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An argument that names two options rests entirely on there being no third, and that is a claim like any other.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will test a two-option claim by trying to describe a case that is neither option, and say whether the pair is complete or leaves something out. You will write both kinds in symbols — one that can never come out false and one that can — and use the difference between them as the test.
You can break an argument by describing one case where the premises hold and the conclusion fails. This lesson uses the same move on a premise instead: a claim that there are only two possibilities is broken by describing a third.
| Term | What it means |
|---|---|
| Complete alternatives | Options that cover every possibility relevant to the question and stated conditions. |
| Mutually exclusive | Options that cannot both hold in the same relevant case. |
| False dichotomy | Treating two options as exhaustive when a relevant possibility lies outside both. |
| Slippery slope | An argument predicting a sequence of further outcomes from an initial step; each important link needs support. |
Half of an either-or argument is always the same: name two options, rule one out, conclude the other. All the weight is on the first step, and the first step is a claim like any other.
Two kinds of pair are safe. A sentence and its denial — the ferry ran or the ferry did not run — leaves nothing out by construction. And a pair that divides every number between them — more than forty or forty or fewer — leaves nothing out either, because every number is on one side.
Everything else has to be argued for. Either we spend the whole budget on the roof or the building falls down names two of the many amounts that could be spent, and quietly deletes the rest.
The notation makes the difference visible. Write the first as R | ~R and the second as W | D. The first cannot come out false however the world turns out. The second is false whenever both halves are false — spend half, and the building stands — so it is a claim somebody has to support.
Another way: picture
Draw a circle for everything that could happen and shade the two options inside it. A complete pair shades the whole circle. A false dichotomy leaves white space, and every case you can put in the white space breaks the argument.
Another way: steps
Three steps:
| The claim | In symbols | Can it be false? | |
|---|---|---|---|
| The switch is on, or it is not on | R \ | ~R | no, never |
| Whole budget, or the building falls | W \ | D | yes, whenever both fail |
| More than forty, or forty or fewer | M \ | ~M | no, never |
| July, or August | J \ | A | yes, in any other month |
The rows that can never be false are the ones where the second half is the denial of the first, written or not. Forty or fewer is exactly not more than forty, which is why it behaves like the switch.
So the test has a symbolic form: is the second option the denial of the first? If it is, the pair is safe. If it is a different claim that merely sounds opposite, start looking for the white space.
An either-or statement makes a claim about the space of possibilities. To evaluate it, first define the question and the relevant conditions. 'The meeting will be in Room A or Room B' may be complete if those are the only rooms available under a confirmed booking rule. The same pair is incomplete if the group can also meet online or use Room C. Names alone do not tell us whether alternatives have been exhausted. The conditions that restrict the choice need evidence too.
Distinguish complete from mutually exclusive. A pair is complete when every relevant possibility falls under at least one option. It is mutually exclusive when no possibility falls under both. These are different properties. 'The room is booked for music or for drama' could allow both activities on different sessions. 'The temperature is above ten or not above ten', with one fixed reading and unit, gives a complete pair in which the alternatives cannot both hold. If an argument relies on exactly one choice, check both coverage and overlap.
Pay particular attention to the word not. The denial of 'the club meets on Saturday morning' is 'the club does not meet on Saturday morning'. That denial includes meeting on Wednesday, meeting on Sunday, and not meeting at all. Replacing it with 'the club never meets' selects only one part of the denial. The apparent opposition is rhetorically strong but logically incomplete. A Wednesday meeting is a counterexample because it satisfies neither of the two restricted options.
A third option must be a case that falls outside both categories, not merely a criticism of one. Against 'coastal trip or no trip', 'the coast is expensive' is an objection, while 'a trip to the nearby museum' is a candidate alternative. You must still check whether the museum is available, affordable, and compatible with the trip's purpose. An imaginary solution with no connection to the actual constraints does not establish that a practical choice exists. It may reveal a logical omission without being a workable plan.
This is why a roof example needs care. Saying 'we could spend a third of the budget and the building would stand' does not itself establish a safe repair. To support that real alternative, obtain a suitable assessment and costed plan. In a logic exercise we can stipulate that a certified repair costing one third is available. Under that stated condition, 'spend the entire budget or accept collapse' clearly leaves out an option. In an actual decision, the stipulated feasibility would need evidence.
The best response to a forced choice is therefore precise: name the omitted option, show that it lies outside both offered categories, and state which constraints it meets or still needs checked. Do not assume that exposing an incomplete list makes the third option best. A complete comparison might still favor one of the original two. Your criticism establishes that the argument cannot reach its conclusion merely by pretending no other relevant possibility exists.
A slippery-slope argument says that one initial choice will lead through further steps to a significant later result. Such reasoning is not automatically a mistake. Some processes have well-supported mechanisms, and some decisions create incentives or precedents that affect later choices. The problem arises when a conclusion of inevitability outruns the support for the connecting steps. 'Allow one trial, and every restriction will disappear' needs more than a dramatic final picture.
Write the proposed chain as separate links. For example: permit one late club session; other clubs request the same arrangement; the committee approves every request; the building stays open every night; costs become unaffordable. Each arrow is a different claim. The first may make new requests plausible. It does not establish that every request will be approved. Approval rules, staffing limits, a trial end date, and a fixed budget can interrupt the chain. Identify the first link that lacks support rather than arguing only about the final disaster.
A limited trial can be evaluated with these questions visible. Does the proposal include a review date? Who authorizes extensions? Are there criteria that distinguish eligible requests? What resources cap the expansion? Such safeguards do not magically prove that no bad outcome is possible. They bear on the specific claim that the sequence is unavoidable. If the critic says the safeguards will fail, that further claim also needs evidence rather than being treated as a fact built into the word slippery.
Compare a supported physical chain. An overflowing gutter can direct water against a wall; repeated wetting can damage susceptible materials under specified conditions. Evidence about the gutter, weather exposure, and materials can make this a serious maintenance concern. It is still useful to state the conditions and uncertainty rather than promising a precise disaster on a precise date without support. A chain becomes stronger through evidence for its links, not by adding more alarming adjectives to its endpoint.
Slippery-slope rhetoric can also create a false dichotomy. 'Either reject this one trial or accept unlimited permanent opening' deletes the option of conducting a bounded trial with review. The two errors are related but distinguishable. The dichotomy concerns the options allowed in the present decision. The slope concerns the predicted connections between that decision and later events. A good analysis can name both while identifying the exact omitted option and the unsupported link.
When responding, avoid the opposite overstatement that the first step can never have wider effects. A fair reply might say: 'Other clubs may request similar treatment, but approval is not automatic; the proposal limits the trial to ten weeks and requires a new vote for extension.' That statement concedes a plausible consequence and rejects an unsupported inevitability. It lets the discussion compare realistic risks and controls instead of choosing between catastrophe and the equally unsupported claim of zero risk.
A fictional youth club has 900 credits available for a one-day trip. A coach visit to the coast costs 1,200 credits. A leader says, 'Either we go to the coast or cancel the trip. We cannot afford the coast, so we must cancel.' The price comparison supports the claim that the coastal plan exceeds this budget by 300 credits. It does not establish the initial either-or.
The club has also received a confirmed quotation of 700 credits for a nearby museum visit with the same required transport capacity and supervision arrangements. This is a concrete third option under the case's stated conditions. It is a trip, so it does not fall into the cancellation category. It is not a coastal trip, so it does not fall into the other category. The museum option leaves 900 − 700 = 200 credits within the stated budget.
The leader replies, 'If we accept a cheaper local trip once, soon nobody will allow a coastal trip again.' That is a further prediction requiring a connection between this year's decision and future rules. If the club decides each year's trip separately, the claimed permanent prohibition does not follow automatically. Members may still prefer a coastal trip in a later year with a different budget.
The corrected decision is to compare the feasible museum plan with cancellation and any other verified alternatives. Discovering the museum option does not prove it is enjoyable or best; those considerations remain open. It does show that cancellation is not forced merely by the coastal price. The committee can make a real comparison once the omitted possibility and the unsupported future claim are made explicit.
Treating every either-or as a trick. Plenty of them are complete, and an argument built on a complete one is perfectly good.
Offering an objection as a third option. The coast is too far is a remark about one of the two options. A third option is a case that is neither of them.
Thinking two very different options must be all there are. How far apart the two are has nothing to do with whether anything lies outside both; opposites in tone are not opposites in logic.
Write the two offered options.
A: meet Saturday morning. B: never meet.
An explicit list makes the claimed coverage visible.
Write the actual denial of A.
Not A: do not meet Saturday morning.
Denial excludes that time without excluding all other times.
Construct a relevant alternative.
Meet Wednesday evening in an available room.
This option fits neither Saturday morning nor never meeting.
Check both original categories.
Wednesday makes A false and B false.
A third option defeats the claim that at least one of the original two must hold.
State the limited conclusion.
The original pair is incomplete.
This does not yet establish that Wednesday is the best available time.
Fix the quantity being classified.
The number of people in this room at noon.
Both alternatives must concern the same quantity and time.
State the first category.
More than 40 people.
This includes every count above the boundary.
State its exact denial.
40 or fewer people.
This includes the boundary itself and all lower counts.
Test the apparent boundary case.
Exactly 40 belongs to the second category.
Using fewer than 40 instead would leave this case out.
Classify the pair.
Complete and mutually exclusive.
Every possible count falls into exactly one of the two specified ranges.
Record the available budget.
900 credits.
The budget is a stated constraint on current plans.
Check the coastal proposal.
1,200 − 900 = 300 credits over budget.
The calculation rules out that plan under the current funding constraint.
Check the confirmed museum alternative.
700 credits, leaving 200.
The case states that the transport and supervision requirements are met.
Test the forced pair.
Museum trip is neither coastal trip nor cancellation.
A feasible third option makes the original dichotomy incomplete.
Separate the future prediction.
Choosing the museum once does not itself ban future coastal trips.
A connection to future decision rules has not been supplied.
State the corrected decision.
Compare feasible current alternatives and examine any supported future effects.
Neither the forced choice nor an unsupported slope can replace that comparison.
State the immediate proposal.
One extra session for ten weeks, with a new vote required for extension.
The limits are part of the actual decision.
Locate the unsupported link in the objection.
A request from another club does not automatically receive approval.
The stated voting requirement interrupts the proposed inevitable chain.
Name the omitted option in the forced choice.
Four either-or claims. Match each to whether the two options between them cover everything.
| The pair covers everything | The pair leaves something out | |
|---|---|---|
| Either the door was shut or it was not shut. | ||
| Either you buy lunch here or you go hungry until four. | ||
| Either the number is more than nine or it is nine or fewer. | ||
| Either the holiday is in July or it is in August. |
A club has 900 credits. The coast costs 1,200; a confirmed feasible museum trip costs 700. Complete the comparison before judging the claim 'coast or cancellation'.
Calculate the coastal shortfall.
Coastal quotation minus available budget = shortfall credits.
This establishes why the current coastal plan exceeds the available funds.
Calculate the museum plan's remaining budget.
900 − 700 = remaining credits.
The positive remainder shows that the stated alternative meets this budget constraint.
Count the explicitly available current options including cancellation.
Coast, museum, cancellation: count options named.
The original pair omitted the museum even though its feasibility is stated.
Write W for *we spend the whole budget on the roof* and D for *we let the building fall down*. Put the claim *either we spend the whole budget on the roof or we let the building fall down* into symbols, using the bar for *or*.
Answer:
A club has 900 credits. The coastal plan costs 1,200. A confirmed nearby museum plan costs 700 and meets the stated transport and supervision requirements. Construct the supported links; the evidence does not force cancellation or ban future coastal trips.
This task has no paper form; do it on a device.
Write R for *the switch is on*. The switch has two positions and no third. Put the claim *either the switch is on or it is not on* into symbols, using ~ for *not* and the bar for *or*.
Answer:
A notice about the summer trip says: *either the trip goes to the coast or it does not go at all.* Which of these shows that the pair leaves something out?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A proposal permits one late session for ten weeks, after which any extension needs a separate vote. A critic says 'either reject this trial or approve permanent nightly opening'. The secretary confirms another club has submitted a request but that no extension has been approved. Construct the supported links.
This task has no paper form; do it on a device.
You can find the third case that breaks a forced choice, and say why a pair made of a sentence and its denial cannot be broken. Write down in symbols the difference between on or not on and July or August. Next: reasons about what would follow, and the arguments where they belong.
13. A bounded evening-opening trial, step 3
Conduct the bounded trial, then review it.
That option is neither rejecting every trial nor accepting unlimited opening.