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Quantification under Conceptual Covers

Aloni, M. (2001). PhD thesis, University of Amsterdam

An individual concept is a function from worlds to individuals, and a conceptual cover is a set of concepts such that in each world each individual is the value of exactly one of them. Each cover is a way of identifying the individuals across worlds; the name cover identifies each by its constant concept. Aloni lets quantifiers range over the concepts of a contextually given cover, so that a question partitions the worlds by which concepts of the cover have the questioned property, and a de re report is true when some concept of the cover picks out the res and has the property in every belief world.

Main definitions

Aloni2001.IsCover: conceptual covers.
Aloni2001.names: the name cover.
Aloni2001.crossId, Aloni2001.IsProper: the method of cross-identification of a set of concepts, and proper methods.
Aloni2001.whoUnder: a constituent question under a cover.
Aloni2001.DeReUnder: a de re belief report under a cover.

Implementation notes

Worlds and individuals are types and concepts total functions, as in Definition 4.1.1.
A question or a report takes its cover as an argument. The conceptual perspectives of Definition 1.4.1, which assign covers to variable indices, multiple-constituent questions, the dynamic semantics of Chapter 3 and the pragmatics of cover selection of Chapter 4 are not formalized.
A question under a cover is GroenendijkStokhof1984.who with the concepts of the cover as the individuals.

References

Aloni 2001, Quantification under Conceptual Covers
Groenendijk & Stokhof 1984, Studies on the Semantics of Questions and the Pragmatics of Answers formalized
Lewis 1979, Attitudes De Dicto and De Se

Conceptual covers

L57–59

The existence and uniqueness conditions of Section 1.4.1 say that in each world each individual is the instantiation of exactly one concept of the cover.

theorem isCover_iff_existsUnique L61–70

{W : Type u_1} {D : Type u_2} {C : Set (W → D)} :

IsCover C ↔ ∀ (w : W) (d : D), ∃! c : W → D, c ∈ C ∧ c w = d

Built from

The name cover

L93–94

L96–98

L100–102

A cover of rigid concepts is the name cover, the one rigid cover, against which the others identify the individuals descriptively (Section 4.1).

theorem IsCover.eq_names L104–115

{W : Type u_1} {D : Type u_2} {C : Set (W → D)} [Nonempty W] (h : IsCover C) (hr : ∀ c ∈ C, IsRigid c) :

C = names

Built from ,

Over a single world the name cover is the only cover (Corollary 4.1.4).

theorem isCover_iff_eq_names L117–119

{W : Type u_1} {D : Type u_2} {C : Set (W → D)} [Unique W] :

IsCover C ↔ C = names

Built from ,

Covers as cross-identifications

Over finitely many worlds and individuals there are (|D|!)^(|W|-1) covers (Proposition 4.1.3).

theorem card_covers L169–178

{W : Type u_1} {D : Type u_2} [Finite W] [Finite D] [Nonempty W] :

Nat.card { C : Set (W → D) // IsCover C } = (Nat.card D).factorial ^ (Nat.card W - 1)

Built from

Covers as methods of cross-identification (Section 4.2)

L182–184

The name cover identifies each individual of a world with itself in every other world, the rigid method of cross-identification.

theorem crossId_names L186–189

{W : Type u_1} {D : Type u_2} {x x' : W × D} :

crossId names x x' ↔ x.2 = x'.2

Built from ,

Questions under cover

L278–282

L284–286

Under the name cover, a question under cover is the question of Groenendijk & Stokhof 1984, the name cover being a copy of the individuals.

theorem whoUnder_names L288–292

{W : Type u_1} {D : Type u_2} (P : D → Set W) :

whoUnder names P = GroenendijkStokhof1984.who P

Built from ,

De re belief under cover

L296–300

A de re report under a cover holds when the believer ascribes the property to the res under one of the cover's concepts, in the sense of Lewis 1979 with worlds as centers.

theorem deReUnder_iff_exists_ascribesUnder L302–308