Logical Reasoning curriculum

An argument is a set of claims in which some, the premises, are offered as reasons to accept another, the conclusion. "The bakery opens at seven. Its rye is popular. The owner is friendly." is three reports. "The rye sells out by nine, so arrive early" is an argument. Test: ask which claim the others are meant to support; if none is, there is no reasoning to evaluate.

An argument gives a reason for thinking a claim is true; an explanation treats the claim as accepted and gives a reason why it holds. "The lawn is wet because the sprinkler ran" explains a wet lawn nobody doubts. "The sprinkler must have run, because the lawn is wet" argues for something in doubt. The word because appears in both. Test: ask which claim is in question; the doubted one is a conclusion.

A premise indicator is a word telling you that the claim after it is offered as support. In "We should leave now, since the bridge closes at nine," since flags the reason, and "we should leave now" is the conclusion. These words are a helpful sign, not proof: because can also introduce an explanation. Test: ask whether the clause after the word is given to make you believe the other claim; if it is, you have found a premise.

A conclusion indicator signals that the claim after it is what the other claims support. In "The lot is empty and the doors are locked, so the library has closed," so introduces the conclusion. Speakers often drop these words when context makes the reasoning plain: "The library has closed. The lot is empty and the doors are locked." Test: put therefore in front of each claim and keep the one where it reads naturally.

In a deductive argument the premises are meant to guarantee the conclusion: if they are true, it cannot be false. In an inductive argument they are meant only to make it more likely. "Every ticket in this box wins, and yours came from the box, so it wins" is deductive. "Ninety of the hundred tickets checked won, so yours probably will" is inductive. Test: ask whether the arguer claims certainty or likelihood.

Two questions are separate: are the premises true, and does the conclusion follow from them? "All cats are black. Tom is a cat. So Tom is black" follows perfectly, yet its conclusion can be false because the first premise is false. Validity rules out only one combination, true premises with a false conclusion. Test: judge the link as if the premises were true, then judge the premises on their own.

The principle of charity says to choose the interpretation that makes an argument as credible as its words allow. If a neighbor says "That dog has bitten two people, so it should be leashed," read the unstated link as "a dog that has bitten people should be leashed," not the sweeping "every dog should be leashed." Test: supply the weakest missing premise that makes the step work, then judge that version.

Longer arguments stack: some claims support intermediate conclusions, which in turn support one main conclusion. In "Ticket prices rose, so attendance fell, so the theater should cut prices," every claim points toward the last one. Test: for each candidate, ask whether another claim in the passage is offered to support it and whether it supports anything further. The main conclusion is supported but supports nothing else.

An intermediate conclusion is a claim that some premises support and that then serves as a premise for a further conclusion. In "The well tested high in nitrates, so its water is unsafe, so the camp should truck in water," the middle claim is both supported and supporting. Test: if a claim has a reason behind it and also leads somewhere, it is intermediate; follow the chain to its end to find the main point.

Many arguments answer someone. The passage states a view, then argues against it: "Some say the town needs a second parking garage. But the first one is half empty most days, so a second would waste money." The opening sentence is the view being rejected; the author's conclusion is that a second garage would waste money. Test: when a passage turns with but or however, look for the author's conclusion after the turn.

Not every sentence in a passage is a premise. Background tells you who, where or when but gives no reason for the conclusion. In "The council met Tuesday in the old courthouse. Repairs would cost more than a new building, so the council should build new," the first sentence only sets the scene. Test: delete the sentence; if the support for the conclusion is unchanged, it was background.

Independent (convergent) premises are separate reasons; remove one and the others still stand. Linked premises work only as a team. "The trail is closed: there was a rockslide, and the ranger posted a closure" gives two independent reasons. "Anyone under twelve needs a guide, and Mia is ten, so Mia needs a guide" is linked. Test: delete one premise and ask whether the rest still give any support.

When an arguer concedes a point, they admit a consideration against the conclusion and then argue that it is outweighed. "Although the new bus route adds ten minutes, it should be approved, because it reaches the hospital" grants the delay and argues past it. The admitted point is not a reason for the conclusion. Test: when a clause concedes something, set it aside and ask what the actual support is.

Diagramming means numbering each claim and drawing arrows from reasons to what they support. Independent reasons get separate arrows; linked ones are joined before a single arrow. For "(1) The roof leaks and (2) the furnace is old, so (3) the house needs work, so (4) the price should drop," draw arrows from 1 and from 2 to 3, then one from 3 to 4. Test: a claim with no arrow in or out is background.

An argument is valid when it is impossible for all its premises to be true while its conclusion is false. "If the shop is closed, its lights are off. The shop is closed. So its lights are off" is valid even if the shop is open right now. Validity is about the connection, not the facts. Test: imagine the premises true and ask whether the conclusion could still fail.

A sound argument is valid and its premises are actually true, so its conclusion must be true. Validity alone is not enough. "All dogs can talk. Rex is a dog. So Rex can talk" is valid but unsound, since the first premise is false. Test: an argument earns your belief only after it passes both checks, the link and the premises.

To show an argument is invalid, describe one possible situation where every premise is true and the conclusion is false. "All librarians read a lot. Jo reads a lot. So Jo is a librarian." Counterexample: Jo is a nurse who reads a lot. You can also build a parallel argument of the same form with plainly true premises and a false conclusion. Test: one coherent scenario is enough.

An argument is formally valid when the pattern left after you swap its content words for letters, keeping words like if, not, all and some, has no invalid instance. "If it rains, the match is off; it rains; so the match is off" and "If the code is wrong, the door stays locked; the code is wrong; so it stays locked" share one such pattern. Test: write out the pattern; if it is valid, so is every argument that fits it.

Validity asks a hypothetical question: suppose every premise held; could the conclusion then fail? A false premise cannot make a valid argument invalid. "All birds are blue. Crows are birds. So crows are blue" is valid with a false first premise. Saying "but birds are not all blue" attacks soundness, not validity. Test: when asked about validity, set the facts aside and grant the premises.

An enthymeme is an argument with an unstated premise. "Lena is a pilot, so she has passed a vision test" works once you add a link such as "every pilot has passed a vision test." The hidden premise ties the stated reason to the conclusion, and it needs checking like any other premise. Test: ask what would have to be true for the reason to support the conclusion, then ask whether it is true.

A deductive argument guarantees its conclusion or it does not. An inductive argument supports its conclusion to some degree, and new evidence can raise or lower that degree. "The last forty buses from this stop were on time, so the next will be" is fairly strong, but news of a road closure weakens it without touching the premises. Test: ask how much support an inductive argument gives, not whether it guarantees its conclusion.

In a conditional, the antecedent is the condition and the consequent is what holds when the condition is met. Word order does not change the roles: in "The alarm sounds if the door opens," the antecedent is "the door opens," even though it comes second. Test: find the word if and take the clause attached to it as the antecedent, wherever it sits. Watch for only if, which works differently.

The contrapositive of "if A, then B" is "if not B, then not A." In classical logic the two are equivalent: each is false in exactly the same case, A true with B false. "If the oven is on, its light is on" becomes "If the light is off, the oven is off." Test: when you learn that a then-part is false, flip to the contrapositive and conclude that the if-part is false too.

From "if A, then B," the converse swaps the parts (if B, then A) and the inverse negates both (if not A, then not B). Neither follows. "If you own a car, you pay for insurance" does not give "if you pay for insurance, you own a car"; you might insure a home. Inferring A from B is affirming the consequent; inferring not-B from not-A is denying the antecedent.

"A only if B" says B is required for A, so standard logic reads it as "if A, then B." "The pump runs only if the tank has water" means that whenever the pump runs, the tank has water; water alone does not make it run. In the same way, unless means if not: "The picnic is on unless it rains" means "if it does not rain, the picnic is on." Test: rewrite each only if and unless as a plain if-then before reasoning.

In classical logic, conditionals that share a part can be chained: "if the river floods, the road closes" and "if the road closes, the bus reroutes" give "if the river floods, the bus reroutes." The shared part must be the then-part of one and the if-part of the other. Two conditionals with the same then-part do not chain. Test: line them up so each then matches the next if.

In classical logic a conditional is false in only one case: the if-part true and the then-part false. So "it is not true that if Sam studies, he passes" asserts that Sam studies and does not pass. The tempting reading, "if Sam studies, he fails," is a different claim. Test: deny a conditional by producing a case where the condition is met and the result is missing.

"A if and only if B" combines "if A, then B" with "if B, then A," so each part is both necessary and sufficient for the other. "The gate opens if and only if the code is entered" lets you reason in either direction. "The gate opens if the code is entered" does not; a guard might open it by hand. Test: before reasoning backward, check that the claim also says only if.

A is sufficient for B when A being true guarantees B. In "if A, then B," the if-part names a sufficient condition. Living in Ohio is sufficient for living in the United States, but people in Texas live there too, so learning that someone does not live in Ohio does not settle whether they live in the country. Test: ask whether A alone settles B; other routes to B may still exist.

A is necessary for B when B cannot happen without A. In "if B, then A," the then-part names a necessary condition. Being a parent is necessary for being a grandparent, yet a parent need not be a grandparent, so the condition is not enough. Test: imagine the condition absent; if the result then becomes impossible, the condition is necessary, whether or not it is also enough.

To translate, ask which part is required and which part settles the matter. "Winning the final requires reaching it" means reaching it is necessary: if you win the final, you reached it. "Paying the fee ensures your spot" means paying is enough: if you pay, you have a spot. Test: rewrite the sentence as if-then, with the guaranteeing part after if and the required part after then.

The error runs both ways. From "a ticket is required to board," you cannot conclude that a passenger holding a ticket boarded; the ticket is necessary, not sufficient. From "a fever is enough to keep a child home," you cannot conclude that a child kept home has a fever. Both mistakes reverse the conditional. Test: ask whether the claim says needed or enough, then use it only in that direction.

"All S are P" leaves no S without P; "no S are P" leaves no S with P; "some S are P" says at least one S is P. "All" contradicts "some are not," and "no" contradicts "some": in each pair exactly one is true. "All" and "no" can both be false, as when some guests ate and some did not. Test: before inferring, name the quantifier and what it rules out.

In logic, "some" means at least one, and it stays true if all turn out to qualify. In conversation, "some guests left early" often suggests that not all did, but that suggestion is an implication a speaker can drop, not part of the meaning. So "some of the apples are ripe" does not tell you any apple is unripe. Test: read some as "at least one, possibly all."

"Most S are P" means more S are P than are not, so more than half. If most club members play chess and most play tennis, at least one member plays both, because two groups each larger than half cannot avoid sharing someone. "Most" with only "some" gives no such overlap. Test: picture the group as 100 people and see whether the numbers force a shared member.

A categorical syllogism draws a conclusion from two premises that share a middle term. Aristotle checked every combination and found patterns: nothing follows from two particular premises or from two negative ones, and a universal conclusion needs two universal premises. "Some runners are vegans; some vegans are bakers" yields nothing about runners and bakers. Test: count the somes and nots first.

Conversion swaps a statement's subject and predicate. "No cats are reptiles" converts safely to "no reptiles are cats," and "some nurses are runners" to "some runners are nurses." But "all cats are mammals" does not give "all mammals are cats," and "some animals are not dogs" does not give "some dogs are not animals." Test: check the quantifier first; from all, carry across at most "some B are A," and only if A has members.

A statement has existential import if it asserts that things of its kind exist. Modern logic reads "all perfect scorers get a medal" as "anyone with a perfect score gets a medal," which stays true if no one scores perfectly. So "some perfect scorer gets a medal" does not follow from it alone, though many speakers hear all as implying members exist. Test: before moving from all to some, confirm the group is not empty.

"Every guest tried a dessert" can mean that each guest tried at least one dessert, perhaps different ones. Read with "a dessert" taking wider scope, it means there is one dessert that every guest tried. The second claim is stronger and does not follow from the first. Test: when every and some appear together, write out both orders and check which one the argument needs.

Modus ponens takes "if A, then B" and "A" and concludes "B." "If the package weighs over a pound, it needs extra postage. It weighs two pounds. So it needs extra postage." The form is valid: with both premises true, the conclusion cannot fail. Test: check that the second premise matches the if-part; if it matches the then-part instead, you are looking at a fallacy.

Modus tollens takes "if A, then B" and "not B" and concludes "not A." "If the dough had yeast, it would have risen. It did not rise. So it had no yeast." The form is valid because the conditional rules out A without B. Test: when a predicted result fails to appear, the condition that predicted it is ruled out, provided the conditional itself is true.

Disjunctive syllogism reasons from "A or B" and "not A" to "B." "The keys are in the car or in my coat. They are not in the coat. So they are in the car." The form is valid, but the conclusion is only as good as the either-or premise; if the keys could be on the counter, that premise is false. Test: before eliminating, ask whether any option is missing from the list.

In logic, "A or B" is inclusive: true when at least one part is true, including both. Everyday speech often suggests "but not both," yet a speaker can cancel that by adding "or both." The difference matters when one option is confirmed: "Ana brings salad or bread" plus "she brings salad" rules out bread only on the exclusive reading. Test: confirm the not-both reading, never assume it.

De Morgan's rules say that denying "A and B" means "not A or not B," and denying "A or B" means "not A and not B." "It is false that the cafe has both wifi and outlets" means it lacks at least one, not that it lacks both. "It is false that the cafe has wifi or outlets" means it has neither. Test: when you push a not inside, swap and with or.

Affirming the consequent reasons from "if A, then B" and "B" to "A." "If the pipe burst, the floor is wet. The floor is wet. So the pipe burst." It is invalid, since a spilled bucket also wets the floor. It works only when the conditional runs both ways, and it can still lend some support when B seldom happens without A. Test: name one other way B could be true.

Denying the antecedent reasons from "if A, then B" and "not A" to "not B." "If you water the fern daily, it will stay green. You did not water it daily. So it will not stay green." It is invalid: a humid room might keep it green. The conditional names one route to B, not the only one. Test: ask whether B could happen without A; if so, missing A proves nothing about B.

An inductive generalization reasons from a sample to a whole group: "Of 200 shoppers surveyed at the market, 70 percent wanted longer hours, so most of its shoppers want longer hours." Its strength depends on how many cases were seen and how well they mirror the group, and the conclusion can still be false. Test: ask how many cases there were and how they were picked.

With a randomly drawn sample, more cases let you pin the group's true rate into a narrower range with more confidence. A coach who sees three of five randomly chosen league players miss a free throw knows little about the whole league; the same rate across 500 randomly chosen players says much more. Size cannot fix a biased sample, though. Test: ask how far a few odd cases could swing the result.

A representative sample shares the traits of the group it is meant to describe. Surveying 5,000 people at a marathon to learn how much adults in general exercise gives a big number but a skewed one, since runners chose to be there. Adding more marathoners only repeats the skew. Test: ask how the cases were selected and whether that method could favor one answer; the statistical guarantees assume a random draw.

Hasty generalization draws a broad conclusion from evidence too thin or too unusual to carry it. "My two neighbors with electric cars had battery trouble, so electric cars are unreliable" rests on two cases that may not be typical. When the thin evidence is a vivid anecdote, some textbooks call the error misleading vividness. Test: ask how many cases support the claim and how much ground the claim covers.

A predictive inference projects an observed pattern onto the next case: "The farmers market has opened at eight every Saturday this year, so it will open at eight this Saturday." It assumes that unobserved cases resemble observed ones, an assumption Hume argued cannot be proved by reasoning. Test: ask whether anything could change the conditions behind the pattern, such as a holiday or a new schedule.

Citing an expert can be good evidence. It holds up when the person is a genuine authority in the field the claim belongs to, can be trusted, bases the claim on evidence, and makes a claim consistent with other experts in that field. A cardiologist on heart medication passes the field check; the same cardiologist on tax law does not. Test: ask whether the claim sits inside the expert's field and whether other experts agree.

A raw total counts events; a rate divides that count by the size of the group, such as cases per 100,000 people. Because it accounts for group size, a rate is the fair way to compare places or years whose populations differ. If a town grows from 20,000 to 40,000 people while burglaries rise from 100 to 150, the total climbed but the rate fell from 5 to 3.75 per 1,000. Before accepting more or fewer, ask: per how many?

A correlation means two things tend to occur together, which can support a prediction but not, by itself, a causal verdict. Suppose towns with more fire stations report more fires. Fires may bring stations, the reverse direction; town size may drive both, a common cause; or a small sample may show the link by chance. Before reading 'causes' into 'goes with', check which of those three the argument has ruled out.

Self-selection happens when people decide for themselves whether to be counted, so whatever drives that choice can also shape the results. A gym posts an online poll and finds 80 percent of respondents exercise daily, but the people who bother to answer a gym's poll are likely its keenest members. Before trusting a figure like that, ask who chose to be counted and whether that choice could relate to the answer.

An average, or mean, is one number standing in for a whole spread of values, and a few extreme values can drag it far from most members. If nine staff earn $40,000 and the owner earns $940,000, the mean pay is $130,000, higher than every salary but the owner's. The median, the middle value, stays at $40,000. When an argument leans on the average member, ask how skewed the data are and what the median says.

A causal conclusion says one thing produces another, not merely that the two occur together. Spot it by verbs such as causes, leads to, results in, prevents. Once you read 'the new lighting reduced shoplifting', you know questions to ask: could something else have reduced it, could the arrow run the other way, could one factor drive both, could the change be chance? A claim of mere association escapes the first three.

An alternative cause is another factor that could have produced the same result. A manager says sales rose because of the new website, but the rise came in December, when shoppers buy more anyway. The claim holds up only if sales beat a normal December, such as last year's. To weaken a causal claim, find a factor present at the right time that could produce the effect; to strengthen it, rule out or account for such factors.

Reverse causation means the arrow runs the other way: the supposed effect produced the supposed cause. A survey finds that people who own fitness trackers are fitter and concludes the trackers made them fit. Yet fit people may be the ones who buy trackers. A link measured at a single moment cannot show which came first. Ask whether the effect could have existed before the cause and helped bring it about.

A common cause is a third factor that produces both things being linked, so they rise and fall together though neither affects the other. On days when more umbrellas are sold, more car accidents occur; umbrellas do not cause crashes, rain drives both. The trap is reading the pairing as a direct link. Name a factor that could produce each side, then ask whether the link survives once that factor is held fixed.

Post hoc reasoning, short for the Latin 'after this, therefore because of this', treats one event as the cause of another merely because it came first. A city adds bike lanes in March and traffic deaths fall by June, so the lanes get the credit. If a speed-camera program also began in April, either change could explain the drop. Coming first is at most part of the case for a cause; look for what else changed meanwhile.

A control group is a comparison group that did not get the treatment, so it shows what happens without it. If every student who used a new study app passed the exam, that proves little until you know how many passed without the app. Assigning people to each group at random also tends to spread hidden differences, called confounders, evenly across the groups. Faced with any claimed effect, ask: compared with whom?

Causal claims differ in strength. A sole cause would produce the effect by itself; a contributing cause is one factor among several that together produce it; an enabling condition, like an unlocked door in a burglary, makes the effect possible without being what we single out as its cause. 'Stress was one cause of the illness' is easier to support than 'stress alone caused it'. Match the evidence to the claim's strength.

An argument from analogy reasons that because a source case and a target case share certain features, the target probably also has a further feature known in the source. 'My last car of this model ran 200,000 miles; this one has the same engine, so it will too.' Lay it out as known similarities, then the feature carried across. Then ask what produced that feature in the source; the answer shows which similarities matter.

Only similarities connected to the inferred feature give an analogy any force. A new restaurant shares a street, a color scheme and an opening month with a famous one, so a blogger predicts the food will be as good. None of those matches bears on cooking, so ten more like them would add nothing. Ask of each similarity: if this were different, would the conclusion be any less likely? If not, set it aside.

A disanalogy is a difference between the source and target cases, and it weakens an analogy when it touches the connection the argument relies on. 'Students may use calculators in math class, so they should be able to use them on the mental arithmetic quiz.' The quiz exists to test the very skill a calculator replaces. Find the feature that made the source case work and ask whether the target lacks it.

A false, or faulty, analogy rests on surface likeness. The cases differ in what actually produces the outcome. 'A company is like a family, so a manager should never let anyone go.' Both are groups that cooperate, but family ties do not depend on work done, while a company exists to produce goods for pay. Test an analogy by asking what causes the feature in the source and whether that cause is present in the target.

An analogy can carry a rule rather than a fact. You already accept a principle in one case, so consistency asks you to accept it in a relevantly similar one. 'You let staff work from home while the office was painted, so allow it during the road closures.' The reply is to name a difference that changes whether the rule fits. Courts reason this way with past rulings, setting aside those whose facts differ in a way that matters.

Inference to the best explanation, also called abduction, reasons from evidence to the hypothesis that would account for it best. Your car will not start, the dashboard lights glow dimly and the clock has reset, so you conclude the battery is weak. Unlike deduction, the conclusion can be false while all the evidence is true. Judge such an inference by asking which rivals were weighed and why this one won.

Calling an explanation the best only ranks it against the options someone thought of; if the true account was never on the list, you have picked the best of a bad lot. An investigator weighs a burglar against a jealous neighbor for a broken window and picks the neighbor, never considering the hailstorm that night. Before accepting 'the best explanation', ask what other accounts could produce the same evidence.

When several explanations fit the evidence, people rank them by explanatory virtues: simplicity, breadth of what they account for, and fit with well-established knowledge. If your houseplants wilted, 'I forgot to water them' beats 'a rare soil fungus plus a heat spike', since it adds fewer assumptions and fits what you know. Philosophers still dispute how to measure these virtues, so treat them as guides, not proofs.

A hypothesis is unfalsifiable when no possible observation could count against it. 'The lucky charm works: when you win it helped, and when you lose, you would have lost worse without it.' Every outcome fits, so no outcome can serve as evidence for it. The philosopher Karl Popper argued that this power to absorb any result is a weakness, not a strength. Ask the holder: what result would show you were wrong?

A testable explanation tells you something you have not yet checked, and it could fail. If a ceiling stain comes from the upstairs shower, it should grow only after someone showers and stay dry on days the shower goes unused. Watching for that pattern puts the idea at risk. Popper held that agreement counts as support only when the prediction might have failed. A story that predicts nothing new cannot be checked this way.

An explanation takes a fact as given and says why it holds; a justification gives reasons to believe the fact at all. 'The bridge closed because inspectors found cracks' explains a closure you already accept, but if you doubt it closed, a plausible why-story does not show it did. The reverse fails too: a wet sidewalk is evidence of rain without explaining the rain. First settle whether the claim is true, then ask why.

An ad hominem argument, Latin for 'against the person', rejects a view by attacking its holder. The abusive form cites a flaw ('his shop failed, so ignore his budget'); the circumstantial form cites self-interest; the tu quoque form cites the speaker's failure to follow their own advice. A conflict of interest can be a fair reason to check evidence closely. Ask whether the trait bears on the reasons given.

A straw man restates an opponent's view as a weaker or more extreme one, refutes that version, and claims victory over the original. A council member proposes fewer parking spaces downtown; a rival replies, 'She wants to ban cars from the city,' and attacks the ban. Even a perfect case against a car ban leaves the parking proposal untouched, because nobody proposed the ban. Check any rebuttal against the opponent's own words.

An appeal to popularity, or ad populum, offers widespread belief or practice as evidence that something is true or right. 'Millions take this sleep supplement, so it must work.' Wide use shows the product is popular, not that it works, since crowds can share a mistake. Popularity counts only where the crowd has special access to the question, such as how locals pronounce their town's name. Ask: how would these people know?

An appeal to emotion tries to win belief by stirring feelings instead of giving evidence: fear of a threat, pity for someone, or a crowd's passions. 'This app was built by a single parent working nights, so its privacy promises must be true.' Sympathy for the maker is real but says nothing about how the app treats your data. Feelings can rightly shape what you choose to do; keep them apart from whether a claim is true.

Citing an authority supports a claim only if the person is a genuine expert in the field the claim belongs to and other experts there do not dispute it. It fails when the expertise is borrowed from another field, as when a famous surgeon endorses a tax plan, or when experts are divided and the arguer quotes one side. Ask three things: expert in what, does the claim fall inside that field, and what do other experts there say?

A red herring brings in an irrelevant point that pulls attention away from the question at issue. Asked whether a new stadium will repay its public cost, an official praises the team's loyal fans and proud history. Both may be true, yet neither speaks to repayment. Proving some other claim than the one in dispute, which logicians call an irrelevant conclusion, leaves that claim untouched. Keep the exact question in view.

An appeal to ignorance treats a lack of disproof as proof: 'No one has shown the new additive is unsafe, so it is safe.' A lack of evidence against a claim does not establish it, least of all when nobody has looked. Whoever makes a claim carries the burden of proof, the duty to support it when challenged. Shifting that duty onto doubters is the move to watch for. Ask whether anyone has actually checked, and what they found.

The genetic fallacy judges a claim by where it came from rather than by the evidence for it. 'That proof was first sketched on a napkin in a bar, so it cannot be sound.' Where an idea started is a question of history; whether it holds is a question of support. Origins matter only when they bear on the evidence itself, as when the seller of a product runs the study. Ask what supports the claim now.

Begging the question uses the conclusion, often reworded, as a premise. 'This newspaper is trustworthy, because its reports are reliable.' Reliable and trustworthy say nearly the same thing, so the premise gives no independent reason. When the loop runs over several steps, A because of B and B because of A, it is called circular reasoning. Ask whether someone who doubts the conclusion would already accept each premise.

A false dilemma presents two options as the only ones, then rules one out to force the other. 'Either we cancel the school trip or we accept that a student will get hurt.' Smaller groups, more chaperones or a safer site go unmentioned. An either-or premise is guaranteed to exhaust the cases only when one option is the plain denial of the other, such as pass or not pass. Ask what a third option would look like.

A complex question carries a disputed claim inside its wording, so any direct answer seems to grant it. 'Did the auditors find out that you moved the money?' Yes and no both concede that you moved it. Such a hidden claim, called a presupposition, usually survives whether the sentence is asked, denied or asserted. The fix is to split the question: first ask whether the buried claim is true, and settle that before answering.

A slippery slope argument rejects a first step because it will supposedly lead, link by link, to a bad outcome. 'If we allow phones at lunch, students will use them in class, grades will fall, and the school will lose funding.' Every step needs its own support, and here none has any. Some chains are well founded, so do not dismiss the form on sight. Check the evidence for each step.

Composition wrongly assumes a whole has a property because its parts do; division wrongly assumes each part has a property because the whole does. 'Every song on this playlist is short, so the playlist is short' commits composition. 'The orchestra is world famous, so each of its violinists is' commits division. Before letting a property pass between whole and parts, ask whether it depends on how the parts combine.

Picking your evidence by the very property an argument means to prove is a form of begging the question: the conclusion is built into the sample. 'Every real fan I know stayed to the final whistle; anyone who left early was not a real fan.' The rule for who counts as a real fan already settles the result, so surveying fans adds nothing. Ask how the members were chosen and whether that choice assumes the claim in dispute.

False precision treats a number's many digits as a sign that it is accurate. 'Our survey of 40 visitors shows 67.5 percent prefer the new layout.' The decimal looks exact, but one person changing their mind moves the figure by 2.5 points, and a sample that small carries a wide margin of error. The digits show how finely a figure is reported; accuracy is whether it matches the truth. Ask how it was measured.

These appeals treat being natural, or long established, as proof that something is good or right. 'This remedy is all natural, so it is safe' overlooks that natural substances can do harm; the National Institutes of Health cautions that natural does not always mean safe. 'We have always graded this way' cites a habit, not a reason. Ask what the practice actually does, and whether you would defend it if it were new.

Equivocation uses one word in two senses within a single argument, so the steps seem to connect when they do not. 'You have a right to say what you like, so whatever you say is right.' The first right means an entitlement; the second means morally correct, so the conclusion does not follow. Pin down the meaning of the repeated word in every line; if the meaning shifts, the argument fails.

Amphiboly is ambiguity in a sentence's grammar rather than in any one word, so the same words support two different claims. A pool sign reads 'Members may bring guests and children under 12 free.' Are all guests free, or only children under 12? A member who says an adult friend gets in free relies on a reading the sign never settled. When a sentence carries weight, rewrite it both ways and see which the argument needs.

A vague term, like tall, rich or crowded, has borderline cases where neither yes nor no is clearly right. The sorites argument exploits this: one more dollar never makes a poor person rich, so by tiny steps no one ever becomes rich. Each step looks harmless; the conclusion is absurd. Not being able to say where the line falls does not erase the difference between clear cases, so resist demands for an exact cutoff.

Loaded language uses words that carry approval or disapproval to do the work of an argument. Calling a tax a cash grab, or a policy common-sense reform, tells you how to feel before any reason arrives. The philosopher Charles Stevenson called this pull a word's emotive meaning, as distinct from what the word describes. Restate the claim in neutral terms, such as a tax or a policy change, and see whether any reason is left.

The fallacy of accent changes meaning through emphasis: stressing one word, or quoting a few words without what surrounded them. Say 'The manager did not approve the trip' with stress on approve, and it hints that she did something short of approving it. A review calling a plot 'brilliant only if you enjoy confusion' becomes 'brilliant' on a poster. Before relying on a quotation or contrast, find the full sentence.

Some arguments look factual but turn on which definition of a word to use. Two friends agree on every fact about a film with three songs and no dancing, yet argue over whether it is a musical. No further fact can settle that; they must agree on a definition or drop the label. A definition that keeps a word's approving tone while changing what it covers is persuasive, not neutral. Ask what is left once the facts are agreed.

The stem is the question attached to a passage, and it names your job: find the conclusion, find a gap, weaken, or draw an inference. Each job calls for a different reading. If the stem asks what the argument assumes, you hunt for the jump from premises to conclusion; if it asks what follows, you treat every sentence as a premise and look for what they guarantee. Read the stem first, then read the passage for that one job.

You can sort argument tasks into a few families. Part tasks ask for the conclusion or a sentence's job. Assumption tasks ask what unstated premise is needed. Strengthen and weaken tasks ask what new fact makes the conclusion more or less likely. Inference tasks ask what follows from the passage. Structure tasks ask how the argument proceeds. The family tells you what to seek: a part, a gap, a fact, a consequence, or a pattern.

Inference and structure tasks are closed: the right choice must follow from, or describe, only what the passage says, so an outside fact disqualifies it. Strengthen, weaken, and evaluate tasks are open: they turn on new information, judged by how it changes the support for the conclusion. Before choosing, ask which kind you face. A choice that brings in new information is wrong in the first kind and expected in the second.

Prephrasing means saying, before you read the choices, what the right answer has to do. A bookstore's sales rose after it added a reading nook, so the nook caused the rise. Your prediction for a weakener: something else changed at the same time. Then you measure each choice against that job instead of judging how reasonable it sounds. Keep the prediction loose, since the right choice may do the job in words you did not expect.

Time spent on one item is time taken from every other, so triage is a choice under uncertainty: which use of the next minute is likely to earn the most? A dense passage stacked with conditionals, or a task type you find slow, may be worth deferring. Make the call early. If you cannot say what the task wants after one read and the format lets you come back, mark it, move on, and return if time remains.

Words like depends on, requires, and presupposes ask for a necessary assumption: an unstated claim that must be true for the reasoning to work. It need not prove the conclusion; the argument just cannot do without it. Someone says a park needs more benches because older visitors avoid it. One requirement: more benches would bring more older visitors. Test a candidate by asking whether the argument fails without it.

To test whether a claim is a necessary assumption, deny it and see whether the argument survives. Argument: the new oven explains why the bakery's bread has been ready earlier this month. Deny 'the new oven was in use on the mornings the bread came out early' and the explanation collapses, so that claim is needed. Deny 'the oven was costly' and nothing changes. Negate with care: the denial of 'all' is 'not all', not 'none'.

A term shift happens when the conclusion talks about something no premise mentions. Dana trains every morning, so Dana will win Saturday's race. The premise is about training; the conclusion is about winning. Nothing connects them until you assume something like: morning training gives Dana an edge over the other runners. Circle the new term in the conclusion, find the premise term it should attach to, and state the link.

Some assumptions do not push the conclusion forward; they block a way it could fail. More people borrowed audiobooks after the library extended its hours, so the longer hours raised borrowing. The argument assumes that something else, such as a new lending app, did not cause the whole rise. That assumption protects the conclusion by ruling out a rival. List the obvious objections and ask which the argument needs to be false.

A necessary assumption is the least the argument must take for granted, so a choice that claims more fails even when it would help. The town should build a second library, since the current one is packed every weekend. The argument needs a second library to ease the crowding, not to end it. Negate 'a second library would end the crowding' and you get 'it would not end it entirely', which leaves the argument standing.

An irrelevant comparison brings in something the reasoning never relies on. Argument: cycling injuries fell after our town added bike lanes, so the lanes are worth keeping. 'A nearby city's lanes cost less' changes nothing. 'Injuries fell as much in a town with no new lanes' also brings in a new group, yet it matters: it points to another cause. Judge a comparison by whether it bears on the conclusion, not by what it mentions.

A sufficient assumption is a claim that, once added, makes the conclusion follow with certainty. To find one, look for a term that appears only in the premises and one that appears only in the conclusion. Rosa is a certified lifeguard, so Rosa can perform CPR. The loose ends are certified lifeguard and can perform CPR. The claim that joins them, every certified lifeguard can perform CPR, makes the argument valid.

Support comes in degrees. A fact can make a conclusion more likely without proving it, and of two helpful facts one may help far more. Conclusion: a school's new reading program raised scores. One choice says the program drills skills the test measures; another says scores rose only in classes that used it. Both help; the second does much more. Rank helpful choices by how much they raise the conclusion's likelihood.

A non-deductive argument leaves a gap between what the evidence shows and what the conclusion claims. A strong strengthener aims at that gap, not at the topic. Evidence: shoppers who took free bread samples bought more bread. Conclusion: the samples caused the extra sales. The gap is causal, so the best help shows that sample-takers were otherwise like other shoppers. Name the gap, then find the choice that narrows it.

A causal conclusion competes with other explanations of the same data. Remove a rival and the remaining explanation gains, even though nothing new was said in its favor. Claim: the clinic's shorter waits came from its new booking system. If staffing and patient numbers stayed the same, two rivals are gone, and the booking system is better supported. Ask what else could explain the result, then find a choice that shuts it out.

In a weaken task the stated facts are not up for debate; what you attack is the step from those facts to the conclusion. The gym's membership doubled after it repainted, so the new colors drew people in. Disputing that membership doubled is off limits. Showing that a rival gym closed that month attacks the link instead. A good weakener accepts the evidence and gives you a reason to doubt that it supports the conclusion.

A universal conclusion says something holds in every case, so one exception refutes it. A counterexample is a situation in which every premise holds yet the conclusion fails. Claim: every plant here needs bright light, since all the ones by the window are thriving. A healthy fern in the dim back corner sinks it. Against an all, never, or always claim, look for one case that keeps the premises true and breaks the conclusion.

When an argument concludes that one thing caused another, a rival cause of the same result weakens it. Power bills in a neighborhood fell after the city mailed energy-saving tips, so the tips worked. If that month was unusually mild, the weather alone could explain the drop. The strongest rival fits all the evidence, including its timing. Ask what else would produce that result, and check whether a choice supplies it.

A generalization reaches past its evidence, from the people surveyed to everyone. It is only as good as the match between the sample and the larger group. Claim: most residents want later bus service, based on a survey taken at the late-night bus station. Riders at that hour likely differ from residents overall. To weaken a generalization, show that the sample differs from the population in a way that bears on the question.

A flaw task says the argument's own reasoning contains an error and asks you to name it. You add nothing new; the mistake is already in the passage. Errors of this kind include ignoring other possible causes, generalizing from a skewed sample, and assuming what the argument sets out to prove. Find the conclusion, find the evidence, and say in your own words why the evidence falls short before you read any choice.

A flaw can be described with no mention of the topic, as in 'treats a necessary condition as sufficient'. To use it, translate it into the passage's content. If an argument says a permit is required to fish here, so anyone with a permit may fish here, the abstract version is that a necessary condition was treated as sufficient. Map each part of the description onto the argument; if any part fails to fit, reject it.

'Fails to consider' says the argument ignores a possibility that would undercut it. 'Takes for granted' says it leans on a claim it never supports. They are one gap from two sides: ignoring a possibility means assuming it false. The team lost more under a new coach, so the coach is worse. It fails to consider tougher opponents; equally, it takes for granted the schedule was no harder. Either phrasing fits if it names that gap.

The composition error carries a property up from the parts to the whole; the division error carries it down from the whole to each part. 'Each episode is short, so the series is short' is composition, since forty short episodes can add up to a long series. 'The hospital is famous, so each of its nurses is famous' is division. When a property moves between parts and whole, ask whether it actually transfers.

Some flaws swap one concept for a nearby one. Correlation for causation: two things rise together, so one must drive the other, ignoring a third factor behind both. Possibility for certainty: a new road could ease traffic, so it will. Others swap a necessary condition for a sufficient one. Abstractly, each treats X as if it were Y. Name the two concepts the passage blurs, and check that a description names the same pair.

A flaw choice can describe the argument accurately and still be wrong, because what it describes is not a mistake. Citing an expert is not an error in itself; it goes wrong when, for example, the source lacks relevant expertise. Drawing a conclusion from a sample is fine when the sample matches the group. For each choice, ask two questions: does the argument do this, and is doing it an error here?

A flaw task and a weaken task can share an argument and still want different things. Flaw asks you to describe a mistake already in the reasoning, using only the passage. Weaken asks for a new fact that makes the conclusion less likely. Sales rose the week a new ad ran, so the ad caused the rise. A flaw description says it overlooks other causes; a weakener names one, such as a price cut that same week.

A must-be-true task turns the usual job around: instead of judging a conclusion, you treat the passage as premises and pick the claim they guarantee. A choice must be true when no situation makes every statement true and that choice false. Every member of the choir sings at the fall concert, and Ana is in the choir; so Ana sings at the fall concert. Test each choice by imagining the passage true and the choice false.

In an inference task the passage is your only evidence, so a choice that sounds sensible but leans on something unstated fails. Passage: the bakery sells out of rye every Saturday. 'Rye is the bakery's most popular bread' needs facts about the other breads, so it does not follow. Ask of each choice which sentences guarantee it. If you have to add even a reasonable assumption, set the choice aside.

Two statements that share a term can be chained into a conclusion neither states alone. Everyone on the night shift gets a parking pass, and Omar is on the night shift, so Omar gets a parking pass. Chain conditionals in their own direction: if A then B and if B then C give if A then C, but not if C then A. Scan for a term that appears in two sentences; that shared term is where an inference can form.

'Most' means more than half. Two 'most' statements about one group force an overlap: if most club hikers own boots and most own poles, at least one owns both, since two majorities of a group must share a member. A 'most' plus an 'all' chains: if most hikers own boots and all boot owners wax them, most hikers wax boots. But 'most A are B' and 'most B are C' guarantee nothing about A and C.

Some inference tasks ask which claim the passage most strongly supports rather than which must be true. The bar drops from certainty to high likelihood. The pond has frozen by mid-January in each of the last ten winters, so it will probably freeze by then this winter: well supported, not guaranteed. Still reject choices that go past the evidence, such as a claim about why the pond freezes.

A choice can point the right way and still claim more than the passage supports. If most new members renew within a year, 'every new member renews' overshoots. If most renewals come from new members, 'only new members renew' overshoots the same way. A stronger claim is true in fewer situations, so it needs more support. Compare the strength of each choice's words with the passage's: some, most, all; sometimes, usually, always.

A claim is consistent with a passage if both could be true together. It follows from the passage only if it is true in every situation where the passage is true. Passage: all the ferries leave from Pier 3. 'The noon ferry is late today' is consistent with that, but nothing in the passage supports it. When a choice seems fine, ask whether the passage forces it or merely leaves room for it.

A specific judgment often rests on a general rule. To find it, keep the features the judgment depends on and drop the names. Judgment: Mia should repay her friend's loan early, since her friend now urgently needs the money and Mia can afford to. Rule: when a lender urgently needs a loan repaid and the borrower can afford it, the borrower should repay early. Check that the rule yields the judgment from exactly those facts.

A principle applies to a case only when the case meets every condition the principle names. Rule: a tenant may keep a pet if the pet weighs under ten kilograms and the landlord has been told in writing. A tenant with a small cat who told the landlord only by phone does not qualify. List the conditions, check each one against the case, and treat a single unmet condition as enough to block the rule.

A principle supports a judgment only if the case falls inside it. Judgment: the diner should refund Sam, who was served the wrong dish and asked for his money back. 'Refund any order that arrives cold' is too narrow, since Sam's food was not cold. 'Refund anyone who asks' covers Sam but also commits you to refunding people who changed their minds. Prefer the principle whose conditions match the features the judgment rests on.

A principle about a type of action can be rewritten as an if-then claim: if an action has these features, it is allowed, wrong, or required. The if-part is the trigger; the then-part is the result. Rule: if a volunteer has finished training, they may lead a group. A trained volunteer may lead. An untrained one is not covered, since the rule never forbids it. Missing the trigger does not produce the opposite result.

Two principles conflict when one case triggers both and they demand incompatible actions. Rule one: grade every student by the same standard. Rule two: make allowances for a seriously ill student. A sick student's late essay triggers both: dock it or excuse it. An argument can settle the clash by claiming that, here, one rule overrides the other, or by narrowing one rule's scope. Look for the priority claim, stated or assumed.

Two arguments are parallel when they share a logical form: the same pattern of claims, whatever the topic. Some dogs are shy and all shy animals hide, so some dogs hide. A parallel: some coats are wool and all wool items itch, so some coats itch. A choice about dogs that reasons in a different pattern is not parallel, however close its topic. Compare patterns, not subjects.

To compare structures, replace each repeated term with a letter. If the shop is open, its lights are on; the lights are off; so the shop is closed. That becomes: if A then B; not B; so not A. Do the same to each choice and look for the same skeleton. Watch the details that change form: some versus all, the direction of each if-then, and whether a conclusion says must or probably.

When the original is flawed, a parallel choice must make the same mistake, not merely reach a similar conclusion. If it rains the match is canceled; the match was canceled; so it rained. In letters: if A then B; B; so A. That affirms the consequent: it treats the then-part as true and infers the if-part, an invalid form. A choice that reasons validly, or errs differently, does not match. Name the flaw in letters first.

Every sentence in an argument has a job. A premise is offered as support. The main conclusion is what everything else supports. A subsidiary conclusion is supported by premises and in turn supports the main conclusion. An opposing view is a claim the arguer sets up in order to reject it. To name a sentence's role, ask what supports it and what it supports; a claim that does both is a subsidiary conclusion.

When two people argue, the second may attack the first one's evidence, conclusion, or the link between them, or may answer a point the first never made. Ana: the library should open on Sundays, since students need study space. Ben: the budget cannot cover Sunday staff. Ben does not deny that students need space; he raises a cost. Pin down which claim the reply engages, and watch for replies aimed at a distorted version.

A method task asks how an argument reaches its conclusion, not what it concludes. Methods include analogy, where two cases are alike so what holds for one probably holds for the other; counterexample, where one case refutes a general claim, as one rainy Saturday refutes 'it never rains on Saturdays'; elimination, where all options but one are ruled out; and appeal to a principle. Then match the description to the passage.

An apparent paradox is two facts that seem to clash but can both be true. The clash can rest on assuming they concern the same group, time, or measure. The zoo sold a record number of tickets this year, yet it had fewer visitors. If it dropped its annual pass, so regulars now buy a ticket each visit, both hold. Look for a fact showing the statements are about different things, or one that adds a factor linking them.

Before you look for a resolution, state the puzzle in one sentence of the form 'X, yet Y'. Umbrella sales rose, yet it rained less than usual. Naming both halves shows what a resolution must do: explain how sales could rise in a drier season, perhaps because a heat wave sent people buying umbrellas for shade. Without it, a choice that merely mentions umbrellas or rain can look like an answer while explaining nothing.

A real resolution explains how both facts hold together; a tempting wrong choice explains only one. Puzzle: the cafe served more customers this month, yet its revenue fell. 'A new office opened next door' explains the extra customers but not the lost revenue. 'The cafe began a half-price weekday deal that drew crowds' explains both. After choosing, reread each half of the puzzle and confirm the choice speaks to it.

An evaluate task asks which information would best help you judge an argument. The useful question is one whose answers would push the conclusion in opposite directions. Ken's tomatoes grew bigger after he switched fertilizers, so the new fertilizer did it. Useful: did Ken also water more this season? A yes points to a rival cause; a no removes it. A question whose every answer leaves the argument where it was is useless.

To test a candidate question, answer it both ways and see whether the support shifts in opposite directions. Argument: accidents fell after a town added speed bumps, so the bumps cut accidents. Did accidents also fall in nearby towns without bumps? Yes points to a regional cause; no favors the bumps. Did the bumps cost more than planned? Either answer leaves the argument unchanged, so that question fails.

Two speakers disagree about a claim when one accepts it and the other rejects it; mentioning a topic is not enough. Priya: the new bridge will cut commute times, so it is worth building. Tom: it will cut commutes, but the cost is too high to justify. Both accept the commute claim. They disagree about whether the bridge is worth building. For each candidate claim, ask whether one speaker would say yes and the other no.

Two variants flip the usual task. An agreement task asks for a claim both speakers are committed to, so check each choice against each speaker separately. A cannot-be-true task asks for the claim that could not hold if every statement in the passage were true. Passage: every room on the second floor has a window. 'Room 204, on the second floor, has no window' cannot be true; 'Room 104, on the first floor, has no window' can.

In these puzzles each speaker is a knight, who always tells the truth, or a knave, who always lies. Kit says 'Jo and I are both knights', and Jo says 'Kit is a knave'. If Kit were a knight, Jo would be one too and could not call Kit a knave. So Kit is a knave, which makes Jo's claim true, so Jo is a knight. Assume a role, follow its consequences, and drop any assumption that leads to a contradiction.

A claim about its own speaker's honesty can rule out roles by itself. Among truth-tellers and liars, nobody can say 'I am a liar': a truth-teller saying it would be lying, and a liar saying it would be telling the truth. Now Lu tells Ming, 'Of the two of us, I am the only liar.' A truth-teller could not say that, so Lu lies. The claim is then false, yet Lu is a liar, so he cannot be the only one: Ming lies too.

When a puzzle says exactly one of several statements is true, you cannot judge any one alone. Note A: the key is in the drawer. Note B: the key is not in the drawer. Note C: the key is not in the box. A and B contradict each other, so exactly one of them is true. That forces C to be false, so the key is in the box, which makes B the true note. Test combinations, not single claims.

With a few unknowns, you can list every possible assignment and strike out each one that breaks a condition. Ari, Bea, and Cy ordered tea, coffee, and juice, one drink each. Bea did not order tea, and Cy ordered neither tea nor juice. Cy must have coffee, so Bea has juice and Ari has tea. Of six possible arrangements, one survives. If none survives, the conditions are inconsistent and cannot all be true.