Our first example deals with multiple negations.

Thinking about this example from a different context, consider the compound statement

¬¬p ∧ q

Seeing as p and ¬¬p share the same truth values, we could perhaps substitute p for ¬¬p to get

p ∧ q

because we replace one part (¬¬p) with another part (p) that has the same truth value. Because they have the same truth values, a conjunction with q would result in the same truth value. We prefer to work with the second compound statement (p ∧ q) because it is much simpler than the first (¬¬p ∧ q.)

Before continuing, let’s look at an example that may be somewhat ambiguous. Consider a statement like

p ∨ q ∧ r.

Do we evaluate the above expression as (p ∨ q) ∧ r or as p ∨ (q ∧ r). Does it matter which of the two expressions we use to evaluate? As always, let’s look at a truth table. First, we create a table with intermediate results (that is, a table tracking the values of p, q, r, (p ∨ q), and (q ∧ r).)

Now we construct a final truth table with the expressions we’re interested in.

Notice that (p ∨ q) ∧ r and p ∨ (q ∧ r) have very different rows, so apparently the order in which we evaluate the expression

p ∨ q ∧ r

matters. As such, we would need parenthesis in order to avoid any ambiguity.

So far, we’ve seen three pairs of statements that share truth values;

p and ¬¬p
¬(p ∧ q) and ¬p ∨¬q
¬(p ∨ q) and ¬p ∧¬q

As discussed after the first example, it appears as if we can essentially replace one statement of each pair above with the other statement in its pair without affecting the truth value of anything. We’ve also seen that when two statements s₁ and s₂ always have the same truth values, we have that the statement s₁ ↔ s₂ is a tautology. This observation yields the following definition:

This gives us a concise way to describe the relationship between two statements that share truth values, as we’ll see in the next example.