Semantics week 12

LING2100

What is intentionality?

Intentionality refers to contexts where substituting co-referential expressions change the truth value. These are contexts sensitive to more than just reference - they care about modes of presentation or possibilities (e.g., beliefs, necessity).

What is extensionality?

Extensionality is when the truth of an expression only depends on the reference of its parts (their extensions), not on how they are described. Most of classical logic is extensional.

What is the difference between extension and intension?

Extension: Actual reference or truth value in the actual world

Intension: A function from possible worlds to extensions (truth values or referents in each world)

What is possible in worlds semantics?

A framework where intensional expressions are interpreted relative to possible worlds. It allows modelling statements about what could be, should be, or someone thinks is true.

What is the intension of a sentence?

A function from worlds to truth values:

⟦φ⟧ᴵ = λw.⟦φ⟧ʷ

So a sentence’s meaning is its truth-conditions across all possible worlds.

What is the extension of a sentence in a world w?

Just its truth value in that world: true or false

How do we model propositional attitude verbs like ‘believe’?

By treating them as operators that take intensional arguments (i.e., prepositions as functions from worlds to truth values)

E.g.,

⟦believe⟧ = λp.λx. x believes p

p is of type ⟨s, t⟩, a proposition varying across worlds.

What is the de dicto reading?

The belief is about the description of the object.

Example: ‘Anna believes that the winner is talented’

De dicto: Anna believes that whoever won is talented

What is the de re reading?

The belief is about the individual, regardless of description.

Same example, de re: There is a person (the winner) Who Anna believes is talented.

Why is the de re / de dicto distinction important?

It shows that meaning is not purely referential - syntactic structure, scope, and world-sensitive evaluation all affect interpretation.

What is the semantic type of a modal operator like ‘must’?

It takes a proposition and returns another proposition:

Type : ⟨⟨s, t⟩, ⟨s, t⟩⟩

e.g.,

⟦must⟧ = λp.λw.∀w′ ∈ Acc(w): p(w′) = 1

(‘‘p is true in all accessible worlds from w’’)

What are accessible worlds in modal logic?

They are the set of worlds considered possible given some conditions (e.g., epistemic, deontic).

E.g., for epistemic modals: worlds consistent with what is known.

How do we interpret ‘it must be raining’

True in world w iff it’s raining in all accessible worlds from w.

What are epistemic and deontic modals?

Epistemic: related to knowledge or belief (e.g., ‘it must be true’’)

Deontic: related to obligation or permission (e.g., ‘you must go’)

What type are prepositions in intensional semantics?

Type: ⟨s, t⟩

They are functions from possible worlds (s) to truth value (t)

What does it mean for a sentence to be intensionally opaque?

Substituting co-referential terms may change truth value

E.g., ‘Lois believes that superman flies’ ≠ ‘Lois believes that Clark Kent flies’

What challenges doe intentionality pose for traditional semantics?

It requires expanding our semantic model to include worlds and contexts, and to distinguish ways of identifying things - not just their extensions.