practice_gen_nl

Practice Questions (Generated): Arguments & Propositions in Natural Language

These questions are newly generated for extra practice. They do not appear in the original exam bank (practice_extracted.md). All example sentences, passages, and syllogisms are different.

Reading guide: each question has the question text, a short Answer, then a Worked solution in numbered steps. Lines starting with “Note (check)” flag a point in the original answer that looks dubious; the original answer is kept unchanged.

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Q1 (3 marks): Fill in the blanks

Fill in each blank with the correct technical term.

“An ________ is a group of statements in which one or more statements, called the ________, are offered as reasons for the truth of another statement, called the ________. An argument in which the arguer claims the conclusion follows with absolute necessity from the premises is called a ________ argument; one in which the arguer claims only that the premises make the conclusion probable is called an ________ argument.”

Answer: argument; premises; conclusion; deductive; inductive.

Term table

Blank	Term	Cue in the sentence
1	argument	the whole group of statements
2	premises	statements offered as reasons
3	conclusion	the statement the reasons are meant to establish
4	deductive	”absolute necessity”
5	inductive	”only probable”

Worked solution

1. Recall the definition of argument

   • An argument is a structured set of statements with two functional roles.
   • Some statements give reasons (premises); one is what the reasons are meant to establish (conclusion).

2. Fill blanks 1 to 3

   • The passage follows the standard textbook definition.
   • Group of statements = “argument”; reasons or evidence = “premises”; the statement claimed to follow = “conclusion”.

3. Fill blanks 4 and 5

   • Conclusion follows with necessity: “deductive”.
   • Conclusion follows only with probability: “inductive”.

4. Confirm the ordering

   • The definition of deductive comes before inductive in the passage.
   • The stronger claim (necessity) is always the deductive one; the weaker claim (probability) is always the inductive one.

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Q2 (4 marks): Mark reason (R) and conclusion (C)

For each passage below, mark the reason(s) with R(…) and the conclusion(s) with C(…). In cases where one statement serves as both a conclusion of one inference and a premise of another, label it accordingly (e.g., C1-R2).

(a) Since the temperature has been below freezing all week, it follows that the lake will be frozen over by morning.

(b) Air travel has become cheaper and more accessible than ever before. Cheaper air travel has in turn led to a dramatic rise in international tourism. We can therefore conclude that the globalisation of leisure travel is accelerating.

(c) In the long run, a policy of appeasement will only embolden aggressors. History teaches us this, because it shows that every concession made to a hostile power in the twentieth century was followed by an escalation of demands.

(d) Regular physical activity is associated with improved mental health outcomes. Furthermore, diet quality has been linked to mood stability and cognitive function. A healthy lifestyle should therefore be understood as essential to psychological wellbeing, not merely physical health.

Answer:

• (a) Since R(the temperature has been below freezing all week), it follows that C(the lake will be frozen over by morning).
• (b) R1(Air travel has become cheaper and more accessible than ever before). C1-R2(Cheaper air travel has in turn led to a dramatic rise in international tourism). C2(The globalisation of leisure travel is accelerating).
• (c) C(In the long run, a policy of appeasement will only embolden aggressors). R1(History teaches us this), because R2(every concession made to a hostile power in the twentieth century was followed by an escalation of demands).
• (d) R1(Regular physical activity is associated with improved mental health outcomes). R2(Diet quality has been linked to mood stability and cognitive function). C(A healthy lifestyle should be understood as essential to psychological wellbeing, not merely physical health).

Argument maps (solid arrow = “is offered as a reason for”)

(a)  R  temperature below freezing all week
         │   "Since" marks the reason
         ▼   "it follows that" marks the conclusion
     C  lake frozen over by morning


(b)  R1    air travel cheaper and more accessible
        │
        ▼   "in turn led to"
     C1-R2 cheaper travel -> dramatic rise in international tourism
        │
        ▼   "therefore ... conclude"
     C2    globalisation of leisure travel is accelerating


(c)  C     appeasement only emboldens aggressors        (stated first)
        ▲                  ▲
        │                  │
     R1  "History      R2  every concession in the 20th century
          teaches          was followed by escalating demands
          us this"         ("because" introduces R2)


(d)  R1  physical activity ~ better mental health ──┐
                                                     ├──▶  C  healthy lifestyle is
     R2  diet quality ~ mood and cognition ──────────┘        essential to wellbeing
         ("Furthermore" adds a second reason)                 ("therefore")

Worked solution

1. List the indicator words

   • Conclusion indicators: “it follows that”, “therefore”, “hence”, “we can conclude that”.
   • Premise indicators: “since”, “because”, “furthermore”, “given that”.

2. Tag (a)

   • “Since” opens the reason; “it follows that” opens the conclusion.
   • Structure: R → C.

3. Tag (b)

   • There are two inferential steps.
   • “Cheaper air travel leads to more tourism” is a conclusion of step 1 (supported by the price fact) and a reason for step 2.
   • “Therefore” marks the final conclusion. Structure: R1 → C1-R2 → C2 (a chain).

4. Tag (c)

   • The conclusion comes first, before the reasons: an order-reversed argument.
   • “Because” then introduces the evidence.
   • R1 and R2 together support C.

Note (check): “because” attaches R2 to R1 (“history teaches us this, because every concession … was followed by escalation”), so R2 → R1 → C is arguably a more faithful map than R1 and R2 both feeding C directly.

5. Tag (d)
   • No indicator marks the first two sentences as conclusions; they are two parallel reasons.
   • “Furthermore” accumulates a second premise; it does not signal a conclusion.
   • “Therefore” near the end marks the conclusion.

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Q3 (4 marks): Extract the main conclusion from an editorial passage

Identify the main conclusion of each passage. The conclusion may not be stated word-for-word and may require paraphrase.

3.1 “Urban cycling infrastructure in most cities remains dangerous and fragmented. Cyclists frequently have to share roads with fast-moving vehicles, and the absence of protected lanes makes every journey a calculated risk. Meanwhile, investment in public transit continues to lag. It is no wonder that car ownership remains the only viable option for most residents. Until cities fundamentally redesign their streets to prioritise people over vehicles, sustainable transport will remain a dream rather than a reality.”

3.2 “Every year, billions of plastic bottles are produced worldwide, and a significant fraction ends up in the ocean, where it breaks down into microplastics that enter the food chain. Recent studies have detected microplastics in human blood and lung tissue. Voluntary corporate pledges to reduce plastic production have consistently fallen short of stated targets. Regulation has failed to keep pace with the scale of the problem. Binding international agreements with enforceable targets are no longer optional, they are the only path forward.”

Answer:

• 3.1 Cities need to fundamentally redesign their streets to prioritise people over vehicles in order to make sustainable transport viable (or equivalently: car-centred street design is the root cause of unsustainable transport, and only a structural redesign can fix this).
• 3.2 Binding international agreements with enforceable targets on plastic production are necessary to address the plastic pollution crisis.

Labelled sentences

Passage	Sentence (short)	Role
3.1	cycling infrastructure dangerous and fragmented	evidence
3.1	cyclists share roads; no protected lanes	evidence
3.1	transit investment lags	evidence
3.1	”no wonder” cars remain the only option	evidence (car dependence)
3.1	”Until cities redesign streets … a dream”	main conclusion (prescription)
3.2	billions of bottles; microplastics in the food chain	evidence
3.2	microplastics found in human blood and lungs	evidence
3.2	voluntary pledges fell short	evidence
3.2	regulation has not kept pace	evidence
3.2	”Binding international agreements … only path forward”	main conclusion

Worked solution

1. Define the main conclusion

   • In an editorial it is the one claim the author most wants accepted.
   • Every other statement is evidence for it, a sub-conclusion building toward it, or a conceded objection.

2. Apply the “what is everything else supporting?” test

   • Is the sentence evidence for something else, or the final target of all the evidence?
   • The claim that everything supports, but that supports nothing further, is the main conclusion.

3. Apply to 3.1

   • Dangerous cycling, missing protected lanes, weak transit and car dependence are all evidence of a structural failure.
   • “Cycling is dangerous” is evidence, not the conclusion.
   • The conclusion is the prescription in the last sentence: redesign streets to make sustainable transport real.

4. Apply to 3.2

   • The pollution statistics, microplastics in human tissue and failed voluntary pledges are premises.
   • The last sentence (“Binding international agreements … are no longer optional”) is explicitly the author’s conclusion.

5. Expect paraphrase

   • In both passages the conclusion is a paraphrase of the overall prescriptive claim, not one verbatim sentence.
   • This is normal for editorial reasoning.

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Q4 (5 marks): Is this passage an argument?

For each passage, state (i) whether it is an argument and (ii) briefly explain why or why not. If it is an argument, identify the conclusion.

(a) “If governments implement a four-day working week, worker productivity will increase.”

(b) “Isn’t it obvious that anyone who drives while using a phone is putting other road users in danger?”

(c) “Students are strongly advised to proofread their essays before submitting them.”

(d) “The candidate filed her nomination papers on Tuesday morning. By Wednesday the papers had been processed, and by Thursday her name appeared on the official ballot.”

(e) “Deforestation in the Amazon has accelerated every year for the past decade, and satellite data confirm the loss of millions of hectares of forest annually. Climate scientists have documented a measurable reduction in local rainfall patterns as a direct result. It is clear, therefore, that continued deforestation will permanently destabilise the regional climate.”

Answer: (a) to (d) are not arguments; (e) is an argument. Conclusion of (e): “Continued deforestation will permanently destabilise the regional climate.”

Verdict table

Passage	Argument?	Deciding feature
(a)	No	conditional “If P then Q”: neither P nor Q is asserted
(b)	No	rhetorical question: no premise-conclusion structure
(c)	No	recommendation or directive: no reasons offered
(d)	No	chronological narration: temporal words, not inference words
(e)	Yes	”therefore” marks a conclusion backed by two factual claims

Worked solution

1. Recall the two requirements

   • Every constituent must be a declarative claim (capable of being true or false).
   • At least one statement must be offered as a reason for another (the conclusion).

2. Test (a)

   • “If P then Q” asserts neither P nor Q; it only claims a hypothetical relationship.
   • With no claim actually asserted there is no premise and no conclusion.

3. Test (b)

   • A rhetorical question implies an answer but does not state a premise or draw a conclusion.
   • It makes no categorical assertion and offers no evidence for a stated claim: a form of expression, not argumentation.

Note (check): a rhetorical question usually does assert its implied answer (“anyone who drives while using a phone endangers others”), so “no categorical assertion” is loose; the verdict still holds because no reasons are given for that claim.

4. Test (c)

   • “Students are advised to …” is a directive, not a claim with a truth value in the standard sense.
   • No reasons are given for the advice, so there is no premise-conclusion structure.

5. Test (d)

   • Events are linked by temporal language (“by Wednesday”, “by Thursday”).
   • There is no inference language (“therefore”, “because”, “it follows that”): a narrative reports what happened, not what follows from what.

6. Test (e)

   • Two factual claims are offered as evidence: accelerating deforestation and documented rainfall reduction.
   • “It is clear, therefore, that” is an explicit conclusion indicator.
   • The conclusion is the statement the evidence is designed to establish.

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Q5 (4 marks): Deductive or inductive?

Classify each argument as deductive or inductive and justify your answer. For any case you consider genuinely ambiguous, explain why it could reasonably be classified either way.

(a) “All prime numbers greater than 2 are odd. The number 97 is a prime number greater than 2. Therefore, 97 is odd.”

(b) “Every patient who received the experimental drug in the clinical trial recovered within a week. Every patient who received the placebo took at least three weeks to recover. Therefore, the drug accelerates recovery.”

(c) “The word ‘homicide’ derives from the Latin ‘homo’ (human) and ‘caedere’ (to kill). Therefore, homicide means the killing of a human being.”

(d) “No amphibian is a reptile. The axolotl is an amphibian. Therefore, the axolotl is not a reptile.”

Answer: (a) Deductive. (b) Inductive. (c) Ambiguous: either is defensible (no penalty for either, provided the reasoning is explained). (d) Deductive.

Test: if all premises are true, can the conclusion still be false?

Passage	Verdict	Deciding feature
(a)	Deductive	universal rule applied to an instance: no way to be false
(b)	Inductive	finite sample generalised to a causal claim
(c)	Ambiguous	definitional (deductive) or etymology-based inference (inductive)
(d)	Deductive	valid syllogism, form EAE

Worked solution

1. Recall the distinction

   • Deductive: the conclusion follows with necessity; true premises make a false conclusion impossible.
   • Inductive: the premises only make the conclusion probable; it could still be false.

2. Test (a)

   • A universal rule (“all primes > 2 are odd”) is applied to a specific instance (97).
   • This is universal instantiation: the conclusion follows with logical necessity.

3. Test (b)

   • The argument generalises from a finite sample to a universal causal claim.
   • The sample may be unrepresentative, there may be confounders, or the effect may be specific to this trial.
   • The conclusion is probable, not necessary: an inductive generalisation with a causal claim.

4. Test (c)

   • Reading 1 (deductive): “homicide” just means what its Latin roots say, so the conclusion follows by definition.
   • Reading 2 (inductive): etymologies do not always preserve original meanings, so deriving a current definition from roots is a probabilistic inference.
   • Both readings are legitimate, so either classification is defensible.

5. Test (d)

   • This is a categorical syllogism: universal negative (“No A is B”) plus an affirmative premise about the axolotl (“X is A”) gives a negative conclusion (“X is not B”).
   • This is the valid form EAE, so the conclusion follows necessarily.

Note (check): “The axolotl is an amphibian” is a singular statement, not a particular one; the original calls it a “particular affirmative”. The verdict is unaffected (singular premises are usually treated as A in the Celarent form EAE-1).

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Q6 (5 marks): Best categorical form (A/E/I/O) for each NL proposition

Restate each sentence as a standard categorical proposition (A: “All S are P”, E: “No S are P”, I: “Some S are P”, or O: “Some S are not P”). Identify the subject class S, the predicate class P, and the form letter.

(a) “A whale is a mammal.” (b) “Not all students who study hard receive top marks.” (c) “Hardly anyone left the theatre before the end of the performance.” (d) “Only licensed pharmacists may dispense prescription medication.” (e) “Everybody except the very young has heard of Darwin.”

Answer:

• (a) All S are P (A); S = whales, P = mammals.
• (b) Some S are not P (O); S = students who study hard, P = people who receive top marks.
• (c) Some S are not P (O); S = audience members, P = people who left before the end. (Equivalently: almost all S are not P, which is captured by O as the closest standard form.)
• (d) All S are P (A); S = people who may dispense prescription medication, P = licensed pharmacists. (Equivalently, E: No S are P where S = non-licensed pharmacists, P = people who may dispense.)
• (e) All S are P (A); S = people who are not very young, P = people who have heard of Darwin.

Answer table

Sentence	Keyword trap	S	P	Form
(a) A whale is a mammal.	”a” means “every”	whales	mammals	A
(b) Not all students who study hard receive top marks.	”not all” means “at least one is not”	students who study hard	people who receive top marks	O
(c) Hardly anyone left before the end.	”hardly anyone” means “almost no one”	audience members	people who left before the end	O (see note)
(d) Only licensed pharmacists may dispense.	”only” reverses the terms	people who may dispense prescription medication	licensed pharmacists	A
(e) Everybody except the very young has heard of Darwin.	”except” removes a group from S	people who are not very young	people who have heard of Darwin	A

Cheat sheet: NL pattern to A/E/I/O

NL pattern	Standard form	Rule of thumb
a / an X is Y (generic)	A: All X are Y	the article acts as “every”
Only X are Y	A: All Y are X	swap: the “only” term becomes the predicate
None but X are Y	A: All Y are X	same as “only”
Not all X are Y	O: Some X are not Y	negation of A
All X are not Y (idiom)	O: Some X are not Y	”not everything”; the literal reading is E
Hardly any X are Y	O (per the original answers)	few X are Y; see the notes
Every X except Z is Y	A: All non-Z X are Y	the exception leaves S
There are X that are (not) Y	I (or O when negated)	existential reading
Anyone who is not X may not Y	E: No non-X are Y	universal negative

Worked solution

1. Recall the four forms

   • A “All S are P”: universal affirmative. E “No S are P”: universal negative.
   • I “Some S are P”: particular affirmative. O “Some S are not P”: particular negative.
   • Every NL sentence must be mapped onto one of these before formal work.

2. Apply to (a)

   • “A” or “an” before a class noun often works as a universal quantifier.
   • “A whale is a mammal” means “Every whale is a mammal”, not “exactly one whale is a mammal”.

3. Apply to (b)

   • “Not all” means “at least one is not”.
   • Maps directly to O: some students who study hard are not top-mark recipients.

4. Apply to (c)

   • “Hardly anyone” means “almost no one” or “very few”: only a small number have the property, most do not.
   • Closest standard form is O: some audience members are not people who left before the end.
   • Strictly, “hardly anyone left” means few people left, so S = audience members and P = people who left; the sentence says most S are not P, captured as O.

Note (check): “hardly any S are P” is not a standard O. It is closer to a weak E (“almost no S are P”) or to the pair “some S are P and some S are not P” (I and O together). “Some S are not P” is true but loses the point of the sentence.

5. Apply to (d)
   • “Only” reverses the direction of the universal quantifier.
   • “Only seniors may take this course” says “All who may take the course are seniors”, not “All seniors may take it”.
   • So “Only licensed pharmacists may dispense” means “All dispensers are licensed pharmacists”: S = people who may dispense, P = licensed pharmacists, form A.

Note (check): S and P are named correctly, but the E equivalent in the original says S = “non-licensed pharmacists”; it should read S = people who are not licensed pharmacists, P = people who may dispense.

6. Apply to (e)
   • “Everybody except X” means “All non-X”.
   • Restated: “All people who are not very young have heard of Darwin”. S = people who are not very young, P = people who have heard of Darwin. Form A.

Note (check): an exceptive statement is strictly compound (it also implies the excluded group is not covered, e.g. not all of the very young have heard of Darwin); the original keeps only the A part.

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Q7 (5 marks): Syllogism validity by Venn diagram

Determine whether the following syllogism is deductively valid by constructing the 7-region Venn diagram. Show each diagramming step explicitly.

“No historians are fiction writers. Some journalists are historians. Therefore, some journalists are not fiction writers.”

Answer: Deductive (valid).

Terms and regions

Term	Meaning here	Role
M	historians	middle term (in both premises, not in the conclusion)
S	journalists	subject of the conclusion
P	fiction writers	predicate of the conclusion

Region	R1	R2	R3	R4	R5	R6	R7
Meaning	S only	S∩P only	P only	S∩M only	P∩M only	S∩P∩M	M only

Key: # = region shaded (empty), * = something exists in this region. Panels are boxes, not circles: S left, P right, M bottom.

Diagrams (universal premise first)

0. blank                    1. P1: No M is P            2. P2: Some S is M       
+--------+-----+--------+   +--------+-----+--------+   +--------+-----+--------+
| S      |     |      P |   | S      |     |      P |   | S      |     |      P |
|   1    |  2  |   3    |   |   1    |  2  |   3    |   |   1    |  2  |   3    |
|   +----+-----+---+    |   |   +----+-----+---+    |   |   +----+-----+---+    |
|   | 4  |  6  | 5 |    |   |   | 4  |##6##|#5#|    |   |   | 4  |##6##|#5#|    |
|   |    |     |   |    |   |   |    |#####|###|    |   |   | *  |#####|###|    |
+---+----+-----+---+----+   +---+----+-----+---+----+   +---+----+-----+---+----+
    |      7       |            |      7       |            |      7       |
    | M            |            | M            |            | M            |
    +--------------+            +--------------+            +--------------+

 P1 No M is P    : M and P overlap is empty       -> regions 5, 6 shaded
 P2 Some S is M  : star in S and M = 4 or 6      -> 6 is shaded, so star in 4
 Conclusion: Some S is not P needs a star in 1 or 4.  Found: star in 4.

Worked solution

1. Assign the terms

   • M = historians, S = journalists, P = fiction writers.
   • Draw three overlapping circles S, P, M with the 7 regions above (panel 0).

2. Order the premises

   • Premise 1 (No historians are fiction writers) is a universal negative (E): draw it first.
   • Premise 2 (Some journalists are historians) is a particular affirmative (I): draw it second.
   • Rule: universal before particular.

3. Diagram premise 1

   • “No M is P” means the M and P overlap is empty.
   • Shade R5 (P∩M but not S) and R6 (S∩P∩M) (panel 1).

4. Diagram premise 2

   • “Some S is M” needs an asterisk in S∩M, so in R4 or R6.
   • Step 3 already emptied R6, so the asterisk goes unambiguously into R4 (S∩M but not P) (panel 2).

5. Read off the conclusion

   • “Some S is not P” is true exactly when an asterisk lies in S outside P, i.e. in R1 or R4.
   • The asterisk is in R4 (conclusion line), so the diagram guarantees the conclusion with no extra assumption.
   • The syllogism is valid.

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Q8 (5 marks): Syllogism validity by Venn diagram (intuitively valid but formally invalid)

Determine whether the following syllogism is deductively valid by constructing the 7-region Venn diagram. Show each diagramming step.

“All mathematicians are logical thinkers. All chess grandmasters are logical thinkers. Therefore, all chess grandmasters are mathematicians.”

Answer: Not deductive (invalid).

Terms and regions

Term	Meaning here	Role
M	logical thinkers	middle term
S	chess grandmasters	subject of the conclusion
P	mathematicians	predicate of the conclusion

Regions are the same as in Q7: R1 S only; R2 S∩P only; R3 P only; R4 S∩M only; R5 P∩M only; R6 S∩P∩M; R7 M only. Same key as Q7: [n] = shaded (empty); ?n? = region still open that the conclusion needs empty.

Diagrams (universal premises first)

0. blank                    1. P1: All P is M           2. P2: All S is M        
+--------+-----+--------+   +--------+-----+--------+   +--------+-----+--------+
| S      |     |      P |   | S      |#####|######P#|   |#S######|#####|######P#|
|   1    |  2  |   3    |   |   1    |##2##|###3####|   |###1####|##2##|###3####|
|   +----+-----+---+    |   |   +----+-----+---+####|   |###+----+-----+---+####|
|   | 4  |  6  | 5 |    |   |   | 4  |  6  | 5 |####|   |###| 4  |  6  | 5 |####|
|   |    |     |   |    |   |   |    |     |   |####|   |###|    |     |   |####|
+---+----+-----+---+----+   +---+----+-----+---+----+   +---+----+-----+---+----+
    |      7       |            |      7       |            |      7       |
    | M            |            | M            |            | M            |
    +--------------+            +--------------+            +--------------+

 P1 All P is M : P outside M is empty            -> regions 2, 3 shaded
 P2 All S is M : S outside M is empty            -> regions 1, 2 shaded (2 already)
 Conclusion: All S is P needs S outside P empty  -> regions 1 and 2 AND 4 empty.
   Region 4 (S and M, not P) is still open, so a grandmaster who is not a mathematician fits.

Worked solution

1. Assign the terms

   • M = logical thinkers, S = chess grandmasters, P = mathematicians.
   • Draw the three circles with the 7 regions (panel 0).

2. Order the premises

   • Both premises are A-form (“All …”), so both only shade regions.
   • There is no particular premise to defer, so either order works.

3. Diagram premise 1

   • “All P is M” means every part of P outside M is empty.
   • Shade R2 (S∩P but not M) and R3 (P only) (panel 1).

4. Diagram premise 2

   • “All S is M” means every part of S outside M is empty.
   • Shade R1 (S only) and R2 (already shaded in step 3) (panel 2).

5. Check the conclusion

   • “All S is P” is true only if every part of S outside P is empty, i.e. R1 and R4 both shaded.
   • R1 was shaded in step 4, but R4 (grandmasters who are logical thinkers but not mathematicians) is untouched by either premise.
   • R4 stays open (conclusion line), so the conclusion is not guaranteed: the argument is invalid.

6. Name the fallacy

   • This is the fallacy of the undistributed middle.
   • Two sets that are both subsets of a third set need not overlap each other.
   • Counterexample: a grandmaster who is a superb logical thinker but has never studied mathematics.

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Q9 (4 marks): Suppressed (implicit) premise

Each argument below has at least one unstated premise. Supply the missing premise(s) and rewrite the argument in standard form to make it deductively valid.

9.1 “The old water pipe must be corroding, since it is made of iron.”

9.2 “You should see a doctor. You have been running a fever for three days.”

Answer: the missing premise is (2) in each block below.

9.1

  (1) This water pipe is made of iron.
  (2) Iron pipes corrode over time.        <-- suppressed premise
  ─────────────────────────────────────
  ∴   This water pipe must be corroding.

9.2

  (1) You have been running a fever for three days.
  (2) Anyone who has had a fever for three or more days should see a doctor.
                                           <-- suppressed premise
  ─────────────────────────────────────
  ∴   You should see a doctor.

Worked solution

1. Recall the enthymeme

   • An enthymeme is an argument with one or more premises left unstated.
   • The speaker assumes the audience already accepts the missing premise.

2. Use the method

   • Identify the stated premise and the stated conclusion.
   • Ask: which general rule or background fact, added as a premise, makes “stated premise → stated conclusion” necessary?

3. Apply to 9.1

   • Stated premise: the pipe is made of iron. Stated conclusion: the pipe is corroding.
   • The bridge is the general claim that iron pipes corrode.
   • Without it, the material tells us nothing about the pipe’s physical state.

Note (check): “Iron pipes corrode over time” is weaker than “must be corroding” (now). For strict deductive validity the bridge should be closer to “Every iron pipe of this kind is corroding”.

4. Apply to 9.2

   • Stated premise: fever for three days. Stated conclusion: see a doctor.
   • The bridge is the medical guideline that a fever lasting three or more days warrants a visit.
   • Without it, the duration alone does not require seeing a doctor.

5. Check deductive validity

   • With the suppressed premise added, both have the form “All F is G; X is F; therefore X is G”.
   • This is a standard valid syllogism (Barbara), so the inference is deductively valid.

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Q10 (5 marks): Argument map: extract 2+ arguments from a passage

For the passage below: (i) identify the main conclusion and any sub-conclusions, (ii) write out at least two distinct arguments in standard form, (iii) identify any counter-considerations and note how they are handled.

“Schools should ban the use of smartphones during class time. Research consistently shows that students who use their phones during lessons retain significantly less of the taught material than those who do not. Furthermore, even students who intend to use their phones only for legitimate purposes, such as looking up definitions, are easily distracted by notifications, messages, and social media feeds. One might object that banning phones takes away a useful educational resource. But this objection only holds if students are able to use phones in a self-disciplined way, and the evidence clearly shows they cannot.”

Answer: main conclusion = “Schools should ban smartphones during class time”; sub-conclusions = C1 (phone use harms learning) and C2 (students cannot self-regulate); counter-consideration = “banning removes a useful resource”, rebutted by C2.

Full argument map (solid arrow = supports; dashed arrow = objects to or rebuts)

 P1  Students who use phones in            P2  Even students with legitimate
     lessons retain much less                  aims are easily distracted
     ("Research consistently shows")           ("Furthermore")
        │                                         │
        ▼                                         ▼
 C1  Phone use during class                C2  Students cannot use phones in
     harms learning                            a self-disciplined way
        │                                         │
        │                                         ├ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┐
        └───────────────┬─────────────────┘       ┆ rebuts
                        │                         ┆ ("But ... the evidence
                        ▼                         ┆  shows they cannot")
 MC  Schools should ban smartphones               ▼
     during class time  ◀ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┄ ┄ CC  Banning phones removes a
                                  objects       useful educational resource
                                  ("One might object")

Standard-form arguments

Arg 1 (retention):
  (1) Students who use phones during lessons retain significantly less taught material.
  ∴ Smartphone use during class is harmful to learning.            [C1]

Arg 2 (distraction):
  (1) Even students intending to use phones for legitimate purposes are easily
      distracted by notifications and social media.
  ∴ Students cannot reliably use phones in a self-disciplined way during class.   [C2]

Main argument:
  (1) Smartphone use during class is harmful to learning.           [from Arg 1]
  (2) Students cannot reliably use phones in a self-disciplined way. [from Arg 2]
  ∴ Schools should ban the use of smartphones during class time.   [MC]

Counter-consideration:
  Banning phones removes a useful educational resource.
  Rebutted by Arg 2's conclusion: the objection only holds if students can
  self-regulate, and the evidence (Arg 2) shows they cannot.

Worked solution

1. Scan for indicators

   • “Research consistently shows that” introduces a factual premise; “Furthermore” adds another.
   • “One might object” signals a counter-consideration.
   • “But … the evidence clearly shows” rebuts it and reaffirms the main line.

2. Find the main conclusion first

   • The opening sentence states it: “Schools should ban the use of smartphones during class time.”
   • Everything else supports it or deals with an objection to it.

3. Identify Arg 1

   • The retention-loss claim is a reason why smartphones harm learning.
   • It functions as a sub-conclusion feeding the main argument.

4. Identify Arg 2

   • The distraction claim is a separate supporting reason.
   • It also rebuts the counter-consideration: the objection assumes self-discipline is possible, and Arg 2’s conclusion says it is not.

5. Handle the counter-consideration

   • “Banning phones removes a useful resource” is acknowledged (“One might object”) and then defeated (“But this objection only holds if …”).
   • It is a conceded objection that is ultimately overridden: a classic “nevertheless” move.

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Q11 (3 marks): Odd one out: which passage is not an argument?

Identify which of the following passages is not an argument and explain why. The others are arguments; for those, briefly indicate what makes them arguments.

(a) “Exercise reduces the risk of cardiovascular disease, so people who want to protect their heart health should exercise regularly.” (b) “The flight was delayed by two hours. Passengers waited in the terminal, and eventually the plane departed at midnight.” (c) “Since no valid deductive argument can have true premises and a false conclusion, and this argument has true premises, its conclusion must be true.” (d) “Organ donation rates are higher in countries with opt-out systems than in countries with opt-in systems. Switching to an opt-out system in this country would therefore likely increase the number of available donors.” (e) “The suspect had motive, means, and opportunity. It is therefore reasonable to conclude that the suspect is guilty.”

Answer: (b).

Verdict table

Passage	Argument?	Deciding feature
(a)	Yes	”so” marks the conclusion drawn from the health fact
(b)	No	events in sequence, joined by temporal “and” and “eventually”
(c)	Yes	”since” reasons apply a general logical principle to this case
(d)	Yes	”therefore” marks a prediction backed by the opt-out comparison
(e)	Yes	”therefore” marks guilt, backed by motive, means, opportunity

Worked solution

1. Apply the test

   • A passage is an argument if and only if at least one statement is offered as a reason for another.
   • The key is an inferential claim, not the mere presence of connecting words.

2. Test (a)

   • “Exercise reduces the risk of cardiovascular disease” is the premise.
   • “People who want to protect their heart health should exercise regularly” is the conclusion; “so” is the conclusion indicator.

3. Test (b)

   • The three sentences describe a sequence: delay, waiting, departure.
   • No sentence is evidence for another; “and” here is temporal, not logical: a narrative.

4. Test (c)

   • A general logical principle (valid deductive arguments preserve truth) is applied to a specific case.
   • The premise that this argument has true premises is offered as evidence that its conclusion is true.

5. Test (d)

   • The opt-out versus opt-in comparison is evidence for a predictive conclusion about this country.
   • “Therefore” marks the conclusion.

6. Test (e)

   • The three factual premises (motive, means, opportunity) support the conclusion about guilt.
   • “Therefore” marks the conclusion.

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Q12 (3 marks): Odd one out: which is the only deductive argument?

One of the following arguments is deductive; the others are inductive. Identify the deductive one and explain your reasoning.

(a) “Every morning this week, the café has opened at 7 a.m. It will probably open at 7 a.m. tomorrow.” (b) “Professor Nakamura has published fifty papers on this topic and has won three international awards for her research. She says the drug is safe, so it probably is.” (c) “No even number greater than 2 is prime. The number 248 is an even number greater than 2. Therefore, 248 is not prime.” (d) “Nine out of ten dentists surveyed recommend this brand of toothpaste. It must be effective.” (e) “This fossil was found in a rock layer dated to 65 million years ago, so the organism it belonged to probably lived around that time.”

Answer: (c).

Verdict table

Passage	Verdict	Deciding feature
(a)	Inductive	prediction from a finite past pattern (“probably”)
(b)	Inductive	appeal to authority (“probably”)
(c)	Deductive	universal mathematical truth plus a checkable fact
(d)	Inductive	statistical claim from a limited survey
(e)	Inductive	probabilistic scientific dating (“probably”)

Worked solution

1. Test each argument for necessity

   • Deductive: true premises make a false conclusion impossible.
   • Inductive: true premises make the conclusion probable but not certain.

2. Test (a)

   • The premise is a pattern over a finite set of past days.
   • Inferring tomorrow from the past is an inductive prediction; patterns can break.

3. Test (b)

   • This is an appeal to authority.
   • Even a highly credentialed expert can be wrong, so the premises do not guarantee the conclusion.

4. Test (c)

   • “No even number > 2 is prime” is a universal mathematical truth; “248 is even and > 2” is a verifiable fact.
   • By universal instantiation “248 is not prime” follows necessarily.

5. Test (d)

   • “Nine out of ten” is a statistical claim from a limited survey sample.
   • The conclusion (“it must be effective”) would need much stronger grounds for necessity: a statistical generalisation with a hedged universal conclusion.

6. Test (e)

   • Dating rock strata is a probabilistic scientific inference.
   • Dating methods have error margins and organisms can be redeposited.

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Q13 (3 marks): Which sentence cannot be expressed as a conjunction of two simpler sentences?

Identify which sentence below cannot be accurately broken into a conjunction “S1 and S2” without changing the meaning, and explain why.

(a) “Lions and tigers are both carnivores.” (b) “Keiko and Yumi both passed the final exam.” (c) “Ricardo and Elena divorced last year.” (d) “The prime minister and the president issued a joint statement.” (e) “The shop sells books and stationery.”

Answer: (c).

Verdict table

Passage	Can split?	Deciding feature
(a)	Yes	each animal is independently a carnivore
(b)	Yes	”both” makes the distributive reading explicit
(c)	No	one bilateral event: they divorced each other
(d)	Yes (borderline)	shared object “statement” gives a distributive paraphrase
(e)	Yes	two independent facts about the shop

Worked solution

1. Recall the conjunction test

   • “A and B” splits into P ∧ Q if both parts are independently meaningful with truth values, and the conjunction has the same truth conditions as the original.
   • If the predicate is relational, needing both subjects to take part jointly, splitting changes the meaning.

2. Test (a)

   • “Lions are carnivores” and “tigers are carnivores” are each true, and together say exactly the original.

3. Test (b)

   • “Keiko passed the final exam” and “Yumi passed the final exam” are independent statements.
   • “Both” makes the distributive reading explicit.

4. Test (c)

   • “Ricardo divorced last year” and “Elena divorced last year” are individually meaningful.
   • But their conjunction does not say they divorced each other: they might each have divorced different people.
   • The original describes a single bilateral event needing two parties in relation, so the meaning changes on splitting.

5. Test (d)

   • “Issuing a joint statement” is a collective action.
   • The original paraphrases it as “the PM issued the statement and the president issued the statement”, the jointness being carried by the single shared object.
   • Borderline, but the distributive reading holds.

Note (check): (d) is arguably also collective (“joint” says they issued one statement together), so it resembles (c); splitting it loses the jointness. The original keeps (d) as splittable and calls it borderline.

6. Test (e)
   • “The shop sells books” and “the shop sells stationery” are independent facts.

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Q14 (5 marks): Best categorical form (A/E/I/O): harder cases with idiomatic quantifiers

Restate each sentence in the best standard categorical form. State the form letter (A/E/I/O), the subject class S, and the predicate class P.

(a) “None but the brave deserve the fair.” (b) “Hardly any of the proposed amendments survived the committee vote.” (c) “There are snakes that are not venomous.” (d) “Anyone who is not a citizen may not vote in this election.” (e) “All that glitters is not gold.”

Answer:

• (a) All S are P (A); S = people who deserve the fair, P = brave people. (Equivalently, E: No S are P where S = non-brave people, P = people who deserve the fair.)
• (b) Some S are not P (O); S = proposed amendments, P = amendments that survived the committee vote.
• (c) Some S are not P (O); S = snakes, P = venomous animals.
• (d) No S are P (E); S = non-citizens, P = people who may vote in this election.
• (e) Some S are not P (O); S = things that glitter, P = gold. (Despite the “all… not” surface form, this means “not all glittering things are gold”, i.e., some are not.)

Answer table

Sentence	Keyword trap	S	P	Form
(a) None but the brave deserve the fair.	”none but” means “only”: reverses the terms	people who deserve the fair	brave people	A (or E)
(b) Hardly any of the proposed amendments survived.	”hardly any” means “very few”	proposed amendments	amendments that survived the committee vote	O (see note)
(c) There are snakes that are not venomous.	”there are … that” means “some”	snakes	venomous animals	O
(d) Anyone who is not a citizen may not vote.	negation inside the universal	non-citizens	people who may vote in this election	E
(e) All that glitters is not gold.	”all … not”: negation scopes over “all”	things that glitter	gold	O

The cheat sheet of NL patterns is in Q6.

Worked solution

1. Handle “none but” in (a)

   • “None but X are Y” means “only X are Y”, which means “all who are Y are X”.
   • So “None but the brave deserve the fair” = “All who deserve the fair are brave”: S = people who deserve the fair, P = brave people, form A.
   • Equivalent from the negation: “No non-brave person deserves the fair” gives E.

2. Handle “hardly any” in (b)

   • “Hardly any” means “very few”: some do not have the property, and most do not.
   • Closest standard form is O: “Some proposed amendments are not amendments that survived the vote”.
   • “Hardly” adds a scalar implicature (very few survived); the standard form captures only the existential claim.

Note (check): same doubt as Q6(c): “hardly any S are P” is closer to a weak E or to I plus O, and O alone is only its weakest consequence.

3. Handle “there are … that” in (c)

   • “There are snakes that are not venomous” equals “Some snakes are not venomous”.
   • This is a direct O form.

4. Handle “anyone who is not” in (d)

   • It means: for every person, if they are not a citizen, then they may not vote.
   • Equivalently “No non-citizens may vote”: S = non-citizens, P = people who may vote, form E.

5. Handle “all … not” in (e)

   • In everyday English the idiom means “not everything that glitters is gold”, i.e. “Some glittering things are not gold”.
   • The logical content is a particular negative (O), not a universal negative.
   • This is a scope-of-negation trap: the negation falls on the universal quantifier, not the predicate.

Note (check): read literally, “All that glitters is not gold” is E (nothing that glitters is gold); O is the idiomatic reading only, so the answer depends on choosing the idiom.

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Q15 (4 marks): Mark reason (R) and conclusion (C): complex multi-premise passage

For the passage below, mark all reasons with R(…) and all conclusions with C(…). Identify which conclusion is the main conclusion and which (if any) is an intermediate conclusion that also serves as a premise for a further step.

“The northern ice sheet has lost more mass in the last thirty years than in any comparable period in recorded history. Sea levels have consequently been rising at an accelerating rate. Coastal cities that were once considered safe from flooding now face regular inundation during storm events. We must therefore conclude that climate change poses an existential threat to human settlements along coastlines worldwide.”

Answer: R1(The northern ice sheet has lost more mass in the last thirty years than in any comparable period in recorded history). C1-R2(Sea levels have consequently been rising at an accelerating rate). C2-R3(Coastal cities that were once considered safe from flooding now face regular inundation during storm events). C3(We must therefore conclude that climate change poses an existential threat to human settlements along coastlines worldwide).

The main conclusion is C3. C1 and C2 are intermediate conclusions: each follows from the previous step and simultaneously serves as a reason for the next step.

Argument map (a three-step chain)

 R1     northern ice sheet has lost more mass than ever recorded
   │
   ▼    "consequently"
 C1-R2  sea levels rising at an accelerating rate
   │
   ▼    (no indicator word; flooding follows from rising seas)
 C2-R3  once-safe coastal cities now face regular inundation
   │
   ▼    "We must therefore conclude"
 C3     climate change is an existential threat to coastal settlements
        (main conclusion)

Worked solution

1. Read the indicator words
   • “Consequently” marks C1 as following from R1.
   • The third sentence (“now face”) describes an outcome flowing from rising seas, making it C2, supported by C1-R2.
   • “Therefore” marks the main conclusion C3, supported by C2-R3.

Note (check): “now face” is not really an indicator word; C2 is identified by content (flooding as a result of rising seas), not by an indicator.

2. Establish the chain

   • R1 → C1: melting ice causes rising seas.
   • C1 becomes R2: rising seas cause coastal flooding.
   • C2 becomes R3: coastal flooding shows an existential threat to settlements.
   • Each link passes a conclusion forward as a premise.

3. Identify the main conclusion

   • It is the final claim: climate change poses an existential threat to coastal settlements.
   • All other statements build toward it, and nothing further is derived from C3.

4. Name the structure

   • This is a three-step chain (sometimes called a sorites or polysyllogism).
   • Every intermediate conclusion is a conclusion of the step behind it and a premise for the step ahead.
   • Recognising this is essential for drawing the argument map correctly.

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