Two souls of disjunction Towards a state-monadic update semantics Patrick D. Elliott July 3, 2019 Asymmetries in Language: Presuppositions and beyond – Berlin 1
Two souls i Two broad traditions addressing semantics and pragmatics of disjunction, with little-to-no overlap: Scalar implicature literature concerned with deriving exclusive readings and ignorance inferences while retaining inclusive disjunction as the basic meaning of natural language “or” (Sauerland 2004). Dynamic semantics literature concerned with deriving facts concerning presupposition projection in disjunctive sentences (Heim 1983, Beaver 2001). 2
Two souls ii • Both approaches to disjunction seem necessary, but it’s not obvious that the two are even compatible – the scalar implicature literature takes as its starting point that “or” is ∨ , whereas dynamic semantics departs from this orthodoxy. • Relatedly, dynamic semantics has been criticized (see, e.g., Schlenker 2009), because the dynamic entry for disjunction can’t be derived from logical disjunction. Our goal In this talk, we’ll sketch a way of systematically lifuing a fragment into the dynamic world. Tiis will be successful, with the exception of disjunction. We’ll suggest that this tension can be resolved by integrating exhaustifjcation . 3
Roadmap • A brief recap of the Heim-Karttunen projection rules, update semantics, and the explanatory problem for dynamic semantics. • Tracking information growth via the State monad. • A type-shifu from propositional to dynamic connectives ( dlift ), and a bad prediction for disjunction. • Resolving the tension via exhaustifjcation, and some possible empirical payofgs. 4
Tie dynamic approach to presupposition projection
The Heim-Karttunen projection rules Implication presupposes 𝜍 . or B” presupposes 𝜌 , and unless “not A” entails 𝜍 , also If A 𝜌 , and B presupposes B 𝜍 , then a sentence of the form “A Disjunction d. presupposes 𝜌 , and unless A entails 𝜍 , also presupposes 𝜍 . c. (1) presupposes 𝜌 , and unless A entails 𝜍 , also presupposes 𝜍 If A 𝜌 , and B 𝜍 , then a sentence of the form “A and B” Conjunction b. If A 𝜌 , then a sentence of the form “not A” presupposes 𝜌 . Negation a. 5 If A 𝜌 and B 𝜍 , then a sentence of the form “If A then B”
Some illustrations (2) Paul didn’t stop vaping. Paul vaped (3) Paul vaped and Paul didn’t stop vaping. Presuppositionless (4) If Paul and Sophie vaped, then Paul would never stop vaping. Presuppositionless (5) Either Paul never vaped, or Paul stopped vaping. Presuppositionless 6
The dynamic view • Classical dynamic semantics (Groenendijk & Stokhof 1991, Heim 1983, a.o.) builds the projection rules directly into the semantics of the connectives. • Sentences themselves express updates of the common ground. • Presuppositions place defjnedness conditions on updates. • Dynamic connectives manipulate the input of subsequent juncts based on the output of previous juncts, thereby getting the projection facts. 7
A Heimian fragment (6) Partial assertion operator (def.) 𝔹 𝜚 ≔ 𝜇𝑑 . 𝑑 ⊆ dom 𝜚 . 𝑑 ∩ { 𝑥 ∣ 𝜚 𝑥 } (7) (8) . 𝑑 ∩ { 𝑥 ∣ ¬ vapes 𝑥 p } 8 � Paul stopped vaping � = 𝜇𝑥 ∶ vaped 𝑥 p . ¬ vapes 𝑥 p 𝔹 � Paul stopped vaping � = 𝜇𝑑 ∶ 𝑑 ⊆ { 𝑥 ∣ vaped p }
Heim connectives i (9) not 𝑣 ≔ 𝜇𝑑 . 𝑑 ∖ (𝑣 𝑑) Take the result of updating 𝑑 with 𝑣 , and subtract the result from 𝑑 . (10) 𝑣 and 𝑤 ≔ 𝜇𝑑 . (𝑤 ∘ 𝑣) 𝑑 First update 𝑑 with 𝑣 , then update the result with 𝑤 . 9
Heim connectives ii (11) if 𝑣 then 𝑤 ≔ ( not 𝑣 𝑑) ∪ (𝑣 and 𝑤) 𝑑 Update 𝑑 with 𝑣 , and subtract the result from 𝑑 – store this as 𝑑 ′ . Next, update 𝑑 with 𝑣 , and then update the result with 𝑤 – store (12) 𝑣 or 𝑤 ≔ 𝜇𝑑 . 𝑣 𝑑 ∪ 𝑤 ( not 𝑣 𝑑) Update 𝑑 with 𝑣 – store this as 𝑑 ′ . Next, update 𝑑 with 𝑣 , subtract the result from 𝑑 , and update this with 𝑤 – store this as 𝑑 ″ . Union 𝑑 ′ and 𝑑 ″ . 10 this as 𝑑 ″ . Finally, union 𝑑 ′ and 𝑑 ″ .
Illustration for disjunction c. ∪ ((𝑑 ∩ { 𝑥 ∣ vaped 𝑥 p }) ∩ { 𝑥 ∣ ¬ vapes 𝑥 p }) . (𝑑 ∖ (𝑑 ∩ { 𝑥 ∣ vaped 𝑥 p })) (𝑑 ∩ { 𝑥 ∣ vaped 𝑥 𝑞 }) ⊆ { 𝑥 ∣ vaped 𝑥 p } ⏞⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⏞⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⏞ ⊤ = 𝜇𝑑 ∶ (14𝑏) or (14𝑐) . 𝑑 ∩ { 𝑥 ∣ ¬ vapes 𝑥 p } (13) b. a. (14) presuppositionless Paul never vaped or Paul stopped vaping. 11 not 𝔹 � Paul vaped � = 𝜇𝑑 . 𝑑 ∖ (𝑑 ∩ { 𝑥 ∣ vaped 𝑥 p }) 𝔹 � Paul stopped vaping � = 𝜇𝑑 ∶ 𝑑 ⊆ { 𝑥 ∣ vaped 𝑥 p }
Explanatory problem for Dynamic Semantics • Linear asymmetries built into the entry for each individual connective; concomitantly, easy to defjne “deviant” dynamic connectives that are truth-conditionally adequate but get the projection facts wrong. • E.g., reverse dynamic conjunction . (15) 𝑣 rand 𝑤 ≔ 𝜇𝑑 . (𝑣 ∘ 𝑤) 𝑑 • Just by reversing the order of function composition, we predict that a subsequent conjunct could satisfy the presuppositions of previous conjuncts. 12
State-monadic update semantics
The monad slide • Here we’ll attempt to (partially) resolving the explanatory problem by stipulating the linear order of information growth once . • Concretely, we follow Shan (2002), Asudeh & Giorgolo (2016), and especially Charlow (2014) in using a monad to extend a pure, Montagovian fragment. • You don’t have to care about what a monad is for the purposes of this talk. Here is what we’re going to introduce: • We answer the question “what kind of semantic object is an update?” by providing a type constructor for updates. • An injection function, for lifuing values into trivial updates. • A way of doing function application in the update-semantic space. 13
State Type constructor for updates: (16) U a ∷= { s } → ( a ∗ { s }) Injection function from an ordinary value 𝑏 to an update: (17) 14 𝑏 𝜍 ≔ 𝜇𝑑 . ⟨𝑏, 𝑑⟩
State-sensitive application (18) 𝑛 ⊛ 𝑜 ≔ 𝜇𝑑 . ⟨ A 𝑦 𝑧, 𝑑 ″ ⟩ for ⟨𝑦, 𝑑 ′ ⟩ ≔ 𝑛 𝑑 ⟨𝑧, 𝑑 ″ ⟩ ≔ 𝑜 𝑑 ′ • Takes two updates 𝑛 and 𝑜 as inputs. • Tie input context set 𝑑 is fjrst fed into 𝑛 , returning a potentially updated context 𝑑 ′ . • Tie ordinary values contained in the updates undergo ordinary function application. 15 • 𝑑 ′ is fed into 𝑜 , returning a potentially updated output context 𝑑 ″ .
Assert operator (19) for ⟨𝑞, 𝑑 ′ ⟩ ≔ 𝑛 𝑑 𝜇𝑑 . ⟨ 𝜇𝑥 . smokes 𝑥 h ∧ vapes 𝑥 p (𝑑 ∩ { 𝑥 ∣ smokes 𝑥 h } ∩ { 𝑥 ∣ vapes 𝑥 p }) ⟩ ⊛ ... ⊛ ∧ 𝜍 ... 16 𝔹 𝑛 ≔ 𝜇𝑑 . ⟨𝑞, 𝑑 ′ ∩ 𝑞⟩ 𝔹 ( � Paul vapes � 𝜍 ) 𝔹 ( � Hubert smokes � 𝜍 )
Heavy lifting i How do we get the dynamic connectives in this system? To simplify the technical details, we have to assume that the lexical entry for each (classical) propositional connective comes with an additional parameter 𝑠 . Tiis is harmless. not 𝑠 𝑞 𝑞 and 𝑠 𝑟 if 𝑞 then 𝑠 𝑟 𝑞 or 𝑠 𝑟 ≔ 𝜇𝑠 . 𝑠 ∩ (𝑞 ∪ 𝑟) 17 ≔ 𝜇𝑠 . 𝑠 ∖ 𝑞 ≔ 𝜇𝑠 . 𝑠 ∩ 𝑟 ∩ 𝑞 ≔ 𝜇𝑠 . (𝑠 ∖ 𝑞) ∪ 𝑟
Heavy lifting ii propositional connectives. not 𝑠 𝑞 𝐸 s = 𝑞 − (𝑞 and 𝑞 𝑟) 𝐸 s = 𝑞 ∩ 𝑟 ( if 𝑞 then 𝑠 𝑟) 𝐸 s (𝑞 or 𝑠 𝑟) 𝐸 s = 𝑞 ∪ 𝑟 18 If the additional parameter is saturated by 𝐸 s we just get...the ordinary = 𝑞 − ∪ 𝑟
Heavy lifting iii ⟩ get...the Heimian connectives (with provisos). inner-argument is saturated by the input context, in which case we meaning. In the contextual dimension, the propositional connective’s argument saturated by 𝐸 s , returning the ordinary propositional Informally – in the ordinary dimension, the connective 𝑔 has its inner ⟨𝑟, 𝑑 ″ ⟩ ≔ 𝑜 𝑑 for ⟨𝑞, 𝑑 ′ ⟩ ≔ 𝑛 𝑑 𝑔𝑑 ″ 𝑑 ′ 𝑑 Now let’s defjne our function dlift 𝑜 . 𝑔 𝑟 𝑞 𝐸 s , 𝑛 ( dlift 2 𝑔) 𝑜 ≔ 𝜇𝑑 . ⟨ for ⟨𝑞, 𝑑 ′ ⟩ ≔ 𝑛 𝑑 ⟩ 𝑔𝑑 ′ 𝑑 𝑔 𝑞 𝐸 s , ( dlift 1 𝑔) 𝑛 ≔ 𝜇𝑑 . ⟨ 19
Heavy lifting iv ⟨𝑞, 𝑑 ′ ⟩ ≔ 𝑛 𝑑 ⟨𝑟, 𝑑 ″ ⟩ ≔ 𝑜 𝑑 ′ ⟨𝑞, 𝑑 ′ ⟩ ≔ 𝑛 𝑑 (𝑑 ∖ 𝑑 ′ ) ∪ 𝑑 ″ ⟩ dlift 2 ( if…then 𝑠 ) = 𝜇𝑜 . 𝜇𝑛 . 𝜇𝑑 . ⟨ (22) ⟨𝑟, 𝑑 ″ ⟩ ≔ 𝑜 𝑑 ′ (𝑑 ∩ 𝑑 ′ ) ∩ 𝑑 ″ ⟩ (20) 𝑞 ∩ 𝑟 (21) for ⟨𝑞, 𝑑 ′ ⟩ ≔ 𝑛 𝑑 𝑑 ∖ 𝑑 ′ ⟩ 𝑞 − 20 dlift 1 not 𝑠 = 𝜇𝑛 . 𝜇𝑑 . ⟨ dlift 2 and 𝑠 = 𝜇𝑜 . 𝜇𝑛 . 𝜇𝑑 . ⟨ 𝑞 − ∪ 𝑟
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