1
One false row, three true ones
p → q fails only when the hypothesis holds and the conclusion does not. Every other combination leaves the promise unbroken.
Lesson · Logic and Proof · Propositional logic
The Conditional and Its Relatives
p → q is false in exactly one case. Establish which, why a false hypothesis makes the whole conditional true, and how the converse, inverse and contrapositive relate — only one of the three is equivalent to the original.
p → q false ONLY when p is T and q is F
q → p the CONVERSE — not equivalent
¬p → ¬q the INVERSE — not equivalent
¬q → ¬p the CONTRAPOSITIVE — EQUIVALENT
One conditional, three relatives
Core ideas
What this lesson establishes
The conditional is the connective every theorem is stated with, and the one whose truth table looks wrong until you see why it must be that way.
2
A false hypothesis makes it true
This is vacuous truth. "Every unicorn in this room is purple" is true, because there is no unicorn here to be a counterexample.
3
Only the contrapositive is equivalent
p → q and ¬q → ¬p agree in all four rows. The converse and the inverse do not, and confusing either with the original is the most consequential error in this lesson.
The material
If, then
The definition, the reason for it, and the three statements it is confused with.
Definition · Conditional
p → q ("if p then q") is false when p is true and q is false, and true in every other case. p is the hypothesis (or antecedent) and q the conclusion (consequent).
Think of a conditional as a promise: "if you pass the exam, I will buy you dinner." The promise is broken in exactly one situation — you pass and no dinner appears. If you fail, the promise says nothing about what happens and is not broken whichever way things go. That is why both rows with a false hypothesis count as true.
Definition · Vacuous truth
A conditional whose hypothesis is false is vacuously true. "Every element of the empty set is prime" is true, and so is "every element of the empty set is composite".
Vacuous truth is not a loophole to be tolerated; it is what makes general statements behave. "Every prime greater than 100 that is even is a perfect square" has to be true, or a statement about a set could fail merely because the set is empty.
Definition · Converse, inverse, contrapositive
Given p → q: the converse is q → p, the inverse is ¬p → ¬q, and the contrapositive is ¬q → ¬p.
Theorem · Contraposition
p → q ≡ ¬q → ¬p. The converse and inverse are equivalent to each other, and neither is equivalent to the original.
p → q is false only when p is T and q is F. ¬q → ¬p is false only when ¬q is T and ¬p is F — that is, when q is F and p is T. Those are the same single row, so the two formulas are false in the same case and true in the other three.
□
Example · Why the converse is a different claim
"If n is divisible by 4 then n is even" is true. Its converse, "if n is even then n is divisible by 4", is false — take n = 6. Its contrapositive, "if n is odd then n is not divisible by 4", is true, as it must be.
Definition · Biconditional
p ↔ q ("p if and only if q") is true exactly when p and q have the same truth value. It is equivalent to (p → q) ∧ (q → p) — the statement and its converse together.
In a theorem, "if and only if" is an instruction: two proofs are owed, one in each direction. Proving only one is the most common way an otherwise correct answer is incomplete.
Interactive
Truth table
Every assignment, evaluated
Method
Working with a conditional
Identify the parts before doing anything else.
1
Find the hypothesis and the conclusion
English hides them: "q whenever p", "p only if q" and "p is sufficient for q" all mean p → q. "p if q" means q → p.
2
Ask what would falsify it
A conditional is refuted by one case with a true hypothesis and a false conclusion. If you cannot describe that case, you have not understood the claim.
3
Form the contrapositive when the hypothesis is awkward
"If n² is even then n is even" is hard to attack directly; its contrapositive "if n is odd then n² is odd" falls out in one line. “Contraposition and Contradiction” makes this a method.
4
Never assume the converse
Proving p → q gives you nothing about q → p. If you need both, say so and prove both.
Conditionals
Score 0 / 0Choose the answer you can justify.
Four forms of one statement
Let p = "n is divisible by 6" and q = "n is divisible by 3".
original p → q if 6 | n then 3 | n TRUE
converse q → p if 3 | n then 6 | n FALSE (n = 9)
inverse ¬p → ¬q if 6 ∤ n then 3 ∤ n FALSE (n = 9)
contrapositive ¬q → ¬p if 3 ∤ n then 6 ∤ n TRUE
The single value n = 9 refutes both the converse and the inverse, which is not a coincidence: they are equivalent to each other, so any counterexample to one is a counterexample to the other.
Common mistakes
1
Using the converse as if it were the originalFrom p → q and q, nothing follows about p. This is the error behind most bad reasoning from evidence, and “Rules of Inference” names it as a fallacy.
2
Finding vacuous truth absurd and rejecting itIt is a consequence of wanting p → q to be false only when the promise is broken. Any other convention makes universally quantified statements over empty sets behave badly.
3
Proving one direction of a biconditionalp ↔ q is two conditionals. An "if and only if" answer with one argument in it is half an answer, however good that half is.
Check your understanding
✓
Finish when you can write all three relatives of a conditional and say which is equivalent, without a table.
Given "if n is a multiple of 10 then n is a multiple of 5", produce the converse, inverse and contrapositive, and identify which are true — one integer settles the two that are not. The habit of forming the contrapositive is what “Contraposition and Contradiction” turns into a proof technique.
Note: p → q ≡ ¬p ∨ q is worth memorising: it converts every conditional into a disjunction and is how conditionals are eliminated when a formula is put into the normal forms of “Normal Forms and Boolean Algebra”.