Lesson 37 of 42 · Critical thinking
Quantifiers and conditionals
Read all, some, most, only, if, only if and unless with their strict meanings, negate them correctly, and use the contrapositive.
About 22 minutes
In this lesson
- State the strict meaning and the correct negation of all, some, most and none.
- Draw the valid conclusions from a conditional and reject its converse and inverse.
- Translate only if, unless and if and only if into ordinary conditionals.
The idea
Small words, exact meanings
Critical thinking items turn on a few words used with their strict logical meaning. In conversation, "some" often hints "not all", and "unless" is often heard as a two-way promise; in a true / false / cannot be deduced item, you must read them strictly. This lesson fixes the meanings.
Quantifiers
| Word | Strict meaning | Does not mean |
|---|---|---|
| all, every, each | no exceptions | (nothing weaker) |
| some | at least one, possibly all | not all |
| most | more than half | almost all |
| none, no | zero | (nothing stronger) |
| not all | at least one exception | none |
| only A are B | every B is an A | every A is a B |
Two consequences matter most.
Negations. The opposite of "all exporters increased sales" is "at least one exporter did not", not "no exporter did". The opposite of "some banks failed" is "no bank failed". The opposite of "most" is "half or fewer".
Combining "most" with "most". If more than half the branches are in the north and more than half offer Saturday opening, at least one northern branch offers Saturday opening: two groups that are each more than half of the same total must overlap. "Some" with "some" gives no such guarantee.
Conditionals
"If P, then Q" says that whenever P holds, Q holds. From it:
| Given | Conclusion | Valid? |
|---|---|---|
| P | Q | Yes |
| not Q | not P | Yes (the contrapositive) |
| Q | P | No (the converse) |
| not P | not Q | No (the inverse) |
Example: "If a supplier delivers late more than twice in a year, it is removed from the approved list." A supplier still on the list did not deliver late more than twice (contrapositive). A removed supplier may have been removed for something else (the converse fails).
Only if, unless, if and only if
- "P only if Q" means Q is necessary for P: if P, then Q. "A firm can bid only if it has audited accounts": a firm that bid has audited accounts; a firm with audited accounts may still be unable to bid for other reasons.
- "P unless Q" means: if not Q, then P. "The plant will close unless it wins an order" means that without an order, it closes. Read strictly, winning an order does not guarantee that the plant stays open; it only removes this reason for closing.
- "P if and only if Q" runs both ways: each guarantees the other.
- Necessary and sufficient. Q is necessary for P when P cannot happen without Q (P only if Q). Q is sufficient for P when Q alone guarantees P (if Q, then P).
Chains
Conditionals chain: "if P then Q" and "if Q then R" give "if P then R". Their contrapositives chain too: not R gives not Q, which gives not P. When a passage contains several rules, write them as arrows and follow the arrows forwards, or follow the negations backwards. Never run an arrow itself backwards.
Reading strictly without being pedantic
Strict reading applies to the logical words, not to ordinary language in general. "A firm" means one firm; "last year" means the year before the one the passage describes. The aim is not to find loopholes but to avoid adding meaning that the words do not carry.
Worked examples
Example 1: When "all" turns out false
A report claimed: "All the region's banks met the new capital requirement." An audit showed the claim was false. What follows?
"All met it" is false when there is at least one exception.
So at least one bank did not meet the requirement.
The audit does not say how many failed. "None met it" and "most failed" are both possible but not guaranteed.
Answer
At least one of the region's banks did not meet the requirement. "No bank met it" is the tempting overstatement.
Example 2: Only if
A firm can bid for the contract only if it has audited accounts for the last three years.
Judge: (1) Firm X has audited accounts for the last three years, so it can bid. (2) Firm Y bid for the contract, so it has audited accounts for the last three years. (3) Firm Z does not have audited accounts for the last three years, so it cannot bid.
Translate: "can bid only if audited" means "if a firm can bid, it has audited accounts". Audited accounts are necessary, not sufficient.
(1) runs the arrow backwards (audited, so can bid). Other conditions may apply. Cannot be deduced.
(2) follows the arrow: Y could bid, so Y has audited accounts. True.
(3) is the contrapositive: no audited accounts, so cannot bid. True.
Answer
(1) cannot be deduced; (2) and (3) are true. Reading "only if" as "if" is what makes (1) tempting.
Example 3: Most and most
More than half of a bank's branches are in the north. More than half of its branches offer Saturday opening.
Judge: At least one of the bank's branches in the north offers Saturday opening.
Let the bank have branches. Northern branches number more than , and Saturday branches more than .
Together the two groups count more than branches, so at least one branch is counted twice: it is in the north and opens on Saturday.
Answer
True. With "some" instead of "most" the statement could not be deduced, because two small groups need not overlap.
Common traps
"Some" read as "some but not all"
"Some employees work remotely" is taken to imply that some do not. Strictly, "some" allows "all". A statement that relies on "not all" cannot be deduced from "some".
Negating "all" as "none"
When "all X did Y" is false, readers conclude "no X did Y". One exception is enough to make "all" false. The correct negation is "at least one X did not".
Running the arrow backwards
From "if late more than twice, removed" readers conclude "removed, so late more than twice". That is the converse, and it needs the extra premise that lateness is the only reason for removal. Use the contrapositive instead.
"Only if" read as "if"
"Only if Q" names a necessary condition. Meeting it does not guarantee the outcome. Rewrite "P only if Q" as "if P, then Q" before you reason.
Check yourself
Answer each question before you open its answer.
Question 1: A supplier still on the list
Critical thinking
Any supplier that delivers late more than twice in a calendar year is removed from the approved list immediately, and a removed supplier cannot be readmitted in the same year. Ferri Srl supplied the company throughout 2025 and stayed on the approved list for the whole of 2025, without interruption.
Is the following statement true, false, or can it not be deduced from the passage? Ferri Srl delivered late no more than twice in 2025.
- A.True
- B.False
- C.Cannot be deduced
Show the answer
Answer: A. True
Working
The rule: more than two late deliveries in a year, then immediate removal. Contrapositive: a supplier that was never removed during the year did not deliver late more than twice in it. Ferri stayed on the list for the whole of 2025 without interruption, so it was never removed in 2025, and so it delivered late at most twice in 2025. The statement is true.
Why the other options are wrong
- B. The passage gives no sign of more than two late deliveries; if there had been, Ferri would have been removed immediately, and it was never off the list in 2025.
- C. The contrapositive settles it: more than two late deliveries would have meant immediate removal, and Ferri stayed on the list for the whole year without interruption.
What carries over
From "if P then Q" and "not Q", conclude "not P": the contrapositive is always valid.
Question 2: A false claim about exporters
Critical thinking
A trade report claimed: "All of the region's exporters increased their sales last year." The claim later turned out to be false.
Which of the following must be true?
- A.None of the region's exporters increased their sales last year.
- B.Most of the region's exporters saw their sales fall last year.
- C.At least one of the region's exporters did not increase its sales last year.
- D.Some of the region's exporters increased their sales last year.
Show the answer
Answer: C. At least one of the region's exporters did not increase its sales last year.
Working
"All exporters increased their sales" is false exactly when at least one exporter did not increase its sales. That is all the falsity tells us. It does not say how many exceptions there were, whether their sales fell or stayed flat, or whether any exporter's sales rose.
Why the other options are wrong
- A. Negates "all" as "none". One exporter whose sales did not rise is enough to make the claim false.
- B. Nothing says how many exceptions there were; there may have been only one, and its sales may have stayed flat rather than fallen.
- D. Possible, but not guaranteed: the claim is also false if no exporter increased its sales.
What carries over
The negation of "all" is "at least one is not", never "none".
Question 3: Who receives the grant?
Critical thinking
The regional grant is paid only to firms that have been registered for at least two years. Some of the firms that applied for this year's grant had been registered for less than a year when it was paid.
Is the following statement true, false, or can it not be deduced from the passage? Every firm that applied for this year's grant received it.
- A.True
- B.False
- C.Cannot be deduced
Show the answer
Answer: B. False
Working
"Paid only to firms registered for at least two years" means: if a firm receives the grant, it has been registered for at least two years. At least one applicant had been registered for less than a year when the grant was paid, so by the contrapositive it did not receive the grant. Therefore not every applicant received it, and the statement is false.
Why the other options are wrong
- A. Ignores the condition: a firm registered for less than a year cannot meet the two-year requirement, so it cannot receive the grant.
- C. The passage is not silent: "some" guarantees at least one applicant registered for less than a year, and the rule excludes that firm.
What carries over
A necessary condition that some applicants fail rules out "every applicant succeeded".
Key points
"Some" means at least one, possibly all; "most" means more than half; the negation of "all" is "at least one is not".
From "if P then Q", the contrapositive "if not Q then not P" is valid; the converse and the inverse are not.
"P only if Q" means "if P then Q"; "P unless Q" means "if not Q then P".
Two groups that are each more than half of the same total must overlap.
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Written by the UniExamer team with AI assistance. Every worked example and check question was re-solved by an independent checker before publication. No subject expert has reviewed this lesson yet. If you find a mistake, write to support@uniexamer.com.