Denotations vs. senses

• The two sides of the equality have the same denotation
• But they have different senses:
  • 2 is the name of the number
  • 1 + 1 is a process whose result is the number

Denotations vs. senses

Similarly in natural language

is not necessarily a tautology:

Denotations vs. senses

Standard analysis:

• “Hesperus” and “Phosphorus” refer to the same entity (the planet Venus)
• Their senses are different:
  • “Hesperus”  →  the evening star
  • “Phosphorus”  →  the morning star
• Two different descriptions (processes) that lead to the same entity

But…

In certain contexts an expression like (3)

is also not a tautology…

Capgras syndrome

From Wikipedia:

“[Capgras syndrome] is a disorder in which a person holds a delusion that a friend, spouse, parent, or other close family member has been replaced by an identical-looking impostor.”

• Assume Kim suffers from Capgras syndrome
• The following sentence is not necessarily unsatisfiable:

• Saying “Sandy” is ambiguous seems wrong

Capgras syndrome

• Kim’s model: two “Sandy” entities
  • Sandy_σ
  • ImpostorSandy_Kim
• Speaker’s model: one “Sandy” entity
  • Sandy_σ
• ImpostorSandy_Kim ≠ Sandy_σ

Capgras syndrome

• At the time of the utterance, the name “Sandy” refers to ImpostorSandy_Kim for Kim
• For the speaker the name refers to Sandy_σ
• One instance of “Sandy” is interpreted from Kim’s persepective
• The other from the speaker’s perspective

Indiana Pi Bill

In 1897 Dr. Edwin J. Goodwin presented a bill to the Indiana General Assembly for

“[…] introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost”

He had copyrighted that

Thus

Our approach in a nutshell

• Contentious expressions are given denotations that depend on an additional interpretation index

• Perspectives rather than senses

• Default interpretation index = speaker’s index

• Verbs like “believe” can change the default index to another index (e.g. the subject’s index)

• Ontology: real entities + mental entities (relative to interpretation indices)

Other opaque contexts

but not in cases like (11)

• Overlap with intensionality but not the same point of view

Outline of the talk

• Opacity

• Naive implementation of the idea

• 3 problems for simple type extension

• Monads as a solution

Naive implementation:
New types

• A single new type i for interpretation indices

• Simplistically i = e

Naive implementation:
Contentious expressions

• Given an expression of type τ, we signal it is contentious by assigning it the type i → τ

• Example: “Kim doesn’t believe Sandy is Sandy”

• “Sandy” contentious, type assigned i → e

• Example: “Reza doesn’t believe Jesus is Jesus”

• “Jesus” contentious, type assigned i → e

Naive implementation:
Index switching expressions

• “love” : λs.λo.love(s)(o(s)) : e → (i → e) → t

• “believe” : λs.λc.believe(s)(c(s)) : e → (i → t) → t

Grammatical infrastructure

• Applicable basically to any grammatical formalism

• We use a sort of soft LFG / Categorial grammar approach

• Linear logic as model for semantic composition

• Functional types become linear implications (⊸)

• No generalized lifting in the lexicon

Problem 1:
Only contradictory reading

• Only reading available:

• “Sandy” always interpreted in the scope of “believe”

Problem 2:
Implication linearity

Problem 3:
Linearity again

What if the object of a verb like “love” is non contentious?

α ⊢ i → α not a valid inference in linear logic

Problems

• Problem 1 is a scopal problem

• Problem 2 and 3 seem to stem from linearity of implication / function type constructor

• Rejecting linearity seems too strong:

  15. John punches Bill
16. punch(John)(John)

Solution

• Same pattern emerges (for problem 2 and 3) in unrelated contexts:

  • Multidimensional semantics (e.g. conventional implicatures)
  • Semantics of questions
  • Optional arguments

• Monads as a generalized model for all these phenomena

Monads

• Monad = ⟨M, η, μ⟩
• M is a functor, mapping between types: τ ↦ i → τ

• η (“unit”) lifts object to the monadic type, solves problem 3:

  17. η = λx.λi.x : τ → i → τ

• μ (“join”) compresses multiple monadic layers into a single one, solves problem 2:

  18. μ = λf.λi.f(i)(i) : (i → i → τ) → i → τ

Technical detail:
 ⋆  (“bind”) rather than μ

• μ impractical when writing lexical entries

•  ⋆  can be defined in terms of μ

•  ⋆  can be interpreted as sequencing too

• New “scope” mechanism, solves problem 1

Monads in the logic

• New unary connective ◊, but ◊α ≠ i → α

Online theorem prover

Monads, intuitively

• Monads as a model of side effects: ◊τ computation that results in value of type τ possibly producing side effects

• η creates a trivial computation without side effects

•  ⋆  combines computations and side effects, enforces order of evaluation

• In our case, side effect = dependency on context / environment

Capgras example

Readings

Satisfiable reading

⟦Sandy⟧ ⋆ λx.⟦believe⟧(Kim)(⟦Sandy⟧ ⋆ λy.η(x = y)) ⋆ λz.η(¬z)

Unsatisfiable readings

⟦believe⟧(Kim)(⟦Sandy⟧ ⋆ λx.⟦Sandy⟧ ⋆ λy.η(x = y)) ⋆ λz.η(¬z)

⟦Sandy⟧ ⋆ λx.⟦Sandy⟧ ⋆ λy.⟦believe⟧(Kim)(η(x = y)) ⋆ λz.η(¬z)

“love” example

Readings

Satisfiable reading

⟦love⟧(MJ)(η(PP)) ⋆ λp.⟦love⟧(MJ)(⟦Spiderman⟧) ⋆ λq.η(p ∧ ¬q)

Unsatisfiable reading

⟦love⟧(MJ)(η(PP)) ⋆ λp.⟦Spiderman⟧ ⋆ λx.⟦love⟧(MJ)(η(x)) ⋆ λq.η(p ∧ ¬q)

Beyond names

• Use the same approach with any possibly contentious expression

Beyond names

To sum up

• Generalized approach to opacity based on interpretation indices, perspectives

• Not only traditional cases but also Capgras examples

• Monads: weaken linearity of compositionality, additional scope mechanism

• Not limited to synonyms but also applicable to more general entailments

Thanks!