Logic Tutorial, Part 6.
The Conditional and the Biconditional

(c) 2017 by Barton Paul Levenson

One type of statement is the if-then statement: If you do well in school, then I'll take you to Disney World. It contains two propositions: "You do well in school" and "I'll take you to Disney World," with the if-then structure relating the two. Say we have

p: You do well in school.
q: I'll take you to Disney World.

We can translate this into formal logic with yet another logical operator:

→ ("implies" or "the conditional operator")

The statement phrased this way would then be p → q, read "p implies q." Here's a truth table for it:

Note that the operation can only become false if p is true. If you don't do well in school in the first place, the question doesn't arise, so these possibilities are assumed to be true. No promise was broken if you don't get taken to Disney World following dismal performance in school; the trip was only promised to happen if you brought home a good report card. But if you do do well in school, then the implication was true if I take you to Disney World, but false if I don't. Only the latter case counts as a broken promise.

Here p is called the hypothesis and q the conclusion--or p the antecedent and q the consequent. That's why such statements are known as "hypothetical conditions."

If . . .
let us assume that . . .
postulate that . . . you do well.

then . . .
In that hypothetical case . . . I will take you to Disney World.

Note that p → q does not mean p causes q. The conditional operator has a specific, highly restricted meaning in formal logic.

By the way, p → q is logically equivalent to (¬p) ∨ q. Work it out with a truth table. Waner and Costenoble, for some reason, call this "the switcheroo law." Sadly, I haven't been able to find a proper term for it.

The statement q → p, called the converse of p → q, is not logically equivalent to it, which you can see if you lay out a truth table for both of them. Both are conditional statements, each is the converse of the other, and they give two different columns of answers. What is the same as p → q is its contrapositive, (¬q) → (¬p).

Beware arguments that assume the converse is always true! For example, just because "all sexy actors use Ultra-Blight," it does not follow that "all who use Ultra-Blight are sexy actors." If the use of the toothpaste is an effect and not a cause, you can't make yourself a sexy actor by using it, any more than you can make yourself a soldier by wearing an Army jacket.

Consider the case where both of the conditional statements p → q and q → p are true. We can get both facts into one succinct statement with a new operator, the "biconditional:"

↔ ("is logically equivalent to" or "the biconditional operator")

Here's a truth table for it:

Note that it acts as a test for whether the two atomic statements have the same truth value. If they're both true, or if they're both false--and only then--the biconditional is true.

Logic experts made up a new word to read this operator: "iff," which stands for "if and only if." Ways to phrase p ↔ q ("p iff q") in English include:

p if and only if q
p is necessary and sufficient for q
p is logically equivalent to q