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Reasoning Foundations

If-Then and Only-If Rules

If-Then and Only-If Rules teaches a transferable conditional reasoning routine for selective and scholarship reasoning questions. Students learn the exact first move, the proof step, and the main trap to avoid before timed practice.

  • Translate if, only if, all, none and unless into usable rules.
  • Use the contrapositive when it is valid.
  • Avoid the converse and inverse traps.
Free sample lesson — reading only

Lesson overview

What this free sample teaches

Use the conditional reasoning routine to choose an answer and name the trap avoided.

Focus

  • Question family: conditional reasoning.
  • Core move: Write the rule as IF condition THEN result, mark whether it can be reversed, and check each option against the valid directions only.
  • Main trap: Treating 'if P then Q' as though it also says 'if Q then P'.

What gets tested

  • NSW/VIC/scholarship fit: NSW thinking skills, VIC verbal logic and scholarship deduction.
  • Choosing the answer that is justified by the stated information.
  • Explaining why the strongest distractor fails.

Quick guide

  • Read the final question before the options.
  • Name the reasoning family and first move.
  • Use the stated facts or relationship, not outside knowledge.

Success criteria

  • Translate if, only if, all, none and unless into usable rules.
  • Use the contrapositive when it is valid.
  • Avoid the converse and inverse traps.

teach

Teach: If-Then and Only-If Rules

Conditional reasoning is about the direction of a rule.

If-Then and Only-If Rules is a core reasoning lesson for NSW thinking skills, VIC verbal logic and scholarship deduction. The aim is to prove the answer from the question, not to guess from topic familiarity.

Conditional reasoning is about the direction of a rule. Students should treat this as a proof task: the answer must be supported by the facts, rules, data, relationship or argument in the question.

The routine is: Write the rule as IF condition THEN result, mark whether it can be reversed, and check each option against the valid directions only. This routine works across NSW, VIC and scholarship contexts because it focuses on the reasoning structure rather than the test label.

The common trap is Treating 'if P then Q' as though it also says 'if Q then P'. Strong students are often caught because the wrong answer is partly relevant. The fix is to name exactly why that option is not justified.

Reasoning family

conditional reasoningIt tells you the first move.

Proof move

Write the rule as IF condition THEN result, mark whether it can be reversed, and check each option against the valid directions only.It links the answer to the question.

Distractor trap

Treating 'if P then Q' as though it also says 'if Q then P'.It explains why a tempting option fails.
Anatomy of conditional reasoning
  1. Question clueconditional reasoning
  2. Core evidencefacts, rules, data, words or argument links supplied in the item
  3. Reasoning moveWrite the rule as IF condition THEN result, mark whether it can be reversed, and check each option against the valid directions only.
  4. Trap checkTreating 'if P then Q' as though it also says 'if Q then P'.
  5. Answer shapeThe valid direction is ... -> ..., so ...

The moveClassify the task, apply the routine, prove the answer, then reject the closest trap.

  • Translate if, only if, all, none and unless into usable rules.
  • Use the contrapositive when it is valid.
  • Avoid the converse and inverse traps.

show

Show: if-then rules worked end to end

Watch the decisions, not just the answer: what the question is really asking, how the proof is built, which option is designed to catch you, and what a deeper reading of the same answer adds.

The item is on the card above, with its options. Work down the six steps in order. Steps 1 to 3 are the thinking that happens before an option is marked; steps 4 to 6 are the same answer at two depths, and what the deeper one buys you.

Full item: one 'only if' rule, two students, four options

Question: The rule about the school badge uses the words 'only if'. Leila has a badge and Omar completed the service log. Which statement must follow?

The service log

  • A school badge is awarded only if a student completes the service log.
  • Leila has a school badge.
  • Omar completed the service log.

Which statement must follow?

A. Omar has a school badge. B. Leila completed the service log. C. Every student who completes the service log is awarded a badge. D. Leila completed the service log before any other student.

  1. Step 1 — Turn 'only if' into an arrow before looking at anybody

    Two words in the rule decide the whole item, and they are the two most often read backwards.

    The rule is: a school badge is awarded only if a student completes the service log. Before I go anywhere near Leila or Omar I have to know which way that rule points, because everything else follows from it.

    'Only if' does not mean 'if'. 'If you complete the service log, you get a badge' would promise a badge to every student who does the work. 'Only if' promises nobody anything. It sets a gate: without the service log there is no badge. It leaves completely open whether a student who did the log gets one.

    So the arrow runs from the badge to the log, and not the other way:

    • badge → completed the service log

    That is the only direction the rule licenses. It licenses one reversal and one only: no completed log → no badge, which is the same statement read from the far end. Nothing else.

    Now the two students. Leila has the badge, which is the left-hand side of the arrow, so the arrow carries her. Omar completed the log, which is the right-hand side. Arrows do not carry anybody backwards.

    • 'Only if' does not mean 'if'.The whole item is built on this and on nothing else. 'If you complete the service log, you get a badge' hands a badge to everybody who does the work; the rule as written promises nobody anything. Read the two versions aloud one after the other — they sound almost the same and mean opposite things.
    • It sets a gate: without the service log there is no badge.Worth memorising as the plain-English form of the rule: no log, no badge. It bars students; it does not admit them. Whenever a rule tells you what is impossible without something, you have been handed a gate rather than a promise, and a gate cannot be used to let anybody in.
    • Arrows do not carry anybody backwards.Leila stands on the left of badge → completed the service log and Omar stands on the right, and that is the entire difference between them. Sort the people onto the ends of the arrow before you read a single option, and two of the four choices decide themselves.

  2. Step 2 — Put each student on the arrow and see which one moves

    One rule, two students, and only one of them is standing at the end the rule can push from.

    The rule, written once: badge → completed the service log.

    • Leila: the facts give her a school badge, which is the left-hand end. The rule says anyone holding a badge completed the log, and it says so with no exceptions, so Leila completed the service log. Forced.
    • Omar: the facts give him a completed service log, which is the right-hand end. The rule says nothing at all about people at that end. It never promised that finishing the log earns anything. Omar may have a badge; he may not. Not forced.
    • The valid reversal: no completed log → no badge. Neither student is described that way, so it has no work to do here, but it is the only other statement the rule supports.

    What that rules in: B — Leila completed the service log.

    What it rules out: any claim about what completing the log leads to, and any claim about the order in which Leila did things.

    Answer: B.

    • so Leila completed the service log. Forced.The one conclusion here that needs no extra assumption. She holds a badge, and a school badge is awarded only if a student completes the service log, so her log is finished before anybody asks her. Notice that the proof runs from the result back to the requirement, which feels backwards and is not.
    • Omar may have a badge; he may not.Written as two possibilities on purpose, because that is what 'not forced' looks like on the page. The rule says nothing at all about people at that end, so both worlds are legal. An option fails here not because it is false but because the item has given you no way to decide it.
    • no completed log → no badgeIncluded because it is the one reversal that is safe, and students who have been warned about reversing rules often conclude that none of them are. Turning both ends of badge → completed the service log into negatives and then swapping them preserves the meaning exactly.

  3. Step 3 — The option that reads a gate as a promise: A

    Omar did the work the rule names. The rule still owes him nothing.

    The option most students choose: A — 'Omar has a school badge.'

    Why they choose it: it is fair. The item hands Omar the precise thing the rule talks about — he completed the service log — and everywhere outside this question, doing the required task is how people earn things. A feels like the option the item was written for.

    Which step it skips: it never asks which end of the rule Omar is standing at. 'Only if' makes the log necessary for the badge, which is badge → completed the service log. A needs the opposite arrow, and that rule was never written. Picture a school where forty students complete the log and three badges are awarded for the three best records. Badges only went to log-completers, Leila has one, Omar completed the log — every fact in the item is still true, and A is false.

    Corrected reasoning: a requirement tells you what a person must have done, given the result. It never tells you what the result will be, given the requirement. Omar has satisfied a gate, and satisfying a gate is not the same as being let through it.

    • he completed the service logWhy A is a genuine trap rather than a silly one. The item hands Omar the precise thing the rule talks about, so choosing him looks like a reward for reading carefully. Word overlap between a fact and a rule tells you the two are related; it never tells you which way the relation runs.
    • forty students complete the log and three badges are awarded for the three best recordsAn invented school, built to order. To show A is not forced you need one world where badges only went to log-completers, Leila has one, and Omar does not — and this is that world. Nothing in the item forbids it, which is the whole of what a counterexample has to prove.
    • satisfying a gate is not the same as being let through itThe sentence to carry into every conditional item. Omar completed the service log, which is what the rule requires, and the rule's job is to keep people out rather than to wave them in. When you catch yourself thinking 'but that is not fair', you have almost certainly found an only-if rule doing exactly what it says.

  4. Step 4 — The answer, and why it is forced

    Translate the rule, place both students on it, then reject A on direction. Three paragraphs.

    B must follow. 'Only if' makes the service log a requirement for the badge, so the rule is badge → completed the service log. Leila has a school badge, which puts her at the badge end, and the rule carries her to the other end with no exceptions, so Leila completed the service log.

    A is the closest wrong option. Omar completed the service log, which is the far end of the same arrow, and a rule cannot be run backwards. 'Only if' never promised that finishing the log earns a badge: a school could award three badges among forty completed logs and every fact in the item would still hold. Omar may have a badge, but the item cannot show it.

    That is enough to be right: it converts 'only if' into a direction before touching either student, it names which end of the rule each student is standing at, and it rejects A for the specific reason that the arrow does not run that way.

    • the rule carries her to the other end with no exceptionsThe proof runs from the badge she is holding back to the service log she must have completed. Conditional items usually feel forwards — do this, get that — and here the only forced conclusion travels the other way. When a student is described by a result, hunt for the requirements that result guarantees.
    • Omar may have a badge, but the item cannot show it.The rejection is careful not to overshoot. Writing 'Omar does not have a badge' would be a second mistake in the opposite direction, because the rule is silent about him and silence is not denial. The safe charge is that the item cannot decide, and that charge cannot be argued with.

  5. Step 5 — Going deeper: state the rule as a ban, and account for C and D

    The rule does not move and neither does B. What moves is the form the rule is written in, and how many mistakes the three wrong options really are.

    B must follow, and the cleanest way to see it is to say what the rule forbids rather than what it allows. 'A school badge is awarded only if a student completes the service log' rules out exactly one kind of student: a badge-holder with an unfinished log. That is the entire content of the rule. Leila is a badge-holder, so she cannot be that student, so her log is finished.

    Reading a conditional as a ban is safer than reading it as an arrow, because a ban cannot be pointed the wrong way. There is no version of 'no badge-holder has an unfinished log' that manages to say anything about Omar. He is not described as a badge-holder, and the ban only ever speaks about badge-holders.

    The ban also explains the two options nobody bothers to reject. C — 'every student who completes the service log is awarded a badge' — is the rule turned into a promise, which is A's error stated as a general rule instead of about one person. A student who rejects A and then accepts C has caught the mistake in the individual case and missed it in the general one. D — 'Leila completed the service log before any other student' — keeps the direction right and adds timing, but the rule fixes an order of requirement, not an order of events, and nothing in the three facts says when anybody did anything.

    So the four options are not four ideas. B applies the ban to a badge-holder. A and C are the same reversal offered twice, once about Omar and once about everybody. D reads a requirement as a timeline. Only one option uses the rule for what it actually does, which is to make a single combination impossible.

    Nothing above overturns Step 4: the rule is the same rule and B is still the answer. The gain is in the form. Written as a ban — no badge-holder has an unfinished log — the rule cannot be reversed by accident the way an arrow can, and once C and D are both accounted for the three wrong options turn out to be two mistakes rather than three. Converting 'only if' correctly and refusing the nearest rival is enough to be right here. The reading that makes the reversal unavailable in the first place, and can name the single misreading the item quietly offered twice, is the one that is still right on the next conditional.

    • rules out exactly one kind of student: a badge-holder with an unfinished logStep 4 wrote the rule as an arrow; this writes it as a ban, and a ban has no direction to get wrong. Everything the rule contains sits in that one forbidden combination, and Leila is disqualified from it simply by holding a badge. Try converting every conditional you meet into the pair it makes impossible.
    • A student who rejects A and then accepts C has caught the mistake in the individual case and missed it in the general one.The reason C is worth writing about at all. It is A in different clothes: 'every student who completes the service log is awarded a badge' is the same reversal, stated about everybody instead of about Omar. Option sets often test one misreading twice, and the general-sounding version is the one that slips through.
    • the rule fixes an order of requirement, not an order of eventsD is the quiet option, because it keeps the direction right and only adds when. 'Leila completed the service log before any other student' invents a timeline out of a rule about what has to be true. Requirement is not chronology, however comfortably the word 'before' fits both.

  6. Step 6 — What the deeper reading buys you next time

    Read side by side: same rule, same B, and one of them holds the rule in a form that cannot be reversed.

    Enough to be rightEnough to be right next timeWhat it adds
    Converts 'only if' into badge → completed the service log.States the same rule as a ban: no badge-holder has an unfinished log.A ban has no direction, so it cannot be reversed under time pressure.
    Rejects A because the arrow does not run that way.Shows A and C are one misreading offered at two scales.It stops the student catching the error about Omar and missing it about everybody.
    Leaves C and D unmentioned.Names D as a requirement misread as a timeline.It accounts for every option instead of the loudest one.

    Translate the rule, choose B — that much is shared. Count the wrong answers each version can explain and the two come apart.

    • States the same rule as a ban: no badge-holder has an unfinished log.The arrow and the ban say the same thing, and only one of them is safe when you are tired. An arrow has two ends and can be drawn either way on a second reading; a forbidden combination has no ends at all. Rewrite the rule in the form that cannot be misdrawn.
    • Names D as a requirement misread as a timeline.Unglamorous and worth the thirty seconds. Leave C and D unmentioned and you cannot tell whether they were rejected or simply never read, and an unread option is the one that reappears next time. Saying what each remaining option got wrong is the cheapest way to prove the rule was understood rather than guessed.

compare

Compare: weak vs strong reasoning

See the difference between topic matching and proof.

The weak answer reacts to the surface topic. The strong answer proves the relationship or rule.

The whole response

Weak answer

Omar must have a school badge.

Strong answer

Only-if means badge -> completed log. Leila has the badge, so her log completion is forced; Omar may or may not have the badge.

Point-by-point breakdown3
  • First moveWeaker: Looks for a familiar word or number.Stronger: Uses the conditional reasoning routine.The routine keeps the answer tied to evidence.
  • ProofWeaker: Gives a plausible answer without a clear link.Stronger: Shows how the stated information supports the answer.Reasoning questions reward justified conclusions.
  • Trap controlWeaker: Treating 'if P then Q' as though it also says 'if Q then P'.Stronger: Names why the distractor fails.Trap awareness improves transfer to mixed timed sections.

guide

Guide: checkpoint

Do one reasoning move before independent practice.

Answer in one sentence using the lesson's sentence starter.

If a code starts with R, it must be checked twice. Code R47 starts with R. What follows?

Want feedback on your own answer? Get started to practise with instant marking.

  • Translate if, only if, all, none and unless into usable rules.
  • Use the contrapositive when it is valid.
  • Avoid the converse and inverse traps.