Logical Identities

Definition

Logically equivalent propositions can be substituted in place of each other while making arguments or proving results. Such frequently used equivalent propositions are called logical identities.

List of logical identities

Let’s summarize logical identities in the following table. The propositions shown in one row of the table are equivalent to each other. That is, Left $\equiv$ Right.

Identity Number	Left	Right
1	$\neg \left(p\land q\right)$	$\neg p \lor \neg q$
2	$\neg \left(p\lor q\right)$	$\neg p \land \neg q$
3	$p\Rightarrow q$	$\neg q \Rightarrow \neg p$
4	$\neg p\Rightarrow \neg q$	$q \Rightarrow p$
5	$p\Rightarrow q$	$\neg p \lor q$
6	$\neg\left(p\Rightarrow q\right)$	$p \land \neg q$
7	$p \Leftrightarrow q$	$\left(p\Rightarrow q\right)\land \left(q\Rightarrow p\right)$
8	$p \Leftrightarrow q$	$\left(\neg p\lor q\right)\land \left(\neg q\lor p\right)$
9	$\neg\left(p \Leftrightarrow q\right)$	$\neg\left(p\Rightarrow q\right)\lor \neg\left(q\Rightarrow p\right)$
10	$\neg\left(p \Leftrightarrow q\right)$	$\left(p\land \neg q\right)\lor \left(q\land \neg p\right)$
11	$\neg\left(p \Leftrightarrow q\right)$	$p\oplus q$

In the table above, the first two identities are DeMorgan’s laws. Particular fundamental identities need to be kept in mind. The following table contains more named identities or laws of propositional logic. Here “$T$” stands true, and “$F$” is false.

Number	Equivalence or identity	Name of identity
1	$p\land T \equiv p$	Identity Law
2	$p\lor F \equiv p$	Identity Law
3	$p\land F \equiv F$	Domination Law
4	$p\lor T \equiv T$	Domination Law
5	$p\lor p \equiv p$	Idempotent Law
6	$p\land p \equiv p$	Idempotent Law
7	$p\land q \equiv q \land p$	Commutative Law
8	$p\lor q \equiv q \lor p$	Commutative Law
9	$p\Leftrightarrow q \equiv q \Leftrightarrow p$	Commutative Law
10	$p \land \left(p \lor q\right) \equiv p$	Absorption Law
11	$p \lor \left(p \land q\right) \equiv p$	Absorption Law
12	$p \land \neg p \equiv F$	Negation Law
13	$p \lor \neg p \equiv T$	Negation Law
14	$\neg\left(\neg p\right) \equiv p$	Double Negation Law
15	$\left(p \land q\right)\land r \equiv p \land \left(q\land r\right)$	Associative Law
16	$\left(p \lor q\right)\lor r \equiv p \lor \left(q\lor r\right)$	Associative Law
17	$p\land \left(q \lor r\right) \equiv \left(p\land q\right)\lor \left(p \land r\right)$	Distributive Law
18	$p\lor \left(q \land r\right) \equiv \left(p\lor q\right)\land \left(p \lor r\right)$	Distributive Law

Distributive laws

The identity numbers seventeen and eighteen in the table above are distributive laws. Let’s first discuss them intuitively and then, with the help of the truth table, examine why they hold.

Consider,

$$p\land \left(q \lor r\right) \equiv \left(p\land q\right)\lor \left(p \land r\right).$$

If we look at the left-hand side, it will be false if $p$ is false. If $p$ is true, then the truth value of the left-hand side depends on $q$ and $r$. If any of $q$ and $r$ are true, the expression is true.

This means, $p \land \left(q \lor r\right)$ is true if, $p\land q$ is true, or $p \land r$ is true; and it is false, if $p$ is false, or both $q$ and $r$ are false. That is the right-hand side.

Now, let’s look at the truth table of this identity as follows:

$\left(p, q, r\right)$	$q \lor r$	$p\land q$	$p \land r$	$p\land \left(q\lor r\right)$	$\left(p \land q\right)\lor\left(p \land r\right)$
(T, T, T)	T	T	T	T	T
(T, T, F)	T	T	F	T	T
(T, F, T)	T	F	T	T	T
(T, F, F)	F	F	F	F	F
(F, T, T)	T	F	F	F	F
(F, T, F)	T	F	F	F	F
(F, F, T)	T	F	F	F	F
(F, F, F)	F	F	F	F	F

From the last two columns of the above truth table, it is evident that the identity under consideration holds.

Now consider,

$$p\lor \left(q \land r\right) \equiv \left(p\lor q\right)\land \left(p \lor r\right)$$

If $p$ is true, it is easy to see that both the left-hand and right-hand sides are true regardless of the truth value of $q$ and $r$. If $p$ is false, then both $q$ and $r$ have to be true to make the left-hand side true, which also makes the right-hand side true.

Now, let’s look at the truth table of this identity as follows:

$\left(p, q, r\right)$	$q \land r$	$p\lor q$	$p \lor r$	$p\lor \left(q\land r\right)$	$\left(p \lor q\right)\land\left(p \lor r\right)$
(T, T, T)	T	T	T	T	T
(T, T, F)	F	T	T	T	T
(T, F, T)	F	T	T	T	T
(T, F, F)	F	T	T	T	T
(F, T, T)	T	T	T	T	T
(F, T, F)	F	T	F	F	F
(F, F, T)	F	F	T	F	F
(F, F, F)	F	F	F	F	F

From the last two columns of the above truth table, it is evident that the identity under consideration holds.

Examples

Let’s look at some examples:

By distributive law, we know that $p_1 \equiv p_2.$ So to express that, Tora is eating salad with rice or steak, it is logically equivalent to say $p_1$ or $p_2.$

For the second example:

We know that $p_3$ and $p_4$ are equivalent through distributive law.