🧭Logic & Proof

Archivist Lumen keeps the Mirror Library because a mirror shows you a statement back to front, and back to front is exactly where most reasoning goes wrong.

A conditional is a sentence of the shape "If ..., then ...". The part after If is the hypothesis; the part after then is the conclusion.

If a shape is a square, then it is a rectangle. hypothesis: a shape is a square conclusion: it is a rectangle

Neither half keeps the words "If" or "then" — those two words are only signposts marking where each half starts. Notice too that the conclusion usually opens with "it" or "they", because it is still talking about the same shape.

Most sentences in mathematics are conditionals wearing ordinary clothes. "All squares are rectangles" and "Vertical angles are congruent" say nothing about "if", but each one describes a condition and what follows from it, so each one can be rewritten:

All squares are rectangles. -> If a shape is a square, then it is a rectangle. Vertical angles are congruent. -> If two angles are vertical angles, then they are congruent.

Rewriting first is a habit worth building, because the three statements in the next section can only be built once you can see the two halves clearly.

From one conditional you can build three more. Every one of them is made by doing something to the two halves — swapping them, negating them, or both:

statement If p, then q. converse If q, then p. (swap) inverse If not p, then not q. (negate) contrapositive If not q, then not p. (swap AND negate)

Take "If a shape is a square, then it is a rectangle":

converse If a shape is a rectangle, then it is a square. inverse If a shape is not a square, then it is not a rectangle. contrapositive If a shape is not a rectangle, then it is not a square.

Now the important part, and the whole reason this topic exists: these four statements do not all have to be true together. The original is true — every square really is a rectangle. The converse is false: a 2 by 5 rectangle is nothing like a square. The inverse is false for the same reason. The contrapositive is true.

That pattern is not a coincidence. It always holds:

• ◆A statement and its contrapositive always share a truth value. Both true, or both false — always.
• ◆A converse and an inverse always share a truth value with each other, and neither one has to agree with the statement.

So if someone hands you a true statement and asks what else must be true, the answer is its contrapositive, and only its contrapositive. Careful with a statement that already has a "not" in it: negating "is not divisible by 2" gives "is divisible by 2", because two negatives cancel.

A conditional is false the moment a single case fits the hypothesis and breaks the conclusion. That case is a counterexample, and one is enough — you never need a second.

"If a whole number is even, then it is divisible by 4." Counterexample: 6 is even and is not divisible by 4. So the statement is false.

Two traps sit here. The first: an example that fits *both* halves is not a counterexample — 12 is even and divisible by 4, which supports the claim rather than breaking it. The second: "true" does not mean "true most of the time". A conditional is true exactly when no counterexample exists at all.

When the reasoning runs forwards, two laws do the work.

• ◆Law of detachment. If "If p, then q" is true and p is true in a particular case, then q is true in that case. Given "If a quadrilateral is a square, then it is a rhombus" and "Quadrilateral Q is a square", conclude "Quadrilateral Q is a rhombus".
• ◆Law of syllogism. If "If p, then q" and "If q, then r" are both true, then "If p, then r" is true. The middle idea drops out and the chain joins end to end.

And two shapes of argument look like those laws but are not, so the honest answer is no valid conclusion:

• ◆Knowing the conclusion holds tells you nothing about the hypothesis. "Quadrilateral Q is a rhombus" does not make it a square — that is the converse, and the converse need not be true.
• ◆Knowing the hypothesis is false tells you nothing either. "Quadrilateral Q is not a square" does not make it a non-rhombus — that is the inverse.

A proof is a list of statements where every line after the first says why it is allowed. The reason column is not decoration: a line with no reason is a guess.

Only four kinds of thing may ever appear in that column — the given information, a definition, a postulate (something accepted without proof), or a theorem already proved. In an algebraic proof the reasons are the properties of equality:

• ◆Addition / Subtraction Property of Equality — the same amount was added to, or taken from, both sides.
• ◆Multiplication / Division Property of Equality — both sides were multiplied or divided by the same non-zero number.
• ◆Distributive Property — a bracket was multiplied out, as in 3(x + 4) = 3x + 12.
• ◆Combine like terms — nothing was done to the equation at all; two like terms on one side were merged.
• ◆Substitution Property — a quantity was replaced by something already known to equal it.
• ◆Symmetric Property — the two sides changed places, a = b becoming b = a.
• ◆Reflexive Property — a = a, which sounds empty but is exactly how a shared side enters a proof.
• ◆Transitive Property — a = b and b = c give a = c.

In a geometric proof the reasons are the definitions and postulates of the Geometry Kingdom:

• ◆Definition of midpoint — the midpoint of AB gives AM = MB.
• ◆Definition of an angle bisector — a bisector of angle ABC gives angle ABD = angle DBC.
• ◆Segment Addition Postulate — if B is between A and C, then AB + BC = AC.
• ◆Angle Addition Postulate — if D is inside angle ABC, then angle ABD + angle DBC = angle ABC.
• ◆Linear Pair Postulate — two angles forming a linear pair are supplementary.
• ◆Definition of supplementary / complementary angles — supplementary angles add to 180°, complementary angles add to 90°.
• ◆Vertical Angles Theorem — vertical angles are congruent.

The near misses matter more than the list. "Definition of vertical angles" is what lets you *call* two angles vertical; the "Vertical Angles Theorem" is what lets you say they are equal. The Linear Pair Postulate makes two angles supplementary; the definition of supplementary is what turns that into "adds to 180°". Read the line on the blank, ask what new fact it claims, and pick the rule that hands you exactly that fact and nothing more.

A definition is special: it is a conditional whose converse is also true. "A polygon is a triangle" and "a polygon has exactly three sides" imply each other in both directions, so the pair folds into one sentence using if and only if:

A polygon is a triangle if and only if it has exactly three sides.

That sentence is a biconditional. It is allowed exactly when the statement and its converse are both true, so the test is always the same: write the converse, then hunt for a counterexample to it. If you find one, no biconditional. "If a shape is a square, then it is a rectangle" is true but reverses badly, so it stays a one-way conditional and is not a definition of anything.

How the questions here ask you to answer. Every problem in this skill is multiple choice — nobody types a proof into a box. You click a sentence, and the options are built from the mistakes people actually make, so read all of them before choosing:

• ◆Asked for the hypothesis or conclusion, you pick the half as it stands, with no "not" added and no "If" or "then" attached.
• ◆Asked for the converse, inverse or contrapositive, the wrong options are the other two forms and one where only half was negated — check both moves, the swap and the negation.
• ◆Asked whether a statement is true, the false options carry a reason with them. The correct false option names a genuine counterexample; an option that names a case fitting both halves is supporting the statement, not breaking it.
• ◆Asked what follows from premises, "No valid conclusion can be drawn" is on every list, and it is the right answer whenever the argument runs a conditional backwards or starts from a false hypothesis.
• ◆Asked for the reason justifying a step, a proof is printed with each line numbered and its reason beside it; one reason is replaced by a ?, and you pick what belongs there.

Angles are written in degrees with the ° sign, an angle is named "angle ABC" or "angle 1", and a segment from A to B is written AB.