Miscellaneous: Truth Tables and Loose Ends

What you'll get from this section. A handful of ideas that tie the course together: truth tables — a mechanical way to settle any claim we've made — the precise sense in which $P \Rightarrow Q$ means "you can't have $P$ without $Q$", the surprising-but-important idea of vacuous truth, and a couple of other faces of the conditional.

Truth tables

A truth table lists every possible combination of truth values for the basic statements and works out the result in each case. With two statements there are four rows; with three there are eight; in general $2^n$ rows for $n$ statements.

Here are the basic connectives — "not" ($\neg$), "and" ($\wedge$), "or" ($\vee$):

"And" is true only when both parts are; "or" is true when at least one part is.

The truth table for P ⇒ Q

The headline fact: $P \Rightarrow Q$ is false in exactly one case — when $P$ is true and $Q$ is false. In all three other cases it's true.

Put another way, if you are told $P \Rightarrow Q$ holds, the combinations still on the table are

• $P$ and $Q$ (both true),
• not $P$ but $Q$,
• not $P$ and not $Q$,

and the only combination ruled out is $P$ true with $Q$ false.

"P and not Q cannot both be true"

That single forbidden row is worth stating on its own. Saying $P \Rightarrow Q$ is exactly the same as saying the situation "$P$ true and $Q$ false" never arises:

The table confirms it — the column for $\neg(P \wedge \neg Q)$ matches the column for $P \Rightarrow Q$ row for row:

This is a genuinely useful reformulation: anywhere you'd use "$P \Rightarrow Q$", you may instead use "we can't have $P$ without $Q$." It's also the fact we quietly relied on in Section 5 — that the only way an implication fails is $P$ holding while $Q$ fails. Here is its justification.

Another face: ¬P ∨ Q

The last column of that table is a third equivalent form:

"either $P$ fails, or $Q$ holds." And this isn't a new fact — it's just $\neg(P \wedge \neg Q)$ rewritten with De Morgan's law from Section 3: $\neg(P \wedge \neg Q)$ becomes $\neg P \vee \neg(\neg Q)$, which is $\neg P \vee Q$.

Vacuous truth

Look again at the two rows where $P$ is false: $P \Rightarrow Q$ is true in both. So whenever the condition $P$ fails, the implication is automatically true — regardless of $Q$. An implication with a false hypothesis is said to be vacuously true.

This feels odd at first, but it's exactly right. "If $2 + 2 = 5$, then I am the King of France" is a true statement, because its condition never holds, so it never gets the chance to be violated. More usefully in mathematics:

"For every real number $x$, if $x^2 < 0$ then $x = 7$" is true — because $x^2 < 0$ never happens, the implication is vacuously true for every $x$.

TMUA is fond of this. A "for all" claim whose condition is never met is true by default.

Truth tables as a checker

Every equivalence in this course can be settled mechanically: build the table for each side and check the columns agree. The contrapositive from Section 4, for instance:

Identical columns, so $P \Rightarrow Q$ and $\neg Q \Rightarrow \neg P$ are equivalent — exactly what we argued by hand earlier. The biconditional from Section 6 yields a tidy table too, true precisely when $P$ and $Q$ share a truth value:

When an argument in words gets slippery, a truth table is the safety net.

Common mistakes

1. Thinking a false hypothesis makes an implication false. The opposite is true: if $P$ is false, $P \Rightarrow Q$ is true (vacuously).

2. Reading $P \Rightarrow Q$ as "both $P$ and $Q$ hold". It only forbids the one combination $P$ true, $Q$ false. The other three are all allowed — in particular, $P$ being false tells you nothing about $Q$.

3. Misremembering the false row. $P \Rightarrow Q$ fails only at $P$ true, $Q$ false — no other row.

Summary

• A truth table evaluates a statement across every combination of its parts ($2^n$ rows for $n$ statements).
• $P \Rightarrow Q$ is false in exactly one case: $P$ true, $Q$ false.
• Equivalently, $P \Rightarrow Q$ is $\neg(P \wedge \neg Q)$, and also $\neg P \vee Q$.
• A false hypothesis makes an implication vacuously true.
• Truth tables give a mechanical check for any equivalence claim.

Next: Section 8 — TMUA worked examples.