Every farmer who owns a donkey beats it At w actual the donkey sentence is neither true nor false. True Neither False w left ≈ w actual w right yes yes no 52
Every farmer who owns a donkey beats it At w actual the donkey sentence is neither true nor false. But it is true at w left . So it is true enough at w right . True True (enough) False w left ≈ w actual w right yes yes no 53
Every farmer who owns a donkey reports it to the IRS w actual 54
Every farmer who owns a donkey reports it to the IRS judged false w actual 55
Every farmer who owns a donkey reports it to the IRS “Is anyone breaking the law?” w actual 56
Every farmer who owns a donkey reports it to the IRS “Is anyone breaking the law?” w actual yes 57
Every farmer who owns a donkey reports it to the IRS “Is anyone breaking the law?” w left w actual w right no yes yes 58
Every farmer who owns a donkey reports it to the IRS “Is anyone breaking the law?” w left w actual ≈ w right no yes yes 59
Every farmer who owns a donkey reports it to the IRS True Neither False w left w actual ≈ w right no yes yes 60
Every farmer who owns a donkey reports it to the IRS True not true enough False w left w actual ≈ w right no yes yes 61
No man who has an umbrella leaves it home on a rainy day Umbrellas left home Umbrellas taken along are black are grey (and with a house) (and without a house)
No man who has an umbrella leaves it home on a rainy day w actual 63
No man who has an umbrella leaves it home on a rainy day judged true w actual 64
No man who has an umbrella leaves it home on a rainy day “Does everyone have an umbrella with him?” w actual 65
No man who has an umbrella leaves it home on a rainy day “Does everyone have an umbrella with him?” w actual yes 66
No man who has an umbrella leaves it home on a rainy day “Does everyone have an umbrella with him?” w left w actual w right yes yes no 67
No man who has an umbrella leaves it home on a rainy day True True (enough) False w left ≈ w actual w right yes yes no 68
No man who has a 10- year-old son gives him the car keys Sons that get the keys Sons that don’t get will be shown in black them, in grey (and with keys) (and without keys)
No man who has a 10-year- old son gives him the car keys 70
No man who has a 10-year- old son gives him the car keys judged false 71
No man who has a 10-year- old son gives him the car keys “Does every father behave responsibly?” 72
No man who has a 10-year- old son gives him the car keys “Does every father behave responsibly?” w actual no 73
No man who has a 10-year- old son gives him the car keys “Does every father behave responsibly?” w left w actual w right yes no no 74
No man who has a 10-year- old son gives him the car keys True not true enough False ≈ w left w actual w right yes no no 75
The theory so far • Context sensitivity of donkey sentences is central (like Yoon 96, Krifka 96) • Links definite plurals to donkey sentences (like Yoon 96, Krifka 96; building on Kri ž 15) • No commitment to sums (unlike Yoon 96, Krifka 96) • No commitment as to whether truth-value gaps are presuppositions (Barker 96: YES; Kri ž 15: NO) 76
Compositional implementation 77
The bird’s-eye view • True Semantics • Neither input into Pragmatics delivers • False • True (incl. true enough) delivers • False 78
Zooming in on the semantics • True Pragmatics delivers input into • Neither … … • False True (incl. true enough) • … delivers False • Semantics 79
The semantic pipeline Every farmer who owns a donkey True • beats it Neither • False • 80
Tasks for the semantics • Generating and managing anaphora without sums • I will build on PCDRT (Brasoveanu 08). • Generating truth value gaps • I will enrich PCDRT with error states (van Eijck 93) and assume that donkey pronouns produce gaps • Projecting gaps and keeping them under control • Supervaluation quantifiers (van Eijck 96) 81
Our semantic backbone: PCDRT (Brasoveanu 08) • Constituents relate input (I) to output (O) states • A state is a set of assignments i 1, i 2 etc. that relate discourse referents u 1 , u 2 etc. to entities x, y etc. • A state can be seen as a table: u 1 u 2 i 1 i 2 82
Restrictor (not today's focus) Every farmer who owns a donkey True • beats it Neither • False • 83
Restrictor (not today's focus) • [[every u1 farmer who owns a u2 donkey]] • I assume that all indefinites are strong : they introduce as many individuals as they can. u 1 u 2 • For each farmer x , this will generate a state in which every i 1 assignment maps u 1 to x and u 2 to a i 2 di ff erent donkey that x owns 84
Verb phrase Every farmer who owns a donkey True • beats it Neither • False • 85
Error-state semantics { produce VP truth-value gaps u success x u [[ λ x. x beats it u ]] ≈ u failure x u u error x u van Eijck 93 86
DPL with error states (van Eijck 93) • In DPL and related systems, information about the values of variables is encapsulated in a state , passed on from one subterm to the next. • In DPL, states are assignment functions • van Eijck adds error states: special assignments that prevent a formula from having a truth value • Error states can be thrown, passed on, and caught 87
PCDRT with error states • Conventions: • We’ll use the empty table ε as an error state • Most conditions return true on the error state • Most DRSs pass incoming error states onwards • This requires various tweaks for bookkeeping 88
A PCDRT predicate denotes a test on each row • farmer ↝ λ v. λ I λ O. I=O & forall i in I. farmer(i(v)) (true if v=u 1 ) • beats ↝ λ v λ v’. λ I λ O. I=O & forall i in I. beats(i(v),i(v’)) (false if v=u 1 , v’=u 2 ) u 1 u 2 • No trivalence yet i 1 i 2 89
Introducing PCDRT shorthands • farmer ↝ λ v. λ I λ O. I=O ∧∀ i ∈ I. farmer(i(v)) Shorthand: λ v. [ farmer{v} ] • beats ↝ λ v λ v’. λ I λ O. I=O ∧∀ i ∈ I. beats(i(v),i(v’)) Shorthand: λ v λ v’. [ beats{v,v’} ] u 1 u 2 • No trivalence yet i 1 i 2 90
Conditions only have inputs, DRSs also have outputs • A condition is a test on an input state: λ I … • Atomic predicates: R{u} = def λ I. ∀ i ∈ I. R(i(u)) • A DRS relates input to output states: λ I λ O … • Lifting a condition C into a DRS: [C] = def λ I λ O. C(I) ∧ I=O • Random and targeted assignments of discourse referents: [u] = def λ I λ O. ∀ i ∈ I ∃ o ∈ O. i[u]o ∧ ∀ o ∈ O ∃ i ∈ I. i[u]o u:=x = def λ I λ O. [u](I)(O) ∧ ∀ o ∈ O. o(u)=x 91
Success, failure, error • succeeds(D,I) = def ∃ O ≠ε . D(I)(O) D transitions to some non-error state • fails(D,I) = def ¬ ∃ O. D(I)(O) D does not transition to any output state • error(D,I) = def ∃ O. D(I)(O) ∧ ∀ O. (D(I)(O) → O= ε ) D only transitions to error states Mutually exclusive, jointly exhaustive. 92
Static connectives turn DRSs into conditions • DRS negation checks that the DRS fails on any nonempty substate of the input state: • ~D = def λ I. ∀ H ≠ε . H ⊆ I → fails(D,H) • DRS disjunction checks that at least one of the disjuncts succeeds: • D | D’ = def λ I. succeeds(D,I) ∨ succeeds(D’,I) 93
Dynamic connectives turn DRSs into other DRSs • DRS conjunction: apply the two DRSs in sequence • D ; D’ = def λ I λ O. ∃ H. D(I)(H) ∧ D’(H)(O) • Maximalization: store as many di ff erent entities under column u as possible as long as D returns an output • max u (D) = def λ I λ O. (I=O= ε ) ∨ ([u] ; D)(I)(O) ∧ ∀ K. ([u] ; D)(I)(K) → uK ⊆ uJ where uK = def { x : there is an i in K such that x=i(u)} 94
Testing if a DRS treats all rows the same • uniformTest(D) = def λ I. ( D | [~D] ) u 1 u 2 uniformTest([beats{u 1 ,u 2 }]) holds of this state: i 1 i 2 and of this state: but not of this state: u 1 u 2 u 1 u 2 i 1 i 1 i 2 i 2 95
Goal: mixed worlds should trigger error states { O=I and v beats all the u referents of u in I x u or beats it u ↝ O = ε and v beats some λ v. λ I λ O. but not all of the referents u of u in I x u or (in the third case, no u output matches the input) x u 96
The DRS uniform converts failed uniformTest s into error states uniform(D) = def λ I λ O. (uniformTest(D)(I) ∧ I=O) ∨ (¬uniformTest(D)(I) ∧ O= ε ) u 1 u 2 uniform([beats{u 1 ,u 2 }]) succeeds on this state i 1 i 2 and on but maps to the error state u 1 u 2 u 1 u 2 i 1 i 1 i 2 i 2 97
In pronouns, I depart from Brasoveanu 08 • In original PCDRT, it u tests if all assignments in the input agree on some atom as the referent of u . it u ↝ λ P . [atom{u}] ; P(u) where atom{u} = def λ I. ∃ x.atom(x) ∧ ∀ i ∈ I. i(u)=x • This test precludes trivalence, so I’ll drop it. • I don’t use sums, so I’ll drop the atomicity check. 98
I propose that pronouns introduce trivalence via uniform it u2 ↝ λ P . uniform(P(u 2 )) ; P(u 2 ) brays ↝ λ v. brays{v} u 2 it u2 (brays) succeeds on this state and i 1 i 2 u 2 u 2 fails on and maps to the error state i 1 i 1 i 2 i 2 99
Pronouns in object position are type-lifted in the usual way Lift(it u2 ) ↝ λ R λ v. uniform(R(u 2 )(v)) ; R(u 2 )(v) beats ↝ λ v’ λ v. beats{v,v’} u 1 u 2 Lift ( it u2 )(beats)(u 1 ) succeeds on this state i 1 i 2 u 1 u 2 u 1 u 2 fails on and maps to the error state i 1 i 1 i 2 i 2 100
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