Questions: Modal Semantics: Necessity and Possibility

Question 1

A sentence is parsed as ¬□(∀x. attends(x)). Which English sentence does this represent?

• A. Everyone must not attend
• B. It's not necessary that everyone attends
• C. It's not possible for anyone to attend
• D. No one is required to attend

Answer: (B)

¬□(∀x. attends(x)) has negation taking wide scope over necessity, which scopes over the universal quantifier. This reads: it is not the case that (necessarily (everyone attends)) — i.e., 'It's not necessary that everyone attends.' Option A would be ∀x.□(¬attends(x)) — a very different truth condition where each person individually is forbidden to attend. Getting scope interactions right is the main payoff of doing modal semantics formally.

Question 2

According to Kratzer's restrictor analysis, what is the role of the 'ordering source' in modal semantics?

• A. It restricts which possible worlds are in the modal base
• B. It ranks worlds within the modal base by a standard of normality or ideality
• C. It determines which modal flavor (epistemic vs. deontic) the modal expresses
• D. It specifies the worlds in which the embedded proposition must hold

Answer: (B)

In Kratzer's two-parameter analysis, the modal base restricts the relevant worlds (e.g., to what is known, or what the rules require), while the ordering source ranks those worlds by how closely they approximate some ideal. This two-parameter structure explains why modals can behave differently across contexts even when the modal base is the same — and why sentences like 'It might be that you must leave' are coherent (the two parameters can be set independently).

Question 3

The sentence 'She should be home — her lights are on' and 'You should report income over $50,000' both use 'is expected to' to express the same type of modal necessity.

• T. True
• F. False

Answer: False

The first 'must' is epistemic — it expresses what is necessary given what the speaker knows (evidential commitment). The second is deontic — it expresses an obligation imposed by law or rule. In possible-worlds semantics, these differ in what the accessibility relation represents: epistemic necessity ranges over worlds compatible with the speaker's knowledge; deontic necessity ranges over worlds compatible with relevant norms. The same word can express different modal flavors depending on the contextual parameters.

Question 4

In standard possible-worlds semantics, 'possibly P' is true at a world w if and only if P holds in nearly every world accessible from w.

• T. True
• F. False

Answer: False

This is the definition of necessity (□P), not possibility (◇P). Possibility (◇P) is true at w if P holds in AT LEAST ONE world accessible from w. The two operators are duals: □P = ¬◇¬P, and ◇P = ¬□¬P. Confusing these is a fundamental error — necessity requires universal truth across all accessible worlds, while possibility only requires existence of one accessible world where the proposition holds.

Question 5

Why does the possible-worlds framework handle natural language modals by varying the accessibility relation rather than by treating all modals uniformly?

Model answer: Because different modal flavors quantify over different sets of worlds: epistemic modals range over worlds compatible with what is known, deontic modals over worlds where relevant norms are satisfied, and dynamic modals over worlds compatible with an agent's capacities. A single fixed accessibility relation couldn't distinguish these, and the same word (e.g., 'must') can express any flavor depending on context.

This flexibility is the major advantage of the possible-worlds framework over simpler accounts. By relativizing the accessibility relation to contextual parameters (what is known, what the rules say, what an agent can do), the framework explains both the diversity of modal flavors and why a single word like 'must' or 'can' can shift between them. Kratzer's restrictor analysis formalizes this further by decomposing the context into a modal base and ordering source, giving even finer control over truth conditions.