Conditionals ACET Reviewer: Lesson and Practice

TEACHER ABI ACET LOGICAL REASONING

Conditionals

Translate if, only if, if and only if, and unless into rules you can actually use.

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Conditionals

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A conditional is a one-way rule

Start with ordinary language:

Rule: If a parcel is marked fragile, then it receives special handling.Fact: Parcel K is marked fragile.Conclusion: Parcel K receives special handling.

This follows. The condition happened, so the result must happen.

We can shorten the rule only after its meaning is clear:

P = “the parcel is fragile”    Q = “the parcel receives special handling”
If P, then Q: P → Q

The arrow points from the sufficient condition to the necessary result. P is enough to guarantee Q. Q is required whenever P occurs.

Do not read the arrow backward. Special handling does not prove that a parcel was marked fragile; there may be other reasons for special handling.

DO IT FAST

Use TRIGGER → RESULT → CONTRAPOSITIVE

1. TRIGGER: Identify what comes immediately after “if,” or what the sentence says is sufficient.
2. RESULT: Identify what the trigger guarantees.
3. CONTRAPOSITIVE: Reverse both parts and negate them: if not result, then not trigger.

P → Q    is equivalent to    not Q → not P

Never merely reverse it. Q → P is the converse, and it is not guaranteed unless the rule works in both directions.

WORKED EXAMPLES

Six forms you must recognize

1. If P, then Q

If Mara joins the field trip, she submits a consent form.Mara joins the field trip.Therefore, Mara submits a consent form.

The trigger is present, so the result follows.

Valid: P → Q; P; therefore Q.

2. The contrapositive

If a manuscript is accepted, the editor sends a contract.The editor did not send a contract for Manuscript R.Therefore, Manuscript R was not accepted.

No result means no trigger. This is the valid contrapositive.

Valid: P → Q; not Q; therefore not P.

3. The converse trap

If a runner wins the regional race, the runner receives a medal.Tomas received a medal.Therefore, Tomas won the regional race.

Medals may also be awarded for second place, service, or another event.

Invalid: Q does not prove P.

4. “Only if” introduces a requirement

A student may enter the laboratory only if the student wears goggles.

Wearing goggles is necessary for entry. Translate from the thing being restricted:

enter laboratory → wear goggles

“A only if B” means A → B. It does not mean B → A; goggles alone may not be enough for entry.

5. “If and only if” works both ways

The archive opens after hours if and only if the director authorizes access.opens after hours → authorized    and    authorized → opens after hours

Here, each condition is both necessary and sufficient.

Either direction may be used.

6. “Unless” means “if not”

The outdoor program will continue unless lightning is detected.

Read it as: if lightning is not detected, the program will continue.

not lightning → continue

An equivalent form is: if the program does not continue, lightning was detected.

GUIDED TRY

Translate before you answer

A visitor may enter the conservation room only if a curator is present.No curator is present.Can the visitor enter?

1. What action is restricted? Entering the room.
2. What is required? A curator being present.
3. If the requirement is absent, what happens to the action?

No. “Enter only if curator present” means enter → curator present. Its contrapositive is no curator present → cannot enter.

COMMON TRAPS

Watch the direction of the rule

Affirming the consequent: P → Q and Q do not prove P.

Denying the antecedent: P → Q and not P do not prove not Q.

Misreading “only if”: A only if B means A → B.

Weakening “if and only if”: It supplies two directions, not one.

Negating carelessly: The contrapositive reverses and negates both parts.

Forgetting alternatives: Q may happen for a reason other than P.

FIVE-FORM SKILL CHECK

Do you need Foundations or Core Practice?

CHOOSE YOUR PRACTICE

Build the skill in stages

Foundations

Direct rules, arrows, and contrapositives.

Core

Only if, unless, and necessary conditions.

ACET Transfer

Chained conditions and dense wording.

FRESH MASTERY CHECK

Can you decide without the scaffolding?

Score 5/5 to verify this competency. If not, a fresh set loads.

QUICK ANSWERS

Conditionals FAQ

Does “if P, then Q” mean P causes Q?

Not necessarily. It states a logical guarantee, not always a physical cause. The test asks what follows from the rule.

What is the difference between necessary and sufficient?

A sufficient condition is enough to produce the result. A necessary condition is required for the result, but may not be enough by itself.

Why is the contrapositive valid?

If P cannot occur without Q, then finding no Q rules out P. That is why P → Q and not Q → not P express the same restriction.

What exactly does “only if” mean?

It introduces a requirement. “A only if B” means whenever A happens, B must be present: A → B.

Does “unless” always mean “if not”?

In standard test logic, “A unless B” is read as “if not B, then A.” Check whether the sentence includes other qualifiers.

Should I draw arrows during the ACET?

Use short arrows for dense conditions, especially only if, unless, and chains. For a simple rule, translating mentally may be faster.

RELATED LOGICAL REASONING COMPETENCIES

Continue the reasoning pathway

Statements and conclusionsNecessary and sufficient conditionsCertainty levelsSyllogisms and quantifiersValid and invalid deductionsTimed mixed reasoning

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