Aptitude skill lesson
Logical reasoning: skill lesson
Logical reasoning is the ability to take a set of statements (conditionals, quantifiers, ordering and grouping constraints), and work out exactly what they force, what they permit, and what they leave open. It is the backbone of graduate screening batteries, law and policy admissions tests, analyst and investigator selection, and the constraint sections of programming aptitude tests. The same discipline runs through eligibility policy, contract clauses, access-control rules and any specification written as if-then. Practice here is device-local: no account, and nothing leaves this device unless you export it.
Published · last reviewed
Applies to Novus Learn 0.1.0
What changed, and when
- , Replaced the body of all 27 non-judgement lessons with construct-specific material: six worked examples each, carrying the actual arithmetic, the actual inference or the actual procedure, plus expanded objectives, practice tips and glossary. Every 'Related Learn topics' link now points at a real page on this site rather than a generic search. The eight workplace-judgement lessons are unchanged.
- , Rebuilt the lesson page around a sticky contents rail, per-section links, previous/next lesson navigation, and two graded checkpoints drawn from the open practice bank.
- , Repaired the aptitude integrations behind the lessons so each one links to skill-specific practice instead of the unscoped fixture engine.
- , Published one lesson for each of the 35 aptitude skill constructs: objectives, worked examples, practice tips, glossary, Learn topic links, and sources.
These 35 skill lessons are authored and revised as one set, so they share one revision history rather than 35 identical dates.
Objectives
Copy link- Translate everyday phrasing (only if, unless, no A is B, none but) into a single arrow of the form antecedent implies consequent, and get the direction right every time.
- Derive the contrapositive of any conditional and use it to reason backwards from a denied consequent.
- Name and reject the two invalid conditional moves, affirming the consequent and denying the antecedent, in items designed to make both feel natural.
- Solve a linear ordering item to a single arrangement by placing the most restrictive block first and eliminating cases exhaustively.
- Answer a grouping item by identifying who is ruled out, not just who is possible, and distinguish must be true from could be true.
- Apply quantifier rules correctly: recognise that some means at least one and does not imply not all, and that two overlapping some statements never guarantee a third overlap.
Checkpoint: does the idea land?
Copy linkBefore the worked examples, check that the objectives above actually landed.
Two questions from the open practice bank, answered here and scored on this device. Untimed, ungraded, and not added to your practice history, the full bank is where attempts are recorded.
Examples
Copy linkOne conditional, four moves, two of them wrong
Take a warehouse rule: if a pallet is flagged by the scanner, then it is inspected before dispatch. Write it as F implies I. Four things can now happen. You learn a pallet was flagged: F is true, so I is true: inspected. That is modus ponens and it is valid. You learn a pallet was not inspected: not-I, so not-F. It was not flagged. That is modus tollens, equally valid, and it is the move most candidates never reach for. Now the two traps. You learn a pallet WAS inspected and conclude it must have been flagged. Invalid: the rule says nothing about why else a pallet might be inspected: random audit, a customer complaint, a damaged corner. That is affirming the consequent. You learn a pallet was NOT flagged and conclude it was not inspected. Invalid for the same reason; that is denying the antecedent. Both feel right because in ordinary conversation people say if when they mean if and only if. Assessment items exploit exactly that slip, so the fix is mechanical: the instant you meet a conditional, write F implies I on one line and its contrapositive not-I implies not-F underneath. Those two lines are everything the statement gives you. Anything else on the page is a distractor.
Chaining conditionals, then running the chain backwards
Three statements from a release policy. If the build fails, the deploy is blocked. If the deploy is blocked, the release date slips. If the release date slips, the client is notified. Write them: B implies D, D implies S, S implies N. Chaining forwards is easy: a failed build guarantees a notified client. The item almost never asks that. It says: the client was not notified. What follows? Take contrapositives and chain them the other way: not-N implies not-S, not-S implies not-D, not-D implies not-B. So the build did not fail, the deploy was not blocked, and the date did not slip. All three follow. Now the near-miss version, and the one candidates get wrong: the client WAS notified. What follows? Nothing about the build. The chain only runs one way, and a client can be notified for reasons the three statements never mention. Answer: none of the above must be true. A useful habit for chain items is to draw the arrows on one line, B implies D implies S implies N, and remember that you can travel left-to-right when you are given a truth and right-to-left when you are given a falsehood, never the reverse.
The four phrases that flip the arrow
Most lost marks in conditional items come from translation, not from reasoning. Only if introduces the consequent: you may board only if you hold a boarding pass means board implies pass. It does NOT mean a pass gets you on the plane. The pass is necessary, not sufficient, and you still need the gate to be open and your name off the no-fly list. Unless is read as if not: unless you check in you cannot board is not-check-in implies not-board, whose contrapositive is board implies check-in, again the check-in is necessary. None but staff may enter is enter implies staff. No temporary badge grants server-room access is temporary badge implies not server-room access. Compare all four with a plain sufficient condition: swiping a manager card opens the door is card implies open, and here the card really is enough. The discipline is to ask one question of every clause. Is this thing the guarantee or the requirement? A guarantee sits at the tail of the arrow, a requirement sits at the head. Get that one decision right and the rest of the item is bookkeeping.
An ordering item, solved to a single arrangement
Five people present in five consecutive slots, one each: Aisha, Ben, Chen, Dana, Eli. Constraints: (1) Ben presents in the slot immediately before Dana. (2) Aisha is neither first nor last. (3) Chen presents at some point before Ben. (4) Eli is not immediately before or immediately after Dana. (5) Dana is not last. Start with the most restrictive item, the Ben-Dana block, and test each placement. Slots 1-2 is impossible: Chen must come before Ben and there is no slot before 1. Slots 4-5 is impossible because Dana would be last, breaking (5). Slots 3-4: Chen takes 1 or 2, and Aisha must take 2 because she cannot be first or last, so Chen is 1 and Eli is forced into 5, but 5 is immediately after Dana in slot 4, breaking (4). Dead. Slots 2-3: Chen must be 1. Aisha and Eli take 4 and 5, and since Aisha cannot be last, Aisha is 4 and Eli is 5. Check (4): Dana is in 3, the neighbouring slots are 2 and 4, and Eli is in 5: fine. The order is Chen, Ben, Dana, Aisha, Eli, and it is the only one. Two habits made that quick. First, place the block before the singletons; a two-slot block only has four possible positions in a five-slot line, so it does most of the elimination for you. Second, when a case dies, write down which constraint killed it. Later questions in the same set often ask what happens if one rule is dropped, and your dead-case notes answer them instantly.
A grouping item where the real answer is who is excluded
Exactly three of six people are picked: Farah, Gus, Hana, Ivo, Jo, Kit. Rules: (1) If Farah is picked, Gus is not. (2) Hana is picked only if Ivo is. (3) At least one of Gus and Jo is picked. (4) Kit and Ivo are never both picked. The question: if Hana is picked, which of the following must be true? Work it. Rule (2) is Hana implies Ivo, so Ivo is in. That is two of the three seats. Rule (4) then puts Kit out. The third seat goes to Farah, Gus or Jo. Suppose it went to Farah: then neither Gus nor Jo is picked, breaking (3). So the third seat is Gus or Jo, and both of those complete a legal team. Check Gus: rule (1) is vacuous because Farah is out, (2) holds, (3) holds, (4) holds. Check Jo: same. So the answer is not who joins Hana; that is genuinely open. The answer is that Farah is NOT picked, and it must be true in every legal arrangement. Grouping items are built around that distinction. Could be true means one arrangement exists; must be true means no counter-arrangement exists. The fastest way to kill a must-be-true option is to build a single legal team that violates it, which is why sketching two complete valid teams before reading the options is usually faster than reasoning about the options one at a time.
Some, all and none: the overlap nobody gave you
Two premises: all compliance officers have completed the ethics module, and some people who completed the ethics module work in the Leeds office. Does it follow that some compliance officers work in Leeds? No, and the diagram shows why. Draw a big circle for ethics-module completers. Compliance officers sit entirely inside it. Leeds staff overlap it somewhere, but nothing places that overlap on top of the compliance officers, so a world where every Leeds completer is a lab technician satisfies both premises and refutes the conclusion. Two overlapping particular statements never chain. Now three rules worth memorising. First, some means at least one and is silent about the rest, so the premise some invoices are overdue does not rule out all of them being overdue, though it does not establish that either. An option reading all invoices are overdue is therefore could-be-true rather than must-be-true, and a candidate who takes some to mean some but not all will wrongly mark it impossible. Second, some does convert: if some completers are Leeds staff then some Leeds staff are completers, and that is valid. All does not convert. All compliance officers are completers tells you nothing about most completers. Third, a valid chain needs a universal: all A are B plus no B are C gives no A are C, and that one is airtight. When an item mixes quantifiers, sketch three circles before you read a single answer option; the arrangement that satisfies the premises while breaking the conclusion is usually visible in about ten seconds.
Checkpoint: can you apply it?
Copy linkNow apply it. These come from a later section of the bank, so they are not more of the same.
Two questions from the open practice bank, answered here and scored on this device. Untimed, ungraded, and not added to your practice history, the full bank is where attempts are recorded.
Practice tips
Copy link- Notate before you reason. Every conditional gets two lines on your paper, the statement and its contrapositive, written in symbols. Items that look like paragraphs collapse to four or five arrows, and the arrows are what the question is actually about.
- Underline only if, unless, none but and no as you read. Those four phrasings cause more errors than every inference rule combined, and they are the cheapest thing on the page to get right.
- On ordering sets, draw a numbered row of slots and place the largest fixed block first. Keep a note of which constraint killed each dead case; follow-up questions in the same set reuse them.
- On must-be-true options, attack rather than verify. One complete legal arrangement that breaks the option kills it outright, and building one is usually faster than proving the option holds everywhere.
- Train the invalid moves deliberately. Write out an affirming-the-consequent item and a denying-the-antecedent item of your own each session until the shape of them is instantly recognisable under time pressure.
- Work untimed until you can state which rule licensed each step, then add the clock. Attempts are stored on this device only, so a slow, fully-narrated first pass through a constraint set costs nothing but your own time.
Glossary
Copy link- Antecedent and consequent
- In a conditional if P then Q, P is the antecedent and Q is the consequent. Getting the two the right way round during translation is where most conditional items are won or lost.
- Contrapositive
- For P implies Q, the statement not-Q implies not-P. It is always logically equivalent to the original, which is what lets you reason backwards from a denied consequent.
- Modus ponens
- The valid inference from P implies Q together with P to the conclusion Q. The most common inference in assessment items, and the safest.
- Modus tollens
- The valid inference from P implies Q together with not-Q to the conclusion not-P. Under-used by candidates, and the move most backwards-chaining items are built around.
- Affirming the consequent
- The invalid move from P implies Q together with Q to the conclusion P. It is tempting because ordinary speech often means if and only if when it says if.
- Necessary condition
- Something that must hold for an outcome to occur, but does not by itself bring it about. Signalled by only if, unless, none but, and required for.
- Sufficient condition
- Something whose presence guarantees the outcome. Signalled by a plain if, whenever, or any time that. A condition can be necessary, sufficient, both, or neither.
- Must be true versus could be true
- Must be true holds in every arrangement the constraints permit; could be true holds in at least one. Nearly every grouping and ordering answer option is one of these two, and mistaking which is being asked costs the mark.
Sources
Copy link- Worked items written for Novus Learn from standard introductory logic: conditional translation, the contrapositive, categorical syllogisms, and linear ordering under constraints. Every puzzle above was solved and checked for a unique answer before publication.
- Terminology checked against the public Wikipedia articles 'Modus ponens', 'Modus tollens', 'Contraposition', 'Affirming the consequent' and 'Syllogism'. Definitions only; all items are original.
- Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
- Public educational framing only: not affiliated with any official exam board, publisher or employer, and no copyrighted test item is reproduced.
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