Unit quiz: Negation, countermodels and quantifiers

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5 questions · 80% to pass · no timer. This quiz checks the lessons in this unit. These authored questions assess recognition and application of the taught distinctions, not professional qualification. You may review the lessons and retry. Repeat attempts reuse the question bank; a remembered answer is not proof of transfer to a new situation.

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Question 1

“It is not the case that both witnesses signed” is compatible with one signature. “Neither witness signed” excludes even one. If Ada signed and Ben did not, the first assertion is true and the second false. That single case prevents treating the phrases as equivalent.

Does “not both witnesses signed” entail that neither signed?No; one may have signedYes; it excludes every signature

Question 2

Consider P or Q; not P; therefore Q. To make the conclusion false, set Q false. The first premise then needs P true, but the second needs P false. No assignment meets all requirements. Contrast P or Q; P; therefore not Q: setting both true is a countermodel.

Which row would refute an argument’s validity?A premise false and conclusion falseAll premises true and conclusion false

Question 3

In a domain of library items, “all manuscripts are catalogued” means any item that is a manuscript is catalogued. “Some manuscript is catalogued” additionally requires a manuscript. A library with no manuscripts satisfies the universal conditional but not the existential claim.

What extra commitment does “some manuscript is catalogued” make?That at least one manuscript exists in the domainThat every library item is a manuscript

Question 4

In a two-person domain, Ada trusts only Ben and Ben trusts only Ada. Everyone trusts someone. Nobody is trusted by everyone, because neither trusts themself. This relation satisfies the first statement while refuting the second; no claim about actual human psychology is needed.

Why does the two-person case refute the reversed inference?It shows nobody trusts anybodyIt supplies individual witnesses but no shared witness

Question 5

Thesis: every permissible policy is popular. One permissible but unpopular policy would refute it. That refutation would not show that all permissible policies are unpopular. Conversely, refuting “a permissible policy is popular” requires showing there is no such case in the specified domain.

What exactly negates “every permissible policy is popular”?At least one permissible policy is not popularEvery permissible policy is unpopular

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