not q” ~p →~ q
The contrapositive is “if not q, then not p” ~q → ~p
Examples
Consider the conditional “If it is raining, then there are clouds in the sky.” It seems reasonable to assume that this is true.
The converse would be “If there are clouds in the sky, then it is raining.” This is not always true.
The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true.
The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This
statement is true, and is equivalent to the original conditional.
Examples
Suppose this statement is true: “If I eat this giant cookie, then I will feel sick.” Which of the following statements must also be true?
a. If I feel sick, then I ate that giant cookie.
b. If I don’t eat this giant cookie, then I won’t feel sick.
c. If I don’t feel sick, then I didn’t eat that giant cookie.
a. This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.
b. This is the inverse, which is not necessarily true. Again, I could feel sick for some other reason; avoiding the cookie doesn’t guarantee that I won’t feel sick.
c. This is the contrapositive, which is true, but we have to think somewhat backwards to explain it. If I ate the cookie, I would feel sick, but since I don’t feel sick, I must not have eaten the cookie.
Notice again that the original statement and the contrapositive have the same truth value (both
are true), and the converse and the inverse have the same truth value (both are false).
Biconditional
A biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable.
A biconditional is written as p ↔ q and is translated as “p if and only if q”.
Because a biconditional statement p ↔ q is equivalent to (p → q) ⋀ (q → p), we may think of it as a conditional statement combined with its converse: if p, then q and if q, then p. The double-headed arrow shows that the conditional statement goes from left to right and from right to left. A biconditional is considered true as long as the antecedent and the consequent have the same truth value; that is, they are either both true or both false.
Thebiconditionaltells us that, “Either both are the case, or neither is… ” Thus, a biconditional statement is true when both statements are true, or both are false.
Examples
Suppose this statement is true: “The garbage truck comes down my street if and only if it is Thursday morning.” Which of the following statements could be true?
a. It is noon on Thursday and the garbage truck did not come down my street this morning.
b. It is Monday and the garbage truck is coming down my street.
c. It is Wednesday at 11:59PM and the garbage truck did not come down my street today.
a. This cannot be true. This is like the second row of the truth table; it is true that I just experienced Thursday morning, but it is false that the garbage truck came.
b. This cannot be true. This is like the third row of the truth table; it is false that it is Thursday, but it is true that the garbage truck came.
c. This could be true. This is like the fourth row of the truth table; it is false that it is Thursday, but it is also false that the garbage truck came, so everything worked out like it should.
Working with the Conditional Statement
Conditional statements play a very big role in logic and one of the ways we can learn more about them is to study the three related statements.
Example
Consider again the valid implication “If it is raining, then there are clouds in the sky.”
Write the related converse, inverse, and contrapositive statements.
Show Solution
Try It
| Conditional | Converse | Inverse | Contrapositive | ||
|---|---|---|---|---|---|
| p | q | [latex]p\rightarrow{q}[/latex] | [latex]q{\rightarrow}p[/latex] | [latex]\sim{p}\rightarrow\sim{q}[/latex] | [latex]\sim{q}\rightarrow\sim{p}[/latex] |
| T | T | T | T | T | T |
| T | F | F | T | T | F |
| F | T | T | F | F | T |
| F | F | T | T | T | T |
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