Logical Reasoning: Conditional Logic and Quantifiers
The engine under both scored LR sections: arrows from sufficient to necessary, the one valid transformation, the two invalid ones, and quantifier circles.
16 topics · 69 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Sufficient and Necessary Conditions
- How it worksHow does a conditional statement draw as an arrow running from the sufficient condition to the necessary one?
- How it worksWalk me through deciding which half of 'If it rains, the game is cancelled' is sufficient and which is necessary.
- Common mistakeWhy isn't the necessary condition just whichever event happens first in time?
- Worked exampleDraw 'Every member who votes must have paid dues' as an arrow and read off what paying dues fails to guarantee.
Your turneasy1 of 3
The bylaws of the Hartle Society state that every member who votes at the annual meeting must have paid the 96-dollar annual dues. Which one of the following must be true according to that bylaw?Sufficient-Condition Indicators
- How it worksHow does the word 'whenever' tell you which idea gets the tail of the arrow?
- Common mistakeWhy isn't 'when the alarm sounds, the doors lock' a claim about timing at all?
- Worked exampleWalk me through translating 'Any student who enrolls receives a locker' into a single arrow.
- Going furtherWhat changes when a sentence buries its sufficient half at the end, as in 'The doors lock if the alarm sounds'?
Your turneasy1 of 3
College notice: any student who enrols in the 14-week language course receives a locker. Which one of the following is the correct translation of the notice into a single conditional?Necessary-Condition Indicators
- How it worksHow do 'then', 'only', 'must', 'requires' and 'necessary' all point at the head of the same arrow?
- Common mistakeWhy doesn't satisfying a necessary condition make the other thing happen?
- Worked exampleDraw 'A license is necessary for driving legally' and label which side is sufficient and which is necessary.
- Going furtherWhat happens to the picture when one sentence names two necessary conditions at once?
Your turneasy1 of 3
Gallery sign: you must hold a timed ticket for one of the 10 daily entry slots in order to enter the exhibition. Which one of the following is consistent with the sign?The Contrapositive
- How it worksHow does flipping an arrow end to end and negating both terms produce something that must be true?
- How it worksWalk me through taking the contrapositive of 'If Maya attends, Jonah stays home'.
- Worked exampleFind the contrapositive of 'Only licensed drivers may rent a car' and test it on a driver with no license.
- Common mistakeWhy isn't the contrapositive a brand new fact rather than the original fact said backwards?
- Going furtherWhat happens to a contrapositive when the original sentence already contains a 'not' in it?
Your turneasy1 of 3
Rule: if Maya attends the 12-person committee meeting, Jonah stays home. Which one of the following is the contrapositive of that rule?The Invalid Converse: Affirming the Consequent
- How it worksHow does affirming the consequent smuggle a second, unstated arrow into an argument?
- Worked exampleWalk me through spotting the reversal error in 'All thieves lie; Sam lied, so Sam is a thief'.
- Going furtherWhat changes when the original rule genuinely runs both ways, so the reversal happens to be true anyway?
- Common mistakeWhy doesn't 'every winner trained hard' let you conclude that everyone who trained hard won?
Your turneasy1 of 3
Every one of the 16 winners at the regional meet trained at least 20 hours a week. Priya trained 20 hours a week, so Priya was one of the winners. Which one of the following identifies the error in this reasoning?The Invalid Inverse: Denying the Antecedent
- How it worksHow does denying the antecedent differ from the contrapositive it looks almost identical to?
- Worked exampleWalk me through why 'If Ana wins, dues rise; Ana lost, so dues hold steady' is invalid.
- Common mistakeWhy isn't losing the sufficient condition the same as losing the necessary one?
- Going furtherWhat happens to the inverse error when the rule is really an 'if and only if'?
Your turneasy1 of 3
Rule: if the water main freezes, the 23 houses on Bell Lane lose pressure. The main did not freeze this winter. Which one of the following explains why it does not follow that the houses kept their pressure?Unless, Until, and Without
- How it worksHow does a translation machine turn the word 'unless' into 'if not' without changing the meaning?
- Worked exampleWalk me through turning 'The picnic happens unless it rains' into an arrow, then into its contrapositive.
- Worked exampleDraw 'You cannot leave until the bell rings' and read off what the bell does not promise.
- Common mistakeWhy isn't 'unless' just a longer way of writing 'if'?
- Going furtherWhat changes when 'without' takes the place of 'unless', as in 'no entry without a badge'?
Your turneasy1 of 3
Invitation: no one is admitted to the 114-guest reception unless they have sent an RSVP. Which one of the following must be true?Only versus The Only
- How it worksHow does 'Only seniors may apply' put seniority at the head of the arrow and applying at its tail?
- Going furtherWhat happens when 'only if' sits in the middle of a sentence instead of at its start?
- Worked exampleWalk me through diagramming 'The only tickets left are balcony tickets' and naming the sufficient term.
- Common mistakeWhy isn't 'only if' the same thing as 'if' with a spare word bolted on the front?
Your turnhard1 of 3
Two notices are posted at a theatre. Notice A: only patrons over 19 may buy a seat in the gallery. Notice B: the only seats left are the 59 gallery seats. Which one of the following correctly reports what each notice makes sufficient?Conditional Chains and Transitivity
- How it worksHow do arrows link head to tail so that two rules hand you a third one for free?
- Worked exampleWalk me through chaining 'All poets read Keats' with 'Anyone who reads Keats owns a notebook'.
- Worked exampleFind every valid inference in the chain sculptor to artist, artist to dreamer, dreamer to insomniac.
- Common mistakeWhy doesn't a chain hold when the shared term sits on the wrong end of the second arrow?
- Going furtherWhat happens to a long chain when you take the contrapositive of the whole thing at once?
Your turneasy1 of 3
Two rules hold in the 31-member writers' circle: all poets read Keats, and anyone who reads Keats owns a notebook. Which one of the following must be true?Biconditionals: If and Only If
- How it worksHow does 'if and only if' weld a conditional and its reversal into a single rule?
- Worked exampleDraw 'You pass if and only if you score 60' and show exactly what a score of 59 forces.
- Common mistakeWhy isn't every sentence containing 'only if' secretly a biconditional?
- Going furtherWhat changes about a biconditional when you negate both of its halves at the same time?
Your turneasy1 of 3
Course rule: a student passes if and only if the student scores at least 67 on the final. Erez scored 50 on the final. Which one of the following must be true?Negating a Conditional
- How it worksHow does denying a conditional produce a case where the tail is true and the head is false?
- Common mistakeWhy isn't the denial of 'if A then B' the statement 'if A then never B'?
- Worked exampleWalk me through negating 'Every applicant who interviews gets an offer' using a single counterexample.
- Going furtherWhat happens to the negation when the rule being denied is a chain of three linked arrows?
Your turneasy1 of 3
A recruiter states: every applicant who interviews at the firm gets an offer. In the last cycle 62 applicants interviewed. Which one of the following would show the recruiter's statement to be false?Quantifiers: All, Most, Some, None, Few
- How it worksHow do all, most, some and none draw as overlapping circles with different amounts shaded in?
- How it worksWalk me through why 'most' has to mean more than half and never merely 'quite a few'.
- Common mistakeWhy doesn't 'few students cheated' carry the same content as 'some students cheated'?
- Worked exampleDraw 'Most jurors were persuaded' and 'No juror slept' as circles and read off what overlaps.
- Going furtherWhat changes about the shading when a quantified claim is negated, so that 'all' becomes 'not all'?
Your turneasy1 of 3
A report says that some of the 75 jurors in the pool were persuaded by the opening statement. Which one of the following is consistent with the report?Valid Quantifier Inferences
- How it worksHow does 'most A are B' together with 'most A are C' force at least one B to be a C?
- Worked exampleWalk me through why 'most cellists read music' plus 'most cellists tour' gives a touring reader.
- Worked exampleFind what follows from 'All violinists are musicians' plus 'Some violinists are teenagers'.
- Common mistakeWhy isn't 'most' transitive, so that most A are B and most B are C settles nothing about A and C?
Your turnmedium1 of 3
Of the 54 cellists in the ensemble, 38 read music fluently and 38 tour internationally. Which one of the following must be true?Why Some Statements Do Not Chain
- How it worksHow does one dot sitting in an overlap show that two 'some' circles need never touch?
- Worked exampleDraw the counterexample that kills 'some lawyers golf, some golfers cheat, so some lawyers cheat'.
- Common mistakeWhy doesn't adding a third 'some' claim rescue a chain that the first two could not build?
- Going furtherWhat has to be bolted onto a 'some' statement before it can legally link to another one?
Your turneasy1 of 3
Of the 157 birds recorded at the reserve, some fly. Some flying things are insects. A student concludes that some of the birds recorded are insects. Which one of the following best explains what has gone wrong?Combining Conditionals and Quantifiers
- How it worksHow does an arrow hook onto a circle so that 'some A are B' plus 'all B are C' reaches C?
- Worked exampleWalk me through combining 'Some jurors are engineers' with 'All engineers studied calculus'.
- Going furtherWhat happens when the rule you want to attach points into the quantified group instead of out of it?
- Common mistakeWhy doesn't 'some A are B' plus 'all A are C' establish that all B are C?
Your turneasy1 of 3
Of the 15 jurors empanelled, 5 are engineers, and every engineer studied calculus. Which one of the following must be true?Translating Ordinary Language into Conditionals
- How it worksHow do you locate the sufficient term when a sentence contains no 'if' anywhere in it?
- Worked exampleWalk me through translating 'No cat that hunts at night sleeps through the morning' into one arrow.
- Worked exampleDraw 'A recipe fails whenever the oven runs cold' and check your translation against a failed recipe.
- Common mistakeWhy isn't 'people who exercise are healthy' a claim that healthy people exercise?
- Ties togetherHow does a clean translation turn a brutal inference question into three arrows and a reading?
Your turnmedium1 of 3
Field note: no cat that hunts after dark sleeps through the 6 hours of the following morning. Which one of the following is the correct translation of that note into a single conditional?AP®, SAT® and PSAT/NMSQT® are trademarks registered by the College Board (PSAT/NMSQT® jointly with the National Merit Scholarship Corporation). ACT® is a registered trademark of ACT Education Corp. GRE® is a registered trademark of ETS. LSAT® is a registered trademark of the Law School Admission Council. MCAT® is a registered trademark of the Association of American Medical Colleges. ISEE® and SSAT® are registered trademarks of their owners. None of these organizations is affiliated with, or endorses, Jentoo.