Questions: Implicature and Logical Form

Question 1

A speaker says 'Some of the students passed the exam.' A listener infers 'Not all of them passed.' Where does this 'not all' inference come from?

• A. It is part of the logical form of 'some' — the word semantically means 'at least one but not all'
• B. It is a scalar implicature generated by the maxim of quantity: the speaker said 'some' when she could have said 'all,' so she must not believe 'all' is true
• C. It is a presupposition triggered by the definite article 'the students'
• D. It is an entailment — 'some passed' logically entails 'not all passed' in classical logic

Answer: (B)

The word 'some' has the logical form of existential quantification (∃x), which is consistent with all students passing — it sets a floor, not a ceiling. The 'not all' inference is a scalar implicature, not part of the truth-conditional content. It arises from the Horn scale ⟨some, most, all⟩: the speaker could have used the stronger 'all,' and since she didn't, a cooperative listener infers she believes 'all' is false. Option A (the most common misconception) treats the pragmatic enrichment as if it were part of the word's meaning. Option D is wrong because 'some passed' is consistent with all passing — it does not entail 'not all.'

Question 2

A linguist can demonstrate that 'not all passed' is a scalar implicature (not a semantic entailment) by applying which test?

• A. The substitution test: replace 'some' with 'a few' and check whether the inference persists
• B. The cancellability test: add 'in fact, all of them passed' and check whether the result is a contradiction
• C. The scope test: embed the utterance under negation and check whether the inference survives
• D. The presupposition projection test: embed the utterance in a conditional and check whether the inference projects out

Answer: (B)

Cancellability is the diagnostic test that distinguishes implicatures from entailments. 'Some students passed — in fact, all of them did' is perfectly coherent: it corrects and strengthens the implicature without contradiction. Compare with a true semantic entailment: 'All students passed — in fact, some didn't' is a contradiction, because 'all passed' entails 'some passed,' and adding its negation is incoherent. Implicatures can be cancelled because they are pragmatic inferences, not truth-conditional commitments. Option C (negation test / scope under operators) is used to test presuppositions, not implicatures.

Question 3

The logical form of 'Some students passed' is equivalent to 'At least one student passed and at most some students passed.'

• T. True
• F. False

Answer: False

The logical form of 'some' is purely existential: ∃x[student(x) ∧ passed(x)], meaning 'there exists at least one thing that is a student and passed.' There is no upper bound in the logical form — it is fully consistent with all students passing. The 'not all' upper bound is a pragmatic inference (scalar implicature), not part of the semantic content. This is the central insight: logical form underdetermines what is communicated. Pragmatic enrichment via Gricean reasoning adds the upper bound at the level of 'what is meant,' but it is not 'what is said.'

Question 4

Scalar implicatures are cancellable, which means they can be withdrawn without contradiction — unlike semantic entailments.

• T. True
• F. False

Answer: True

This is the defining property of implicatures that distinguishes them from entailments. 'Some students passed — in fact, all of them did' is coherent: the speaker first triggered the implicature 'not all' (from the Horn scale) and then explicitly cancelled it by adding stronger information. No contradiction arises because the implicature was never part of the truth-conditional content. Semantic entailments cannot be cancelled: 'All students passed — in fact, not all did' is a logical contradiction. Cancellability is thus the key diagnostic test for identifying whether an inference is pragmatic or semantic.

Question 5

Explain what a Horn scale is and how it generates scalar implicatures through the maxim of quantity.

Model answer: A Horn scale is an ordered set of expressions ⟨weaker, stronger⟩ where each stronger expression asymmetrically entails all weaker ones (e.g., ⟨some, most, all⟩; ⟨possible, probable, certain⟩; ⟨or, and⟩). When a speaker uses a weaker expression from a scale, the maxim of quantity — 'be as informative as required' — implies she is not in a position to use a stronger one. A listener reasons: the speaker knows which expression applies; she chose 'some' rather than 'all'; if 'all' were true, she would have said 'all'; therefore she must believe 'all' is false. This reasoning generates the scalar implicature 'not all' from the use of 'some.'

The same logic applies across all Horn scales. 'It's possible he's home' implicates 'it's not certain he's home' because the speaker could have said 'certain' and didn't. 'I'll have coffee or tea' implicates 'I won't have both' because the speaker could have said 'and' and didn't. In each case, the choice of a weaker term from a scale of alternatives, combined with the assumption of cooperativity, licenses the inference that the stronger term does not hold. This generalizes Grice's informal notion of relevance into a formal, scale-based mechanism.