Section1.3Conditional and Biconditional Statements
The last two logical connectives deal with implication.
Definition1.3.1.
The compound statement "if \(p\text{,}\) then \(q\)" or "\(p\) implies \(q\)" is called an implication or a conditional statement, denoted \(p \implies q\text{,}\) where \(p\) is referred to as the hypothesis or premise and \(q\) is referred to as the conclusion. The implication is defined by the truth table below.
Table1.3.2.Truth Table for Implication
\(p\)
\(q\)
\(p\implies q\)
T
T
T
T
F
F
F
T
T
F
F
T
An implication in which the premise is false is called vacuously true.
You may be looking at the last two rows of the truth table and wondering what’s going on. For lay people, the statement \(p \implies q\) is meaningless when \(p\) is false. But then \(p \implies q\) wouldn’t be a statement. Statements must have a truth value!
Example1.3.3.
Consider the statement, "If two is odd, then one is even." The constituent statements "two is odd" and "one is even" are both false, and the compound statement is defined to true. Any implication with the premise "two is odd" is vacuously true.
In formal mathematical logic, no causation in implied by an if-then statement. That is, the statement, "If it’s raining, then it’s wet outside" doesn’t the same thing as, "It’s wet outside because it is raining." For all we know, it could be wet outside because someone sprayed everything down with a fire hose. And so, even though the statement, "If \(2+2=5\text{,}\) then Mars is a planet," is a true statement, we don’t infer that the truth of the conclusion is dependent on its causal relationship to the premise ("Mars is a planet" is true regardless of arithmetic!).
Theorem1.3.4.Negating Implications.
For any statements \(p\) and \(q\text{,}\)
\(\lnot(p \implies q) \equiv p \land (\lnot q)\text{,}\) and
\(p \implies q \equiv (\lnot p) \lor q\text{.}\)
Proof.
We see from the truth tables for \(\lnot(p \implies q)\) and \(p \land (\lnot q)\) that these statements are logically equivalent. Indeed, we have
Table1.3.5.
Truth Table Verification for \(\lnot(p \implies q) \equiv p \land(\lnot q)\)
\(p\)
\(q\)
\(p \implies q\)
\(\lnot (p \implies q)\)
\(\lnot q\)
\(p \land (\lnot q)\)
T
T
T
F
F
F
T
F
F
T
T
T
F
T
T
F
F
F
F
F
T
F
T
F
as desired.
For the remaining equivalence, we have
\begin{align*}
p \implies q \amp\equiv \lnot\left(\lnot(p \implies q)\right) \amp\text{by the Law of the Double Negative,}\\
\amp\equiv \lnot\left(p \land(\lnot q)\right) \amp\text{by the previous equivalence,}\\
\amp\equiv (\lnot p) \lor \left(\lnot(\lnot q)\right) \amp\text{by De Morgan's Law,}\\
\amp\equiv (\lnot p) \lor q \amp\text{by the Law of the Double Negative.}
\end{align*}
There are three implications that are related to the statement \(p \implies q\text{,}\) and they are defined as follows.
Definition1.3.6.
Let \(p\) and \(q\) be statements and consider the implication \(p \implies q\text{.}\) The converse of \(p \implies q\) is the statement \(q \implies p\text{.}\) The inverse of \(p \implies q\) is the statement \((\lnot p) \implies (\lnot q)\text{.}\) The contrapositive of \(p \implies q\) is the statement \((\lnot q) \implies (\lnot p)\text{.}\)
Observe that the implication \(p \implies q\) is logically equivalent to its contrapositive \((\lnot q) \implies (\lnot p)\text{.}\) The converse \(q \implies p\) is logically equivalent to the inverse \((\lnot p) \implies (\lnot q)\text{.}\)
Checkpoint1.3.7.
Let \(p\) be the statement, "It’s raining," and \(q\) be the statement, "It’s wet outside." Match the sentences to the correct compound statements.
Implication \(p \implies q\)
If it’s raining, then it’s wet outside.
Converse of \(p \implies q\)
If it’s wet ouside, then it’s raining.
Inverse of \(p \implies q\)
If it’s not raining, then it’s not wet outside.
Contrapositive of \(p \implies q\)
If it’s not wet outside, then it’s not raining.
Checkpoint1.3.8.
\(p \implies q\text{,}\)
Definition1.3.10.
The biconditional of statements \(p\) and \(q\) is the compound statement "\(p\) if and only if \(q\text{,}\)" denoted \(p \iff q\text{.}\) It is defined by the truth table below.
Another way to express \(p \iff q\) is, "\(p\) is necessary and sufficient for \(q\text{.}\)" In particular, "\(p\) is necessary for \(q\)" describes the implication \(q \implies p\text{,}\) and "\(p\) is sufficient for \(q\)" describes the implication \(p \implies q\text{.}\)
Example1.3.13.Mathematical Definitions are Biconditional Statements.
A matrix is symmetric if and only if it equals its transpose.
Example1.3.14.The Contrapositive of the Biconditional.
From the logical equivalences we’ve established, we see:
This means that the biconditional is logically equivalent to its contrapositive.
Definition1.3.15.
A tautology is a compound statement that is true for all truth values of its constituent statements. A contradiction is a compound statement that is fall for all truth values of its constituent statements.
Checkpoint1.3.16.
Verify that \(p \lor (\lnot p)\) is a tautology and \(p \land (\lnot p)\) is a contradiction.
We refer to the tautology \(p \lor (\lnot p)\) as the Law of the Excluded Middle.
