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One case where the reasons hold and the conclusion fails settles whether a premise is missing.
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 decide whether an argument is valid by looking for a single case in which every premise is true and the conclusion is false, and you will write that case down when you find one. You will then say which extra premise shuts the case out, and check that the same premise written the other way round does not.
You can name the sentence an argument leaves out. This lesson checks the claim that it was needed, with a test that settles it rather than a feeling that something is missing: find one case where the reasons hold and the conclusion does not.
| Term | What it means |
|---|---|
| Valid | No possible case makes every premise true and the conclusion false. |
| Counterexample | One case with all premises true and the conclusion false. |
| Conditional | An if-then claim; A → B is false only when A is true and B false. |
| Equivocation | An inference that depends on treating different meanings of a word as the same meaning. |
Valid is a fact about the shape of an argument, not about whether anything in it is true. An argument can be valid and have false premises; it can be invalid with everything in it true.
To show an argument is missing a premise, describe one case where the stated reasons hold and the conclusion fails. Mira has a season ticket, so Mira swims every week — imagine somebody who bought a ticket in January and has been twice. Reason true, conclusion false, argument broken.
Supply the premise — anybody with a season ticket uses it every week — and that case is gone: it is no longer a case where every premise is true. That is what the missing premise was doing.
The notation. Letters stand for whole sentences, and this course writes them as single capital letters. The one symbol used is the arrow: K -> S reads if K then S. You may type it as -> or as →; both are accepted. A breaking case is written by setting each letter true or false.
Another way: picture
Think of the premises as a fence around the conclusion. Valid means there is no gap: no way to be inside every premise and outside the conclusion. Showing an argument invalid is walking through the gap and pointing at where you came in.
Another way: steps
Three steps to break an argument:
K -> S promises that whenever K holds, S holds. It promises nothing at all about what happens when K is false.
| The case | Is K -> S true in it? |
|---|---|
| K true, S true | yes |
| K true, S false | no — this is the only case it rules out |
| K false, S true | yes |
| K false, S false | yes |
That one row is the whole of it, and it is why the arrow written the wrong way round repairs nothing. From B -> C and C you cannot get B: the third row of the table is a case where both premises hold and B is false.
A breaking case, also called a counterexample or countermodel, has a strict job. It must make every stated premise true and the conclusion false in one and the same situation. Finding a false premise is not enough. Finding a situation where the conclusion is false is not enough either, because the premises might also be false there. Keep a small checking sheet with one line for each premise and one for the conclusion. Write true or false beside every line before deciding whether the case succeeds.
For example, take 'The alarm is sounding; therefore there is a fire.' Let A mean the alarm sounds and H mean there is a fire. A test of the alarm gives A true and H false. That is a counterexample to the argument as stated. A quiet building with no fire gives A false and H false, which is not a counterexample because it does not satisfy the premise. Both stories have no fire, but only the first preserves the evidence the argument actually gives.
Now add 'If the alarm sounds, there is a fire', written A → H. The alarm-test case no longer meets every premise. It makes A true but A → H false. This shows what the added premise does: it excludes the very possibility that broke the original inference. It does not show that the added premise is sensible. In an ordinary building that tests alarms, the premise is plainly too strong. Validity evaluates what follows if the premises hold; it does not license us to invent false premises to reach a preferred conclusion.
Why does one case suffice? The claim of validity covers every possible assignment that makes the premises true. One assignment violating the required relation defeats that universal claim. By contrast, collecting ten or ten thousand cases in which an alarm coincides with a fire does not prove that no exception is possible. Those observations might support an empirical prediction, but they do not establish the unrestricted logical consequence by themselves. Keep a deductive guarantee separate from a reliable pattern.
When there are only two letters, list the four assignments: both true; first true and second false; first false and second true; both false. The conditional A → H is false only in the second of those assignments. This truth rule is what the exercises use. It does not claim that every true material conditional describes a causal connection or supplies useful real-world evidence. The notation isolates one part of the reasoning so we can inspect whether the stated premises exclude a false conclusion.
For a longer argument, start with the conclusion false and work backwards through the premises. If the conclusion is H, try H false. If a premise is A → H, keeping that premise true then requires A false. If another premise says A is true, the requirements conflict and this attempted counterexample fails. With these premises there is no way to keep them both true while making H false. Explain the conflict explicitly; simply saying that you cannot imagine a counterexample is weaker than showing why the truth requirements rule one out.
Not every defective bridge is an omitted if-then sentence. Sometimes an argument uses one word in two meanings and behaves as though the meanings were the same. That mistake is called equivocation. Consider: 'The museum visit is free. Everything free is unrestricted. Therefore we can enter any room.' In the first sentence, free concerns the absence of an admission charge. In the second, it means unrestricted. Repeating the sound of the word does not establish a connection between price and permission.
Rewrite the two meanings explicitly. 'The visit costs no admission fee' does not state that every room is open to visitors. A free museum can have locked stores, staff offices, and conservation rooms. That familiar possibility makes the price premise true while the unrestricted-access conclusion is false. Once the meanings are separated, the gap becomes visible. The argument would need an additional access policy connecting a free ticket to permission to enter every room, and that policy cannot be obtained merely from the word free.
Equivocation does not require deliberate trickery. A writer may slide between meanings without noticing. Your response should identify the change, not guess at the writer's motives. Quote the relevant word in each sentence, replace it with its intended meaning, and ask whether the resulting claims connect. If they do not, state the missing connection. This is more informative than simply writing 'equivocation' beside the paragraph, because it shows the reader exactly where the inference fails.
Compare a legitimate use of a repeated word. 'Every free-admission exhibition in this program needs an online booking; this is a free-admission exhibition in the program; therefore it needs an online booking.' Here free-admission retains the same meaning throughout. The presence of a word with several possible dictionary meanings does not make every argument using it fallacious. The problem occurs when the inference actually relies on treating distinct meanings as interchangeable. Consistent use avoids that particular defect.
A related example involves a loan advertised as 'light'. A lightweight folding bicycle is physically easy to lift, while a light repayment burden is financially easy to carry. Neither description entails the other. Before converting a verbal argument into letters, write complete definitions for the letters. If P stands for physically light, do not later let it stand for inexpensive. A truth-table calculation can check the wrong argument perfectly if the translation has already hidden a meaning change.
Use two stages when evaluating ordinary prose. First check that the claims have stable meanings and that the symbols faithfully represent them. Then test whether the premises, with those meanings, leave a case in which the conclusion fails. If you find such a case, distinguish the logical diagnosis from a proposed repair. Adding 'every no-fee ticket grants unrestricted access' would formally bridge the museum inference, but the museum's actual policy may make that addition false. A good repair needs both a valid connection and premises that deserve acceptance.
A fictional school trip has 24 signed permission forms. A student argues, 'My form is signed, so the trip is definitely going ahead.' Let S mean the student's form is signed and G mean the trip goes ahead. The stated argument has premise S and conclusion G. The number of other forms may be useful planning information, but no stated premise yet makes a signed form sufficient for the trip.
Construct one case in which S is true and G false: all 24 forms were signed on Monday, but a transport cancellation on Thursday prevents the Friday trip. This story preserves the signed form and denies the proposed conclusion. It therefore breaks the argument as stated. It does not establish that a cancellation actually happened or is likely; its role is to show that the guarantee does not follow from the given information.
Someone suggests adding 'If the trip goes ahead, the form is signed', or G → S. This says a signature is a required condition for attending under the stated rule. It does not make a signature a guarantee that the trip will happen. In the cancellation case G is false and S is true, so G → S is true, S is true, and the conclusion G remains false. The proposed repair has left the counterexample intact.
The conditional S → G would remove that case, but the cancellation possibility gives us a reason not to accept such an unrestricted policy as a fact. A more careful conclusion is that the signature satisfies one requirement. The trip still depends on transport, staffing, and other arrangements. Testing the missing premise helps the student replace an unwarranted guarantee with an accurate statement about what the form actually establishes.
Answering with a likely case instead of a possible one. A breaking case does not have to be probable, or even sensible. It has to be possible.
Reading valid as true. Every bus is red, this is a bus, so this is red is a valid argument and its first premise is false. Valid is about the shape.
Supplying the arrow backwards. If Mira swims weekly then Mira has a ticket closes nothing: the case where she has a ticket and does not swim is still sitting there, untouched.
Name the two claims.
A: alarm sounds. H: fire exists.
Letters must represent whole claims with fixed meanings.
Record the argument.
Premise A; conclusion H.
Only the alarm observation is stated as evidence.
Make the conclusion false.
H = false.
A breaking case must deny the claim the argument says follows.
Keep the premise true at the same time.
A = true during an alarm test.
The test story allows the stated evidence to hold without a fire.
Check and report the result.
A true, H false: invalid as stated.
This single assignment meets the definition of a counterexample.
Define the window case.
W: window open. F: floor wet.
The symbols distinguish the proposed cause from the observed effect.
Write both premises and the conclusion.
Premises F and W → F; conclusion W.
The added conditional runs from open window to wet floor.
Try a spill with the window shut.
F = true; W = false.
A spill offers a coherent way for the effect to occur without the proposed cause.
Evaluate each premise.
F true; W → F true.
The conditional makes no demand for W to be true merely because F is true.
Compare the conclusion.
W false, while both premises are true: invalid.
The reverse conditional has not excluded the counterexample.
Identify the first use of free.
The museum visit is free: admission costs zero.
This claim concerns price rather than access rules.
Identify the changed use of free.
Everything free is unrestricted: freedom from constraints.
The same word now expresses a different property.
Replace the ambiguous wording.
No admission charge; therefore entry to every room is permitted.
Explicit meanings reveal the missing connection.
Construct a case preserving the price claim.
Admission costs zero, but the conservation room is locked to visitors.
Free admission and restricted access can coexist.
Diagnose the unsupported step.
No-fee entry does not entail unrestricted access; the inference equivocates.
The attempted bridge relied on the repeated word rather than a stated access policy.
State what a genuine repair would require.
An independently supported policy permitting entry to every room.
A verbal substitution alone cannot establish the missing permission.
Name the evidence and conclusion.
S: form signed. G: trip goes ahead.
The two claims concern different requirements of the trip.
Construct the candidate breaking case.
S true, G false after a transport cancellation.
A cancellation can prevent the trip without erasing the earlier signature.
Check the reverse conditional offered as a repair.
Write K for *Mira has a season ticket* and S for *Mira swims every week*. This is the argument exactly as it was given, with nothing added. Is it valid? If it is not, give a case that breaks it.
K
∴ S
valid invalid — countermodel:
A means the alarm sounds; H means there is a fire. The argument has premise A and conclusion H. Complete a counterexample, using the truth-value symbols T (true) and F (false).
Make the conclusion fail.
H = fire.
A counterexample requires denying the conclusion.
Satisfy the observation.
A = alarm.
The assignment must satisfy every stated premise.
Evaluate the missing bridge in this assignment.
A → H is bridge.
The conditional fails exactly when its antecedent holds and consequent fails.
*Mira has a season ticket, so Mira swims every week.* Which extra sentence would make that argument valid?
The same letters: K for *Mira has a season ticket*, S for *Mira swims every week*. This time the missing sentence has been written in as a second premise. Is the argument valid now?
K
K -> S
∴ S
valid invalid — countermodel:
A new argument, and somebody has already written a premise in. Write B for *the bridge is shut* and C for *the traffic comes past the school*. The premises are: if B then C, and C. The conclusion is B. Is that valid? If not, give the case that breaks it.
B -> C
C
∴ B
valid invalid — countermodel:
Four steps of one argument. Mark the first step you could refuse while accepting everything above it.
This task has no paper form; do it on a device.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
R means the room is reserved, L means the event is listed, and O means the event opens. The organizer gives R and R -> L, then concludes O. Decide validity. If invalid, construct a counterexample by assigning all three letters.
R
R -> L
∴ O
valid invalid — countermodel:
You can break an invalid argument with one case and say which premise closes it, and you can explain why an if-then written backwards repairs nothing. Tell somebody why a valid argument can still have a false premise in it. Next: the first of six moves that are mistakes in one setting and good reasoning in another.
13. Test the signed-form guarantee, step 3
G → S remains true, so the argument is still invalid.
Requiring a form when a trip happens does not guarantee a trip whenever a form is signed.