Logical reasoning, oriented

An argument puts forward some claims, the premises, as grounds for believing another, the conclusion. A list of reports is not an argument, and neither is an explanation: "the pipes froze because the heat failed" takes the frozen pipes as given and says why. Words like since and because can flag a premise, and so and therefore usually flag a conclusion, but many arguments carry no marker. A deductive argument claims its premises guarantee the conclusion; an inductive one claims only that they make it more likely.

Longer passages stack. Premises can back an intermediate conclusion, which in turn backs the main one, so the main conclusion receives support and passes none on. Background sets the scene, and you can delete it without changing the support. A conceded point admits something against the conclusion, so it is no reason for it. Independent premises each carry weight by themselves; linked premises carry it only together. Number the claims and draw support arrows, and you see the skeleton.

The trap here is taking the first sentence for the conclusion. "Some residents want the town to ban leaf blowers. But the noise lasts minutes a week and few people complain, so a ban would fix a problem that barely exists. The council meets on Mondays." The opener is the view under attack, the conclusion follows the but, and the meeting day is scenery.

Rule: the main conclusion is backed by the other claims and backs nothing in turn; when a passage turns on but or however, look after the turn.

An argument is valid when no possible case makes every premise true and the conclusion false. Validity is all or nothing, unlike inductive support, which comes in degrees. "All fish can fly; salmon are fish; so salmon can fly" is valid: if the premises held, the conclusion would have to. Its first premise is false, though, so it is unsound. Soundness needs a valid link and true premises, which together force a true conclusion. Pointing out a false premise attacks soundness, never validity.

A single possible case with every premise true and the conclusion false proves invalidity. Validity is also a matter of form: put letters for the subject-matter words, keep logical words like if, not, all and some, and a valid pattern stays valid for any content.

In everyday talk a premise is often left unsaid. "The bridge is closed, so the bus will be late" needs a link such as "when the bridge is closed, the bus runs late." Read charitably: add the least demanding premise that carries the step, then ask whether it is true.

The trap is judging validity by whether the conclusion is true. "Every member of the chess club is a student. Priya is a student. So Priya is in the chess club." Even if Priya is a member, the argument is invalid: picture her as a student who never joined, and every premise stays true while the conclusion fails.

Rule: grant the premises and look for one possible case where the conclusion fails; if there is none, the argument is valid, and only then judge the premises.

A conditional, "if A, then B," has an antecedent, the if-part, and a consequent, the then-part. Clause order does not matter: in "the kettle whistles if the water boils," the boiling is still the antecedent. The if-part states a sufficient condition, enough by itself to guarantee the then-part. The then-part states a necessary condition, one the if-part cannot hold without.

In classical logic, the contrapositive, "if not B, then not A," is equivalent to the original; the converse, "if B, then A," and the inverse, "if not A, then not B," do not follow. The same logic reads "A only if B" as "if A, then B" and "A unless B" as "if not B, then A," chains conditionals when one's then-part is the next one's if-part, and treats denying "if A, then B" as asserting A with not-B, not "if A, then not B." "If and only if" asserts both directions.

In everyday words, requires, must, needs and prerequisite usually mark the necessary part; ensures, guarantees and is enough, the sufficient one.

The trap is running a conditional backward. Say a sign reads "entry to the lab requires a badge." If you entered the lab, you have a badge; by the contrapositive, no badge means no entry. "Lee has a badge, so Lee went into the lab" reverses the conditional, mistaking a requirement for a guarantee.

Rule: turn "only if," "unless," "requires" and "ensures" into a plain if-then, and reason only forward from an affirmed if-part or backward from a denied then-part.

Modus ponens is valid: if A then B; A; so B. Modus tollens is valid: if A then B; not B; so not A. Their look-alikes are not. Affirming the consequent reasons from B to A, and denying the antecedent from not-A to not-B; both ignore that B may arise some other way. Finding B is still some evidence for A when B rarely occurs without A, but it never proves A. Disjunctive syllogism, "A or B; not A; so B," is valid, yet no more reliable than its either-or premise. Logical or is inclusive, so one option holding rules out the other only if both cannot hold. By De Morgan's rules, denying "A and B" means at least one fails, while denying "A or B" means both fail.

Quantifiers have exact strengths. Some means at least one, possibly all, so "some samples passed" does not tell you any failed. Most means more than half, so two "most" claims about one group must share a member. "All A are B" does not yield "all B are A," though no-statements and some-statements swap their terms safely. In a two-premise syllogism, two some-statements yield no conclusion, and neither do two negative ones. Scope matters too: "every guest tried a dessert" is weaker than "one dessert was tried by every guest."

Here is the trap. "If the printer is out of toner, its light turns red. The light is red. So it is out of toner." That affirms the consequent, since a paper jam might also turn the light red.

Rule: name the form before you trust it, and read every quantifier at its exact strength.

Induction moves from observed cases to a claim beyond them, so its conclusion can be false even when every premise is true. A generalization is capped by how large and how representative its sample is. With a random sample, more cases narrow the likely range, but size cannot cure bias: polling thousands of hikers at a trailhead about how much adults exercise just piles up the same skew. Self-selection does similar damage when people choose whether to be counted. Too few or too unusual cases, such as one vivid story, produce a hasty generalization. A prediction assumes unseen cases will resemble those already observed, an assumption Hume argued reasoning cannot prove. Expert testimony counts when the claim lies in the expert's field, the expert is trustworthy and relies on evidence, and other experts there agree.

A total is a raw count; a rate scales that count to the group's size, so rates are what you compare across groups that differ in size. A handful of extreme values can pull a mean well away from most members, while the median, the middle value, resists them.

Suppose you hear this: "Lakeside had 60 bike thefts last year and Hillview had 30, so theft is a bigger problem in Lakeside." If Lakeside has 60,000 residents and Hillview 15,000, the rates are 1 and 2 per 1,000 residents. Hillview's rate is twice Lakeside's.

Rule: before you trust a figure, ask how many cases, chosen how, per how many, and what the median says.

A causal claim says one thing brings about another, not just that the two go together. A correlation alone leaves four rivals open: another factor could have produced the effect, the arrow could run backward, a third factor could drive both, or the link could be chance. Timing alone is never the whole case, since something else may have changed in the same window. So ask of any claimed effect: compared with whom? A control group shows what happens without the treatment, and random assignment tends to spread hidden differences evenly. And a contributing cause is easier to support than a sole one.

An analogy carries a feature from a known case to a new one because the two share other features. Only likenesses tied to that carried feature count, and a difference that bears on it weakens the argument. Inference to the best explanation, also called abduction, picks the hypothesis that would best account for the evidence, but best only among the rivals someone considered. Simplicity, breadth and fit with well-established knowledge are common ranking guides, though how to measure them is disputed. An explanation that fits every possible outcome cannot be tested, which Popper counted as a weakness.

The trap is trusting a before-and-after story. "After the library added armchairs in May, visits rose by a third, so the chairs drew people in." If May brings end-of-term studying, that alone could explain the rise; check whether visits also rose at branches with no new chairs.

Rule: for any claim that X caused Y, name the rivals, then ask what a comparison group shows.

Fallacies of relevance offer a reason that does not bear on the claim. Ad hominem, Latin for "against the person," attacks the arguer instead of the argument. A straw man refutes a distorted version of the opposing view. Appeals to popularity, to emotion or to an expert outside their field belong here, as does a red herring that answers a different question. An appeal to ignorance takes the absence of disproof as proof, and the genetic fallacy trusts or dismisses a claim because of where it started.

Fallacies of presumption build in something unearned. Circular reasoning slips the conclusion, often in new words, into the premises. A false dilemma offers two options when a third exists. A complex question hides a disputed claim in its wording. A slippery slope predicts a run of consequences without backing each step. Composition and division wrongly pass a property from parts to whole or from whole to parts.

Fallacies of ambiguity let language do the work. Equivocation shifts a word's sense partway through, amphiboly exploits loose grammar, loaded language persuades through charged words, and selective quotation cuts away context.

The trap is firing a label on sight. Noting that "the sweetener study was paid for by its maker" is not automatically a fallacy. Origins count when they touch the evidence itself, so this is a fair reason to check the study closely, not to reject it. Likewise, some slippery slopes have well-supported links.

Rule: before naming a fallacy, say in one sentence why this reason fails to support this claim.

Read the question under a passage first, because it tells you what to look for. Part tasks want the conclusion or one sentence's job. Assumption tasks want the unstated premise the reasoning leans on. Strengthen and weaken tasks want a new fact that raises or lowers the conclusion's likelihood. Inference tasks want what the passage guarantees; structure tasks want how the reasoning moves.

The families also differ on outside information. In inference and structure tasks only the passage counts, so an option that needs an outside fact fails. Strengthen, weaken and evaluate tasks run on new facts. Before you read the options, say what a good one has to do, and hold that prediction loosely. Against the clock, an item that looks far slower than the rest may be worth deferring, if you can return to it.

Role tasks name a sentence's job: premise, main conclusion, a subsidiary conclusion that is both supported and supporting, or an opposing view set up to be rejected. Method tasks describe how the argument works, such as arguing from analogy, refuting with a counterexample, ruling out alternatives or applying a principle.

When two people argue, the trap is misreading what the reply targets. Jun: "The city should plant more street trees, because shade cools the sidewalks." Pia: "Tree roots crack pavement, and the repairs are expensive." Pia never disputes the cooling; she raises a cost, so an option saying she denies it misreads her.

Rule: read the task first, predict what a good option must do, and ask exactly which claim each sentence or reply engages.

A necessary assumption is an unstated claim the argument cannot do without. To test one, deny it: if the argument collapses, the claim was needed. Deny carefully, since the opposite of all is not all, not none. If the conclusion introduces a term no premise uses, the assumption has to tie that term back to the premises. A sufficient assumption does a different job: added to the premises, it guarantees the conclusion. To find one, join a premise term the conclusion never uses to a conclusion term no premise uses.

When you strengthen or weaken, you bring in new facts, treat the premises as true and target the link. Support comes in degrees, so prefer the fact that narrows this argument's specific gap; ruling out a rival explanation helps a causal claim. A weakener supplies a rival cause, shows the sample does not match the group, or finds one case where a universal claim fails.

Try it. "Since the bakery began opening at 6 a.m., its morning sales have doubled, so the early opening caused the rise." The argument assumes that nothing else, such as a new office across the street, accounts for the whole rise; deny that and the conclusion falls. "Early opening is the only thing that ever changes sales" is too strong: deny it and the argument survives. You may not dispute that sales doubled.

Rule: name the gap first, deny a candidate assumption to test it, and when you weaken, leave the premises alone and attack the link.

A flaw task tells you the reasoning already goes wrong and asks you to name the mistake using only the passage. Such errors include overlooking other causes, generalizing from an unrepresentative sample, assuming what is to be proved, moving a property between parts and whole, and blurring two nearby concepts: correlation with causation, possibility with certainty, a necessary condition with a sufficient one.

A flaw description can be free of the topic, so match each piece of it to the passage and drop it if any piece finds no match. "Fails to consider" and "takes for granted" name a single gap from two sides. A description can be accurate yet name no error: citing an expert is fine in itself and goes wrong when, for example, the expert lacks relevant expertise. A weakener adds a new fact; a flaw is already in the passage.

Parallel tasks compare structure, not topic. Swap every recurring term for a letter, set the skeletons side by side, and check the details that change the form: all against some, which way each if-then runs, and must against probably.

Try it. "Every successful bakery in town uses local flour. Rosa's bakery uses local flour, so it will succeed." Local flour is required for success but not shown to be enough, so the argument treats a necessary condition as sufficient. "Every licensed pilot has logged training hours; Theo has, so he is licensed" matches. "Theo has not, so he is not licensed" is valid, so it does not.

Rule: name the error and write the skeleton before comparing; a match shares both the form and the flaw.

In an inference task, take the statements as given and find what they guarantee. A claim must be true when no possible situation keeps every statement true while the claim is false. A claim that merely could be true alongside the passage does not follow, and neither does one that needs an outside fact. Strong words need strong support, so if the passage says most, an option saying all or only overshoots. Statements sharing a term can combine: if most A are B and all B are C, then most A are C. But "most A are B" with "most B are C" guarantees nothing. A most-strongly-supported task lowers the bar to high likelihood, never past the evidence.

You can treat a principle as an if-then: if an action has certain features, it is permitted, wrong or required. It applies only when a case meets every condition in its if-part. Too narrow a principle misses the case; too broad a one may bind you to verdicts you would reject. When two principles clash, look for a claim, stated or assumed, that one outranks the other.

The trap is reading a principle's silence as a verdict. Suppose a library may waive a late fee if the book comes back within a week and the borrower has no earlier late returns. Sam returned his book in three days but was late once last year. The rule does not cover him, yet it never says the fee must be charged.

Rule: accept only what holds in every situation the passage allows, and apply a principle only when every condition in its if-part is met.

An apparent paradox pairs two facts that seem unable to hold together; state it as "X, yet Y" first. The tension may rest on assuming both facts describe one group, period or measure.

An evaluate task asks what you would most want to know to judge an argument. Answer a candidate question both ways: it is useful only if the two answers would move the conclusion in opposite directions. A disagreement needs a claim one speaker accepts and the other rejects; sharing a topic is not enough. A cannot-be-true task wants the claim the passage rules out.

Deduction puzzles run on consistency: suppose one possibility, trace what it forces, and drop it if it produces a contradiction. With few unknowns, list every assignment and cross off each that breaks a condition. "Exactly one note is true" means testing combinations, never single claims.

The trap is a half-explanation. "The bookstore sold more books than ever this year, yet its book revenue fell." "A popular author released a new novel" explains only the sales. "It began selling used paperbacks at a quarter of the new price, and many shoppers who once bought one new book now buy two used ones" explains both.

In a knights-and-knaves puzzle, knights always tell the truth and knaves always lie. Ava says, "Ben is a knave." Ben says, "Ava and I are different types." If Ava were a knight, Ben would be a knave telling the truth, which is impossible. So Ava is a knave, her claim is false, and Ben is a knight.

Rule: make a resolution account for both halves, and in a puzzle discard any supposition that forces a contradiction.