Section 2.3 Biconditionals, Logical Equivalence
Subsection 2.3.1 Biconditional
Definition 2.3.1.
The biconditional, \(P \leftrightarrow Q\) is true if \(P\) and \(Q\) are either both true and both false, and is false if one of them is true and one is false.
In other words, \(P \leftrightarrow Q\) is true if both statements share the same truth value.
The truth table for a biconditional is given by,
With a biconditional, either both statements are true, or both are false. If one is true, then so is the other, and if one is false, then so is the other.
Other ways to say \(P \leftrightarrow Q\) include “\(P\) occurs precisely when \(Q\) occurs.”, “If \(P\text{,}\) then \(Q\text{,}\) and conversely.”, “\(P\) is a necessary and sufficient condition for \(Q\text{.}\)”
In English, some conditionals are implicitly biconditionals. In mathematics, we try to be more precise.
Subsection 2.3.2 Logical Equivalence
If two statements have the same truth values in every case, then we say that \(P\) and \(Q\) are logically equivalent, and this is denoted by \(P \equiv Q\text{.}\) Roughly, this means that the two statements are “equal”, in the sense of logic.
This notation allows us to describe some of the previous logical connectives in terms of more basic logical connectives. For example,
For a biconditional, \(P \leftrightarrow Q \equiv (P \rightarrow Q) \land (Q \rightarrow P)\text{.}\) The truth table is given by,
\begin{equation*} \begin{array}{c|c|c|c|c|c} P \amp Q \amp P \leftrightarrow Q \amp P \rightarrow Q \amp Q \rightarrow P \amp (P \rightarrow Q) \land (Q \rightarrow P) \\ \hline T \amp T \amp T \amp T \amp T \amp T \\ \hline T \amp F \amp F \amp F \amp T \amp F \\ \hline F \amp T \amp F \amp T \amp F \amp F \\ \hline F \amp F \amp T \amp T \amp T \amp T \end{array} \end{equation*}Notice that the column for \(P \leftrightarrow Q\) and for \((P \rightarrow Q) \land (Q \rightarrow P)\) have the same truth value in every case.Previously, we saw that a conditional is logically equivalent to its contrapositive.
The definition of conditional says that \(P \rightarrow Q\) is false if (and only if) \(P\) is true and \(Q\) is false. This means that,
\begin{equation*} \neg (P \rightarrow Q) \equiv P \land \neg Q \end{equation*}Recall that the negation of a negation is equivalent to the original statement. In other words,
\begin{equation*} \neg (\neg P) \equiv P \end{equation*}The converse of a condition is logically equivalent to the inverse. This can be seen by comparing the truth tables.
The converse is not logically equivalent to the original conditional. Incorrectly assuming this this is called the fallacy of the converse.
The inverse of a conditional is not logically equivalent to the conditional. Incorrectly assuming this is called denying the antecedent, or fallacy of the inverse.
In terms of the (inclusive) or, the exclusive or can be written as,
\begin{equation*} P \oplus Q \equiv (P \lor Q) \land \neg (P \land Q) \end{equation*}
Subsection 2.3.3 Order of Operations for Logical Operators
Subsection 2.3.4 Alternate Intuition for Implication
Example 2.3.2.
Here is another intuitive reason why \(P \rightarrow Q\) is true when \(P\) is false. First, it makes sense that \(P \rightarrow Q\) is true when \(P\) and \(Q\) are both true, and is false when \(P\) is true but \(Q\) is false. Then, we have the following partial truth table,
Then, there are 4 possible ways we can fill in the remainder of the table: \(TT, TF, FT, FF\text{.}\) This gives us,
Then, the first 3 columns can be recognized as \(P \land Q\text{,}\) \(P \leftrightarrow Q\text{,}\) and \(Q\text{,}\) respectively. This leaves the last case as the most intuitive to define as \(P \rightarrow Q\text{,}\)