17.6 Truth Tables: Conditional, Biconditional

We discussed conditional statements earlier, in which we take an action based on the value of the condition. We are now going to look at another version of a conditional, sometimes called an implication, which states that the second part must logically follow from the first.

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 consquent becomes irrelevant.

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.

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.

In traditional logic, a conditional is considered true as long as there are no cases in which the antecedent is true and the consequent is false.

Note

Truth table for the conditional

p	q	$p \to q$
T	T	T
T	F	F
F	T	T
F	F	T

Again, if the antecedent $p$ is false, we cannot prove that the statement is a lie, so the result of the third and fourth rows is true.

Example 4

Construct a truth table for the statement $open parenthesis m ∧ ~ p close parenthesis → r$

Show solution

We start by constructing a truth table with 8 rows to cover all possible scenarios. Next, we can focus on the antecedent, $m ∧ ~ p$ .

m	p	r
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

m	p	r	$~ p$
T	T	T	F
T	T	F	F
T	F	T	T
T	F	F	T
F	T	T	F
F	T	F	F
F	F	T	T
F	F	F	T

m	p	r	$~ p$	$m ∧ ~ p$
T	T	T	F	F
T	T	F	F	F
T	F	T	T	T
T	F	F	T	T
F	T	T	F	F
F	T	F	F	F
F	F	T	T	F
F	F	F	T	F

Now we can create a column for the conditional. Because it can be confusing to keep track of all the Ts and $F s$ , why don't we copy the column for $r$ to the right of the column for $m ∧ ~ p$ ? This makes it a lot easier to read the conditional from left to right.

m	p	r	$~ p$	$m ∧ ~ p$	r	$open parenthesis m ∧ ~ p close parenthesis → r$
T	T	T	F	F	T	T
T	T	F	F	F	F	T
T	F	T	T	T	T	T
T	F	F	T	T	F	F
F	T	T	F	F	T	T
F	T	F	F	F	F	T
F	F	T	T	F	T	T
F	F	F	T	F	F	T

When $m$ is true, $p$ is false, and $r$ is false- -the fourth row of the table-then the antecedent $m ∧ ~ p$ will be true but the consequent false, resulting in an invalid conditional; every other case gives a valid conditional.

If you want a real-life situation that could be modeled by $open parenthesis m ∧ ~ p close parenthesis → r$ , consider this: let $m =$ we order meatballs, $p =$ we order pasta, and $r =$ Rob is happy. The statement $open parenthesis m ∧ ~ p close parenthesis → r$ is "if we order meatballs and don't order pasta, then Rob is happy". If $m$ is true (we order meatballs), $p$ is false (we don't order pasta), and $r$ is false (Rob is not happy), then the statement is false, because we satisfied the antecedent but Rob did not satisfy the consequent.

For any conditional, there are three related statements, the converse, the inverse, and the contrapositive.

Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.

Truth table comparing a conditional with its converse, inverse and contrapositive. Reading the rows for p and q as true-true, true-false, false-true and false-false: the conditional p implies q gives T, F, T, T; the converse q implies p gives T, T, F, T; the inverse not p implies not q gives T, T, F, T; and the contrapositive not q implies not p gives T, F, T, T. Arrows beneath bracket the converse with the inverse and the conditional with the contrapositive, both brackets labelled Equivalent.

Note

Equivalence

A conditional statement and its contrapositive are logically equivalent.

P_TEXT = "it is raining"                 # EDIT
Q_TEXT = "there are clouds in the sky"   # EDIT

def implies(p, q):
    return (not p) or q          # a conditional lies ONLY when p is true and q is false

print(f"conditional     p -> q        If {P_TEXT}, then {Q_TEXT}.")
print(f"converse        q -> p        If {Q_TEXT}, then {P_TEXT}.")
print(f"inverse        ~p -> ~q       If NOT {P_TEXT}, then NOT {Q_TEXT}.")
print(f"contrapositive ~q -> ~p       If NOT {Q_TEXT}, then NOT {P_TEXT}.\n")

cols = [("p -> q", lambda p, q: implies(p, q)),
        ("q -> p", lambda p, q: implies(q, p)),
        ("~p -> ~q", lambda p, q: implies(not p, not q)),
        ("~q -> ~p", lambda p, q: implies(not q, not p)),
        ("p <-> q", lambda p, q: p == q)]
head = f"{'p':>3}{'q':>3}  |" + "".join(f"{name:>11}" for name, _ in cols)
print(head)
print("-" * len(head))
table = {name: [] for name, _ in cols}
for p in (True, False):
    for q in (True, False):
        vals = [(name, f(p, q)) for name, f in cols]
        for name, v in vals:
            table[name].append(v)
        print(f"{'T' if p else 'F':>3}{'T' if q else 'F':>3}  |"
              + "".join(f"{('T' if v else 'F'):>11}" for _, v in vals))

print("\ncolumns that match all the way down are logically equivalent:")
names = [n for n, _ in cols]
for i in range(len(names)):
    for j in range(i + 1, len(names)):
        if table[names[i]] == table[names[j]]:
            print(f"   {names[i]:<9} is equivalent to {names[j]}")
print("\nso: the original and the contrapositive stand or fall together,")
print("and the converse and the inverse stand or fall together - but the two")
print("pairs are NOT the same. 'If it rains there are clouds' is true; ")
print("'if there are clouds it rains' is not.")

print("\nthe negation of a conditional is NOT another conditional:")
print(f"   ~(p -> q) is equivalent to p ^ ~q")
same = all(((not implies(p, q)) == (p and not q)) for p in (True, False) for q in (True, False))
print(f"   checked on all 4 rows: {same}")
print(f"   'It is not true that if you park here you get a ticket'")
print(f"   means 'you park here AND you do not get a ticket'.")

The converse and inverse of a conditional statement are logically equivalent.

In other words, the original statement and the contrapositive must agree with each other; they must both be true, or they must both be false. Similarly, the converse and the inverse must agree with each other; they must both be true, or they must both be false.

Be aware that symbolic logic cannot represent the English language perfectly. For example, we may need to change the verb tense to show that one thing occurred before another.

Example 6

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?

If I feel sick, then I ate that giant cookie.
If I don’t eat this giant cookie, then I won’t feel sick.
If I don’t feel sick, then I didn’t eat that giant cookie.

Show solution

This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.
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.
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).

Consider the statement “If you park here, then you will get a ticket.” What set of conditions would prove this statement false?

You don’t park here and you get a ticket.
You don’t park here and you don’t get a ticket.
You park here and you don’t get a ticket.

The first two statements are irrelevant because we don’t know what will happen if you park somewhere else. The third statement, however contradicts the conditional statement “If you park here, then you will get a ticket” because you parked here but didn’t get a ticket. This example demonstrates a general rule; the negation of a conditional can be written as a conjunction: “It is not the case that if you park here, then you will get a ticket” is equivalent to “You park here and you do not get a ticket.”

In everyday life, we often have a stronger meaning in mind when we use a conditional statement. Consider “If you submit your hours today, then you will be paid next Friday.” What the payroll rep really means is “If you submit your hours today, then you will be paid next Friday, and if you don’t submit your hours today, then you won’t be paid next Friday.” The conditional statement if t, then p also includes the inverse of the statement: if not t, then not p. A more compact way to express this statement is “You will be paid next Friday if and only if you submit your timesheet today.” A statement of this form is called a biconditional.

Because a biconditional statement $p \leftrightarrow q$ is equivalent to $open parenthesis p → q close parenthesis ∧ open parenthesis q → p close parenthesis,$ 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.

Note

Truth table for the biconditional

p	q	$p \leftrightarrow q$
T	T	T
T	F	F
F	T	F
F	F	T

Notice that the fourth row, where both components are false, is true; if you don’t submit your timesheet and you don’t get paid, the person from payroll told you the truth.

Example 8

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?

It is noon on Thursday and the garbage truck did not come down my street this morning.
It is Monday and the garbage truck is coming down my street.
It is Wednesday at 11:59PM and the garbage truck did not come down my street today.

Show solution

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.
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.
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.

Example 9

Create a truth table for the statement $open parenthesis A ∨ B close parenthesis ↔ ~ C$

Show solution

Whenever we have three component statements, we start by listing all the possible truth value combinations for $A, B,$ and $C .$ After creating those three columns, we can create a fourth column for the antecedent, $A ∨ B$ . Now we will temporarily ignore the column for $C$ and focus on $A$ and $B$ , writing the truth values for $A ∨ B$ .

A	B	C
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

A	B	C	$A ∨ B$
T	T	T	T
T	T	F	T
T	F	T	T
T	F	F	T
F	T	T	T
F	T	F	T
F	F	T	F
F	F	F	F

Next we can create a column for the negation of $C$ . (Ignore the $A ∨ B$ column and simply negate the values in the $C$ column.)

A	B	C	$A ∨ B$	$~ C$
T	T	T	T	F
T	T	F	T	T
T	F	T	T	F
T	F	F	T	T
F	T	T	T	F
F	T	F	T	T
F	F	T	F	F
F	F	F	F	T

Finally, we find the truth values of $open parenthesis A ∨ B close parenthesis ↔ ~ C$ . Remember, a biconditional is true when the truth value of the two parts match, but it is false when the truth values do not match.

A	B	C	$A ∨ B$	$~ C$	$open parenthesis A ∨ B close parenthesis ↔ ~ C$
T	T	T	T	F	F
T	T	F	T	T	T
T	F	T	T	F	F
T	F	F	T	T	T
F	T	T	T	F	F
F	T	F	T	T	T
F	F	T	F	F	T
F	F	F	F	T	F

To illustrate this situation, suppose your boss needs you to do either project $A$ or project $B$ (or both, if you have the time). If you do one of the projects, you will not get a crummy review ( $C$ is for crummy). So $open parenthesis A ∨ B close parenthesis ↔ ~ C$ means "You will not get a crummy review if and only if you do project $A$ or project $B$ ." Looking at a few of the rows of the truth table, we can see how this works out. In the first row, $A, B,$ and $C$ are all true: you did both projects and got a crummy review, which is not what your boss told you would happen! That is why the final result of the first row is false. In the fourth row, $A$ is true, $B$ is false, and $C$ is false: you did project $A$ and did not get a crummy review. This is what your boss said would happen, so the final result of this row is true. And in the eighth row, $A, B$ , and $C$ are all false: you didn't do either project and did not get a crummy review. This is not what your boss said would happen, so the final result of this row is false. (Even though you may be happy that your boss didn't follow through on the threat, the truth table shows that your boss lied about what would happen.)

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.