A biologist states: 'All mammals are warm-blooded.' What would count as a successful refutation of this claim?
AProving that the statement 'No mammals are cold-blooded' is false
BDemonstrating that at least one mammal exists that is not warm-blooded
CEstablishing that most mammals are warm-blooded but some exceptions exist
DShowing that the concept 'warm-blooded' is ambiguous or poorly defined
A universal statement ('All S are P') is refuted by a single counterexample — one instance of S that is not P. Finding even one cold-blooded mammal (an S that lacks P) defeats the claim. Option A is wrong: 'No mammals are cold-blooded' is a much stronger claim (equivalent to the original 'all are warm-blooded'), so disproving it doesn't help. Option C describes confirming the negation but uses informal language; the logical negation requires only one counterexample, not 'most.'
Question 2 Multiple Choice
Which statement is the correct logical negation of 'All students passed the exam'?
ANo students passed the exam
BAll students failed the exam
CAt least one student did not pass the exam
DMost students did not pass the exam
The negation of a universal statement ('All S are P') is an existential statement ('Some S are not P' — i.e., at least one S lacks P). Option A, 'No students passed,' goes far beyond what is needed to make the original claim false — it's a much stronger statement. The original claim fails the moment even one student failed. This is the most common error: people treat the negation of 'all' as 'none,' when it is actually 'some are not.'
Question 3 True / False
Under classical predicate logic, the statement 'All unicorns have golden horns' is true.
TTrue
FFalse
Answer: True
In classical logic, 'All S are P' is formalized as 'for every x, if x is S then x is P.' When S is an empty class (no unicorns exist), the conditional 'if x is a unicorn, then x has a golden horn' is never tested — there are no unicorns to serve as counterexamples. A conditional with a false antecedent is vacuously true. This conflicts with everyday English intuition (which assumes 'all S' presupposes S exists), but in formal logic vacuous truth is the standard interpretation.
Question 4 True / False
The negation of 'Some birds can seldom fly' is 'Some birds can fly.'
TTrue
FFalse
Answer: False
The negation of an existential statement ('Some S are P') is a universal statement ('No S are P,' equivalently 'All S are not P'). The negation of 'Some birds cannot fly' is 'All birds can fly' — i.e., there are no birds that cannot fly. 'Some birds can fly' is actually compatible with 'Some birds cannot fly': both can be true simultaneously. A statement and its negation cannot both be true; only the universal denial achieves that.
Question 5 Short Answer
A student says: 'To disprove that all S are P, I need to show that no S are P.' Why is this reasoning wrong, and what do you actually need to show?
Think about your answer, then reveal below.
Model answer: The negation of 'All S are P' is 'Some S are not P' — you need only one counterexample where something is S but not P. Showing 'No S are P' is a far stronger claim that goes well beyond what is needed. For example, to disprove 'All swans are white,' finding a single black swan is sufficient; you do not need to establish that no swans are white. The student's error conflates the negation of a universal ('some are not') with the contrary universal ('none are').
This error arises from a natural but mistaken symmetry: people expect the negation of 'all' to be another universal statement ('none'). But the negation of a universal is existential — it only claims that the universal fails somewhere, not that the opposite universal holds. In practice this matters enormously: disproving 'all drugs of type X are safe' requires finding just one unsafe drug, not proving all of them are unsafe.